Time Reversal Breaking and Entropy Production in Non-Equilibrium Systems: Insights from Mean Back Relaxation

Time Reversal Breaking and Entropy Production in Non-Equilibrium Systems: Insights from Mean Back Relaxation Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades „Doctor rerum naturalium“ der Georg-August-Universität Göttingen im Promotionsstudiengang Physik der Georg-August University School of Science (GAUSS) vorgelegt von Gabriel Johann Werner Knotz aus Kassel Göttingen, 2025 Betreungsausschuss Prof. Dr. Matthias Krüger Institut für Theoretische Physik, Georg-August-Universität Göttingen Prof. Dr. Timo Betz Drittes Institut für Physik, Georg-August-Universität Göttingen Prof. Dr. Peter Sollich Institut für Theoretische Physik, Georg-August-Universität Göttingen Mitglieder der Prüfungskommission Referent: Prof. Dr. Matthias Krüger Institut für Theoretische Physik, Georg-August-Universität Göttingen Koreferent: Prof. Dr. Peter Sollich Institut für Theoretische Physik, Georg-August-Universität Göttingen Weitere Mitglieder der Prüfungskommission Prof. Dr. Timo Betz Drittes Institut für Physik, Georg-August-Universität Göttingen Prof. Dr. Fabian Heidrich-Meisner Institut für Theoretische Physik, Georg-August-Universität Göttingen Prof. Dr. Ramin Golestanian Max-Planck-Institut für Dynamik und Selbstorganisation, Göttingen Prof. Dr. Tim Salditt Institut für Röntgenphysik, Georg-August-Universität Göttingen Tag der mündlichen Prüfung: 10.11.2025 List of publications [1] G. Knotz, U. Parlitz, and S. Klumpp, Synchronization of a genetic oscillator with the cell division cycle, New J. Phys. 24, 033050 (2022). 1 [2] T. M. Muenker, G. Knotz, M. Krüger, and T. Betz, “Accessing activity and viscoelastic properties of artificial and living systems from passive measurement,” Nat. Mater. 23, 1283–1291 (2024). 2 [3] G. Knotz, T. M. Muenker, T. Betz, and M. Krüger, Entropy bound for time reversal markers, Front. Phys. 11, 1331835 (2024). [4] G. Knotz and M. Krüger, Mean back relaxation for position and densities, Phys. Rev. E 110, 044137 (2024). [5] G. Knotz, T. M. Muenker, T. Betz, and M. Krüger, Evaluating NonEquilibrium Trajectories via Mean Back Relaxation: Dependence on Length and Time Scales, arXiv:2507.05912 (2025) (accepted in Journal of Chemical Physics). 1This work is based on the authors bachelor thesis and not part of the dissertation 2 This work is relevant for this thesis and cited in multiple places. However, no text or figures are taken from this article. Abstract The distinction between equilibrium thermal fluctuations, exemplified by Brownian motion, and active non-equilibrium processes is an essential problem in various fields of non-equilibrium statistical physics and biophysics. While Brownian motion is created by the random collision of a colloidal particle with much smaller surrounding particles, many biological and artificial systems exhibit externally and self-driven motion, for example, induced by external energy sources and molecular motors. This thesis addresses the problem of differentiating these motion types by focusing on the breaking of time reversal symmetry, a hallmark of non-equilibrium dynamics. We investigate a novel quantity, the mean back relaxation (MBR), to detect said time reversal symmetry breaking. The mean back relaxation is a three point correlation function and is defined as the negative ratio of the displacement between [0,𝑡] to the displacement from [−𝜏,0] . We find that the long time value of MBR approaches 12 if later displacements get statistically independent from earlier ones, and if time reversal symmetry is valid. We demonstrate that MBR shows time reversal breakage for an active Brownian particle in a harmonic potential. We derive a solution for the finite time MBR for a Gaussian process in terms of the mean squared displacement (MSD) with a path integral formalism. This allows the analytic investigation of the time dependence of MBR curves in a Gaussian model. Also, we introduce a new density based MBR generalizing MBR to other types of observables. By analyzing the variance of the back relaxation (VBR), we characterize MBRs statistical properties and give estimates on how to select MBR parameters for practical applications. Connecting MBR to fundamental concepts, we derive a new bound for entropy production, a central quantity in non-equilibrium dynamics. The bound is valid for time antisymmetric observables whose absolute value is bounded, and we find that the tightest observable is the sign of entropy production. This leads to an interesting coarse graining scheme that leaves the bound intact by grouping paths according to their stochastic entropy production. We test the entropy bound for a time discrete ring and compare it to other known relations. We apply MBR to experimental data obtained from colloidal particles in living cells. We can qualitatively reconstruct the curves found from cell data with a Random Horse and Cart model. Both in the model and cell data for different cell types, the long time value of MBR is linearly related to effective energy amplitude, which quantifies the violations of the fluctuation dissipation theorem (FDT), a common indirect marker of breaking of time reversal symmetry. Finally, we test the time reversal properties of MBR for cell data. We show that for colloidal particles in cells, MBR detects breaking of time reversal symmetry. This is one of the first direct empirical evidences for observable time reversal breaking of colloidal particles in living cells. The time reversal breaking is greatly reduced by depolymerization of microtubules, which hints that the effect is linked to cytoskeletal activity. We also apply the entropy production bound by inserting the anti-symmetric part of MBR and find that the lower bound given by MBR is correlated to FDT violations for different cell types, quantified by the effective energy amplitude. Contents 1. Introduction 1 1.1. Preamble............................. 1 1.2. Brownian particles as model system . . . . . . . . . . . . . . 2 1.2.1. Brownian motion . . . . . . . . . . . . . . . . . . . . 2 1.2.2. Langevin equation . . . . . . . . . . . . . . . . . . . 3 1.2.3. Pathintegral ...................... 7 1.2.4. Fluctuation dissipation theorem . . . . . . . . . . . . 9 1.3. Time reversal and entropy production . . . . . . . . . . . . . 11 1.3.1. Time reversal symmetry and detailed balance . . . . 11 1.3.2. Stochastic thermodynamics . . . . . . . . . . . . . . 13 1.3.3. Fluctuation theorems . . . . . . . . . . . . . . . . . . 18 1.3.4. Thermodynamic uncertainty relation . . . . . . . . . 19 1.4. Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 20 2. Mean Back Relaxation 23 2.1. Introduction ........................... 23 2.2. Mean Back Relaxation: Motivation . . . . . . . . . . . . . . . 24 2.3. Mean Back Relaxation: Conditioned correlations . . . . . . . 25 2.4. MBR in stationarity . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1. Long time limit of MBR and time reversal . . . . . . 26 2.4.2. Fluctuation theorem . . . . . . . . . . . . . . . . . . 28 2.4.3. Example: Active Brownian particle . . . . . . . . . . 29 2.5. Conditioned correlations and MBR in path integrals . . . . . 32 2.5.1. Conditioned path integrals . . . . . . . . . . . . . . . 32 2.6. MBR for Gaussian systems – MBR-MSD formula . . . . . . . 33 2.6.1. Relating MBR and MSD for Gaussian systems . . . . 33 2.6.2. Example: Overdamped particles . . . . . . . . . . . . 35 2.7. Density Mean Back Relaxation marks broken detailed balance in confinement and bulk . . . . . . . . . . . . . . . . . 37 2.8. Summary............................. 38 3. Variance of Back Relaxation 41 3.1. Introduction ........................... 41 3.2. Formal expression . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3. Gaussian process: Dependence on 𝑡,𝜏and 𝑙......... 41 3.4. Summary............................. 44 4. MBR,VBR and effective Energy in cells and model 47 4.1. Introduction ........................... 47 4.2. Mean Back Relaxation: Model and experiment . . . . . . . . 49 4.2.1. Experiment: Cells . . . . . . . . . . . . . . . . . . . . 49 4.2.2. Model system: "Random Horse and Cart" . . . . . . . 50 4.3. MBR and effective energy . . . . . . . . . . . . . . . . . . . . 58 4.3.1. Effective energy and FDT . . . . . . . . . . . . . . . 58 4.3.2. Effective energy and MBR in RHC model system . . 58 4.3.3. Experiment: Effective energy and MBR relation for various 𝜏........................ 60 4.4. VBR in model and cell data . . . . . . . . . . . . . . . . . . . 62 4.4.1. Variance: RHC model . . . . . . . . . . . . . . . . . . 62 4.4.2. Variance: Experiment . . . . . . . . . . . . . . . . . . 62 4.5. Summary............................. 63 5. Entropy bound for time reversal markers 67 5.1. Introduction ........................... 67 5.2. Setup and fluctuation theorem . . . . . . . . . . . . . . . . . 68 5.3. Entropybound.......................... 69 5.4. Optimal observable: Signum of entropy . . . . . . . . . . . . 70 5.5. Coarsegraining ......................... 72 5.6. Example: Network on a ring . . . . . . . . . . . . . . . . . . 72 5.7. Summary............................. 75 6. Time reversal breaking and entropy production bound in cells 77 6.1. Time reversal symmetry breaking in cells . . . . . . . . . . . 77 6.2. Time reversal breaking of MBR and dependence on length andtimescales.......................... 79 6.2.1. When does MBR detect broken time reversal breaking: Insights from the probability density . . . . . . 79 6.2.2. Model: Random horse and cart with double well . . . 81 6.2.3. Cells: Length and timescales of long time MBR time reversalbreakage.................... 89 6.3. Entropy production bound in cells . . . . . . . . . . . . . . . 92 6.4. Summary............................. 96 7. Summary, Discussion and Outlook 99 7.1. Summary............................. 99 7.2. Discussion and outlook . . . . . . . . . . . . . . . . . . . . . 101 7.3. FinalRemarks ..........................104 A. Mean Back Relaxation 107 A.1. Eq. (2.13) for general observable . . . . . . . . . . . . . . . . 107 A.2. Simulations............................108 A.3. Two coupled particles in harmonic potential . . . . . . . . . 109 B. Variance of Back Relaxation 113 C. Random horse and cart model 117 C.1. Mean Back Relaxation . . . . . . . . . . . . . . . . . . . . . . 117 C.2. MaximumofMBR........................118 C.3. Critical activity . . . . . . . . . . . . . . . . . . . . . . . . . 119 C.4. Effectiveenergy.........................119 D. Entropy bound in cells - Supplementary plots 121 6 1. Introduction properties [28, 29] 𝑊0=0 (1.13a) Increments 𝑊𝑡+Δ𝑡−𝑊𝑡are independent of past 𝑊𝑠,𝑠≤𝑡(1.13b) 𝑃(𝑊𝑡+Δ𝑡−𝑊𝑡)∝(𝑊𝑡+Δ𝑡−𝑊𝑡|0,Δ𝑡)(1.13c) 𝑊𝑡is continuous in 𝑡. (1.13d) Property Eq. (1.13c) means that increments of the Wiener process are Gaussian distributed with zero mean and standard deviation Δ𝑡 . The term 𝑎(𝑋𝑡,𝑡) in Eq. (1.12) is called the drift term, and 𝜎(𝑋𝑡,𝑡) is named diffusion. The general solution of the SDE is given by 𝑋𝑡−𝑋0=∫𝑡 0𝑎(𝑋𝑠,𝑠)d𝑠+∫𝑡 0𝜎(𝑋𝑠,𝑠)d𝑊𝑠.(1.14) The first integral term ∫𝑡 0𝑎(𝑋(𝑠),𝑠)d𝑠 can be easily understood as a Riemann integral and therefore needs no further discussion. The second integral over the Wiener process ∫𝑡 0𝜎(𝑋(𝑠),𝑠)d𝑊𝑠 , however, has some rather peculiar properties, depending on how the integral is defined. There are two main conventions, the Ito convention and the Stratonovich convention. The Ito convention is rather simple ∫𝑡 0𝜎(𝑋𝑠,𝑠)d𝑊𝑠=lim Δ𝑡→0𝑁 ∑ 𝑛=1𝜎(𝑋𝑡𝑛,𝑡𝑛)(𝑊𝑡𝑛+Δ𝑡−𝑊𝑡𝑛)(1.15) where we discretize time into 𝑁+1 time points with are equidistantly separated by Δ𝑡 . So the time point 𝑡𝑛 is given by 𝑡𝑛=𝑡0+𝑛Δ𝑡 . The Ito convention has some mathematical advantages because it is a semi-martingale, which opens the pathway to many mathematical theorems. However, it requires a modification of the chain rule. According to Ito’s lemma for a stochastic process 𝑋𝑡=ℎ(𝑊𝑡) the following relation holds [29] d𝑋𝑡=ℎ′(𝑊𝑡)d𝑊𝑡+12ℎ′′(𝑊𝑡)d𝑡. (1.16) The first term of Eq. (1.16) corresponds to the known chain rule. The term 12ℎ′′(𝑊𝑡)d𝑡 is additional and is referred to as suspicious drift because it is a differential in d𝑡 , reassembling a drift term. While it is technically possible to use Ito calculus and there are in principle no limitations, another integral convention, the Stratonovich convention, is used especially in physics [30]. It is usually denoted as ◦d𝑊 . The stochastic integral is then defined as ∫𝑡 0𝜎(𝑋𝑠,𝑠)◦d𝑊𝑠= lim Δ𝑡→0𝑁 ∑ 𝑛=1𝜎(𝑋𝑡𝑛+Δ𝑡,𝑡𝑛+Δ𝑡)+𝜎(𝑋𝑡𝑛,𝑡𝑛) 2(𝑊𝑡𝑛+Δ𝑡−𝑊𝑡𝑛)(1.17) 1.2. Brownian particles as model system 7 In this case, the chain rule behaves as expected d𝑋𝑡=ℎ′(𝑊𝑡)◦d𝑊𝑡(1.18) and one can apply basic calculus. The Ito and Stratonovich conventions are completely equivalent and can always be transformed into each other if one uses Ito’s lemma Eq. (1.16) . Numerically, one usually uses the Euler-Mayurama method, which is equivalent to directly applying the Ito convention of the stochastic integral for a finite time step Δ𝑡 . There are also higher order schemes as the Runge-Kutta-Milstein method, which is primarily used in the simulations of this thesis [31]. 1.2.3. Path integral The Langevin equations discussed in section 1.2.2 describe the time evolution of a stochastic variable via a SDE. There is an equivalent description in terms of path probabilities. This description will be useful when we discuss entropy production and stochastic thermodynamics in the following. We start by defining a path. A path 𝜔 of 𝑥is a collection of time ordered stochastic values 𝜔(𝑡)=(𝑥𝑡1,𝑥𝑡2,…,𝑥𝑡𝑛)(1.19) with 𝑡1<𝑡2<⋯<𝑡𝑛 , where we generally assume that the distance between two neighboring time points is small. For a Wiener process, this is rather simple, because all increments are statistically independent. The probability of finding a specific Wiener process is therefore given by 𝑃[𝑊(𝑡)]=𝑁 ∏ 𝑛=1(𝑊𝑡𝑛+1−𝑊𝑡𝑛|0,𝑡𝑛+1−𝑡𝑛).(1.20) 𝑃[𝑊(𝑡)] is the probability of finding the path 𝑊(𝑡) and we use square brackets […] to denote a functional dependence. How can we use this to get the probability of a path 𝑥(𝑡)that is described by the SDE 𝑥𝑡=𝑎(𝑥𝑡,𝑡)d𝑡+𝜎(𝑥𝑡,𝑡)d𝑊?(1.21) In the discertized form (Ito convention), we can write this equation as 𝑥𝑡+Δ𝑡−𝑥𝑡=𝑎(𝑥𝑡,𝑡)Δ𝑡+𝜎(𝑥𝑡,𝑡)(𝑊𝑡+Δ𝑡−𝑊𝑡)(1.22) Δ𝑥=𝑎(𝑥𝑡,𝑡)Δ𝑡+𝜎(𝑥𝑡,𝑡)Δ𝑊(1.23) for a small Δ𝑡 . We know the probability of finding an increment 𝑃(Δ𝑊) which is given by a Gaussian distribution (Δ𝑊|0,Δ𝑡) . To get a probability for the increment Δ𝑥 we use the fundamental identity 𝑃(Δ𝑊)𝑑Δ𝑊=𝑃(Δ𝑥)dΔ𝑥 . The differential is given by 8 1. Introduction dΔ𝑊 dΔ𝑥=1 𝜎(𝑥𝑡,𝑡)and with that we get the probability of an increment [24] 𝑃(Δ𝑥)=dΔ𝑊 dΔ𝑥𝑃(Δ𝑊)= 1 √2𝜋𝜎(𝑥𝑡,𝑡)2Δ𝑡exp(−12(Δ𝑥−𝑎(𝑥𝑡,𝑡)Δ𝑡)2 𝜎(𝑥𝑡,𝑡)2Δ𝑡).(1.24) The probability of the full path is given by the product of all displacements Δ𝑥𝑛= 𝑥𝑡𝑛+1−𝑥𝑡𝑛𝑃[𝑥]=𝑁 ∏ 𝑛=1 1 √2𝜋𝜎(𝑥𝑡𝑛,𝑡𝑛)2Δ𝑡exp(−12(Δ𝑥𝑛−𝑎(𝑥𝑡𝑛,𝑡𝑛)Δ𝑡)2 𝜎(𝑥𝑡𝑛,𝑡𝑛)2Δ𝑡)(1.25) =1Ωexp(𝑁 ∑ 𝑛=1−12(Δ𝑥𝑛 Δ𝑡−𝑎(𝑥𝑡𝑛,𝑡𝑛))2 𝜎(𝑥𝑡𝑛,𝑡𝑛)2Δ𝑡)(1.26) The actual path integral, which we can use to evaluate path observables, is then given by ∫𝑥exp(𝑁 ∑ 𝑛=1−12(Δ𝑥𝑛 Δ𝑡−𝑎(𝑥𝑡𝑛,𝑡𝑛))2 𝜎(𝑥𝑡𝑛,𝑡𝑛)2Δ𝑡)(1.27) with the differential 𝑥=∏𝑁𝑛=1 d𝑥𝑡𝑛 √2𝜋𝜎(𝑥𝑡𝑛,𝑡𝑛)2Δ𝑡 . In the limit Δ𝑡→0 we get the OnsagerMachIup functional 𝐴[𝑥]=∫12(𝑥(𝑡)−𝑎(𝑥(𝑡),𝑡))2 𝜎(𝑥(𝑡),𝑥))2d𝑡(1.28) with the total path probability 𝑃[𝑥]=1Ω𝑒−𝐴[𝑥].(1.29) For a Langevin equation like Eq. (1.9) this reads like [32–34] 𝐴[𝑥]=14𝐷∫(𝑥+𝑈′ 𝛾)2d𝑡, (1.30) with the action in the Ito convention, and 𝐴[𝑥]=14𝐷∫(𝑥+𝑈′ 𝛾)2+𝑈′′(𝑥) 𝛾d𝑡(1.31) in the Stratonovich convention. We can now attach probabilities to paths, and by integrating over all possible paths, we get path integrals. In principle, Langevin equations and the path integral are equivalent to each other and describe the same system. However, the latter is more useful to discuss the properties of path ensembles. For completeness, there is also a third description. The probability 𝑊1(𝑥,𝑡) of the particle being at position 𝑥 at time 𝑡 for a overdamped 1.2. Brownian particles as model system 9 Langevin Equation like Eq. (1.9) is given by the Fokker Planck equation [26] 𝜕𝜕𝑡𝑊1(𝑥,𝑡)=𝜕𝜕𝑥[(𝐷𝜕𝜕𝑥+𝑈′ 𝛾)𝑊1(𝑥,𝑡)](1.32) which is a partial differential equation. For 𝑈(𝑥)=0 , a free particle, this equation collapses to the diffusion equation. However, we will primarily work with Langevin equations and path integrals in the following. 1.2.4. Fluctuation dissipation theorem A key technique in describing a system in physics is to analyze its response to a perturbation. For small perturbation forces 𝐹 , the response of an observable 𝑥 can generally be written as [35] ⟨𝑥(𝑡)⟩−⟨𝑥(0)⟩=∫∞ −∞𝜒(𝑡−𝑠)𝐹(𝑠)d𝑠(1.33) with the susceptibility 𝜒(𝑡) . This susceptibility needs to fulfill the causality condition 𝜒(𝑡<0)=0 , otherwise future forces would influence past displacements. A whole field of statistical physics deals with this type of response theory [36–39], both in equilibrium and non-equilibrium. In equilibrium, one can relate the susceptibility to spontaneous fluctuations in the unperturbed system. For a force 𝐹(𝑡) which couples to the phase space observable 𝑦, one finds the following response formula [35] ⟨𝑥(𝑡)⟩=⟨𝑥⟩eq −1 𝑘𝐵𝑇∫∞ −∞d𝑠d d𝑡⟨𝑥𝑦(𝑡−𝑠)⟩eq 𝜃(𝑡−𝑠)𝐹(𝑠),(1.34) where we identify 𝜒(𝑡−𝑠)=−1 𝑘𝐵𝑇d d𝑡⟨𝑥𝑦(𝑡−𝑠)⟩eq 𝜃(𝑡−𝑠)(1.35) as the susceptibility. We can rewrite this relation in Fourier space as 2𝑖𝜒′′(𝜔)=∫∞ −∞d𝑡[𝜒(𝑡)−𝜒(−𝑡)]𝑒−𝑖𝜔𝑡=𝑖𝜔𝐶(𝜔) 𝑘𝐵𝑇(1.36) ⇒𝜒′′(𝜔)=𝜔𝐶(𝜔) 2𝑘𝐵𝑇(1.37) with 𝜒′′ the complex part of the susceptibility, and for 𝑦=𝑥 the correlation function 𝐶(𝜔)=⟨𝑥(𝜔)𝑥(−𝜔)⟩ . This complex part of the susceptibility is also referred to as the dissipative part because the dissipated energy Δ𝐸 due to the perturbation is given by [40] Δ𝐸=12𝜋∫∞ −∞𝜔𝜒′′(𝜔)𝐹(𝜔)𝐹(−𝜔)d𝜔. (1.38) 10 1. Introduction Eq. (1.38) only depends on 𝜒′′ and, thus, the imaginary part of the susceptibility gives the dissipation of the system. Therefore, Eq. (1.37) relates the fluctuations of a system, given by 𝐶 and the dissipation 𝜒′′ , hence the name fluctuation dissipation theorem (FDT). The real part of the response function 𝜒′ is also fixed by the imaginary part, via the Kramers-Kronig relations [41]. Therefore, Eq. (1.37) fully determines the spectrum and the susceptibility. In practice, Eq. (1.37) can be used for determining the mechanical properties of a material by measuring the unperturbed spectrum 𝐶(𝜔) of a prob particle. This approach is usually referred to as passive (micro-)rheology [42]. For this to work, the system, however, has to be in equilibrium. Another way is to use the FDT as a marker for non-equilibrium. If Eq. (1.37) is violated, we can state the system is out of equilibrium 2 . To use this approach, one measures the susceptibility by applying a force to the colloidal particle and separately measures the correlation function in the unperturbed system. In this context, effective temperatures or energies are often mentioned. In the non-equilibrium situation, we can modify Eq. (1.37) as [2] 𝜒′′(𝜔)=𝜔𝐶(𝜔) 2𝐸eff(𝜔)(1.39) where 𝐸eff is a frequency dependent correction factor. If 𝐸eff(𝜔)≠𝑘𝐵𝑇 for any frequency 𝜔 , then the system is out of equilibrium. Because 𝐸eff has the units of energy, we can call it an effective energy. However, on its own, it has no bigger meaning than that it quantifies the FDT violation. Alternatively, 𝑇eff =𝐸eff 𝑘𝐵 we can define an effective temperature, which again has the units of temperature but has no higher meaning. This approach to test if a system is in equilibrium has been applied in multiple experiments, for example, involving hair bundle cells or blood vessels [44–47]. Therefore, FDT violations have proven to be a useful tool to show, for example, that living systems are actually out of equilibrium. The major complication of this approach is usually that one has to determine the susceptibility using active perturbations. Such experimental programs can be very invasive and are sometimes difficult to conduct. Therefore, it would be generally preferred to have methods that just rely on passive observations and not active perturbations. Hence, we will introduce the concepts of time reversal symmetry and entropy production to give a different approach to detecting non-equilibrium. 2 Another possibility is that the weak coupling assumption is violated. During the proof, one generally assumes that the system is weakly coupled to a heat bath. This means that the system and bath exchange energy, but are dynamically independent. In this case, the combined system might be in equilibrium, but one can not split into a separate system and bath. While these technicalities are generally ignored in classical settings, they can be relevant for quantum mechanics [43]. 1.3. Time reversal and entropy production 11 1.3. Time reversal and entropy production 1.3.1. Time reversal symmetry and detailed balance In the following chapters, we will use the breaking of time reversal symmetry and detailed balance breakage as a marker of non-equilibrium features of a system. This means we need to discuss how the two concepts, time reversal symmetry and equilibrium, are related. A (closed) system is in equilibrium if the probability 𝑝(Γ) of finding the system at Γ=({𝑞𝑖},{𝑝𝑖}) is the same everywhere in the accessible phase space. Here 𝑞𝑖 are the generalized coordinates and 𝑝𝑖 the generalized momenta. The accessible phase space is on the energy shell 𝐻(Γ)=𝐸 with 𝐻 the Hamilton function governing the dynamics of the system. The energy 𝐸 is, in this case, a control parameter. The overall concept is referred to as the microcanonical ensemble. The probability is then expressed as 𝑝(Γ)=1Ωwhere Ω= 1 𝑁!ℎ3𝑁∫dΓ𝛿(𝐻(Γ)−𝐸).(1.40) is the phase space volume of the energy shell. From here, one can go on and define thermodynamic quantities like entropy, temperature, etc., the standard textbook procedure in a statistical mechanics course [48]. We want to focus on the implications of the underlying dynamics. Consider a sequence of phase space states Γ1,…,Γ𝑛 , then the system obeys time reversal symmetry if the joint probability 𝑊𝑛 of finding the system at these points at the given times 𝑡1<…𝑡𝑛obeys the following relation [40] 𝑊𝑛(Γ1,𝑡1;…;Γ𝑛,𝑡𝑛)=𝑊𝑛(𝜋Γ𝑛,𝑡1;…;𝜋Γ1,𝑡𝑛)(1.41) with the kinematic reversal operator 𝜋Γ=({𝑞𝑖},{−𝑝𝑖}) flipping all momenta or velocities of the state. The qualitative statement of this is rather simple: The probability of finding the trajectory of a system is the same as if the system follows the same trajectory but backwards. The system is statistically speaking time reversal symmetric because we have no way of distinguishing time forward and time backward trajectories. Therefore, every averaged quantity over 𝑊𝑛(Γ1,𝑡1;…Γ𝑛,𝑡𝑛) must have the same result as if averaged over the time reversed dynamics 𝑊𝑛(𝜋Γ𝑛,𝑡1;…𝜋Γ1,𝑡𝑛). For classical Hamilton dynamics, this property is fulfilled by microscopic irreversibility. This means under the time reversal operation 𝑡→−𝑡(1.42a) 𝑞𝑖→𝑞𝑖(1.42b) 𝑝𝑖→−𝑝𝑖(1.42c) 12 1. Introduction the Hamilton or Newtonian equations of motion are invariant, as long as the (interaction) forces only depend on positions 𝑞𝑖 and not on the momentum or velocities 𝑝𝑖 . Because the dynamics are deterministic, the probability of finding a path is only given by the probability of the initial condition and the fact that the path is a solution of the equations of motion. Because of microscopic reversibility, for every solution of the equations of motion, the time reversed path is also a solution. The probability of finding these paths, therefore, only differs by the probability of the initial conditions 𝑝(Γ0) and 𝑝(𝜋Γ𝑡) . However, in the microcanonical ensemble, every microstate has the same probability; thus, in equilibrium, 𝑝(Γ0)=𝑝(𝜋Γ𝑡) is trivially fulfilled. Hence, all forward paths have the same probability as the backward paths. The final important conclusion is that the classical equilibrium microscopic Hamilton dynamics is time reversal symmetric. Often, one does not observe all degrees (𝑞𝑖,𝑝𝑖) of the underlying Hamilton dynamics but transitions between some coarse grained states 𝑛 . A standard example are so called Markov processes, where the transition from one state 𝑛 to 𝑚 does not depend on any previous transitions. Such processes are typically used to model chemical reaction networks, like molecular motors, and transitions between lattice states [49, 50]. For a continuous time discrete Markov process, the probability of finding the particle in state 𝑛 is given by 𝑝𝑛 , and this probability changes in time given by the master equation [29] 𝜕𝑡𝑝𝑛(𝑡)=∑ 𝑚≠𝑛[𝑝𝑚(𝑡)𝑤𝑚𝑛−𝑝𝑛(𝑡)𝑤𝑛𝑚](1.43) with 𝑤𝑛𝑚 the transition rate from state 𝑛 to 𝑚 . The terms 𝑗𝑚𝑛=𝑝𝑚(𝑡)𝑤𝑚𝑛−𝑝𝑛(𝑡)𝑤𝑛𝑚 are the probability currents between the Markov states. For these Markov systems, we would like to preserve the time reversal symmetry of the underlying Hamilton dynamics. This leads to the notion of detailed balance, if 𝑝𝑚(𝑡)𝑤𝑚𝑛=𝑝𝑛(𝑡)𝑤𝑛𝑚 (1.44) for all states 𝑛,𝑚 then detailed balance is fulfilled. The practical implication of this property is that all probability currents 𝑗𝑛𝑚 vanish. For a Markov process detailed balance ensures that time reversal symmetry, defined on the 𝑊𝑛 probabilities, is valid. Often, the terms time reversal symmetry and detailed balance are used interchangeably. This is only exactly true for a Markov process, generally time reversal symmetry is a stronger property. As a final remark, for the coarse grained dynamics, we have imposed that if the underlying dynamic is in equilibrium, the coarse grained dynamic has to obey time reversal symmetry. So, detecting broken time reversal symmetry shows that the system is out of equilibrium, which we use as a marker for non-equilibrium. The inverse, however, is not true. A coarse grained system can obey time reversal symmetry even if the underlying dynamic does not. Therefore, detecting the breakage of time reversal symmetry can only serve as a marker for non-equilibrium and not as a marker for equilibrium. 1.3. Time reversal and entropy production 13 1.3.2. Stochastic thermodynamics In this section, we want to discuss entropy production as the natural, physically motivated observable to describe the absence of time reversal symmetry. To define entropy production, we first have a look at thermodynamics. In classical thermodynamics, we consider a system that is coupled to heat bath, meaning that the system can absorb and transfer heat to this bath, which is assumed to be an unlimited source and sink. However, this interaction follows certain phenomenological rules, for example, there is no spontaneous heat transfer from a heat bath at high temperature to a heat bath of low temperature without the investment of work. A central concept here is entropy, or better, the change in entropy. The entropy change in a system due to heat transfer with a bath is given by [51] Δ𝑆𝑄=Δ𝑄 𝑇.(1.45) In addition to entropy exchange via heat transfer, the internal entropy in a system can also change, for example, by mixing processes. For irreversible processes, the total change in entropy, or entropy production, between the system and the bath is always positive Δ𝑆≥0(1.46) with equality if the process is reversible. Irreversible processes often involve mixing or other effects that do not naturally reverse without external intervention, as the application of work. Eq. (1.46) is the famous second law of thermodynamics. These relations were first and foremost phenomenological observations, but in later works have been grounded by the principles of statistical mechanics. In statistical mechanics, one defines the entropy in equilibrium of an isolated system as the so called Boltzmann entropy 𝑆𝐵=log(Ω)(1.47) with Ω the microcanonical partition sum of Eq. (1.40) . The Boltzmann entropy increases if the number of accessible microstates increases. Another, more general access is via the Gibbs or Shanon entropy 𝑆 , which is motivated by information theory. If 𝜌(Γ,𝑡) is the probability density, then 𝑆=− 1 𝑁!ℎ3𝑁∫dΓ𝜌(Γ)log(𝜌(Γ)).(1.48) Here 𝑆𝐵 is the special case for the 𝜌(Γ,𝑡)=1Ω microcanonical ensemble. The advantage of Eq. (1.48) is that we can describe how the system approaches the final equilibrium state, which maximizes the entropy, because the probability distribution is completely flat. This was investigated by Boltzmann, and he could show that for an ideal gas with collisions that the entropy increases but never decreases in the process, as one would expect by the second law. This relation is known as the H-Theorem [52]. 14 1. Introduction In the following, we want to discuss the relation between the breakage of time reversal symmetry and entropy production. This yields a generalization of the entropy production to fluctuating paths, which is important when analyzing systems far from thermal equilibrium. For this discussion, we mainly follow Ref. [53], however, we adapted the notation to be consistent with the previous discussion. To make this relation as general as possible, we start with Hamiltion dynamics with a phase space point Γ=(𝑞1,…,𝑞𝑁,𝑝1,…,𝑝𝑁)∈Ω (1.49) where Ω is the whole phase space on the energy shell with energy 𝐸 . Because energy is conserved, the whole dynamics lies in Ω . Usually, we have no access to the microscopic degrees of freedom, but to a more coarse grained version of phase space Ω . We define coarse grained phase space observable 𝑀(Γ) where we map a microscopic phase space point Γ∈Ω to a macro or mesoscopic phase space observable 𝑀(Γ)∈Ω . We also define for a microscopic phase space region 𝐴⊂Ωthe phase space volume |𝐴|. Next, we need the notion of a phase space trajectory. Let 𝑓 be a map that propagates a phase space point Γ along the Liouville equations for time Δ𝑡 . Then we write a microscopic trajectory as 𝛾=(Γ,𝑓Γ,𝑓2Γ,…,𝑓𝑛Γ) (1.50) and the corresponding macroscopic trajectory 𝜔=(𝑀(Γ),𝑀(𝑓Γ),…,𝑀(𝑓𝑛Γ)).(1.51) To define probability distributions, we make the first assumption. Inside the macroscopic states 𝑀(Γ) , we assume that all belonging microscopic states are distributed by the microcanonical ensemble. In this case, the probability of finding a microstate is 𝜈(Γ)=𝜈(𝑀(Γ)) |𝑀(Γ)|,(1.52) where 𝜈(𝑀)is a probability density on Ω. Entropy is given by the Boltzmann entropy, which is defined as 𝑆𝐵(𝑀)=log|𝑀|(1.53) on the macro states 𝑀 . The corresponding microscopic version is given by 𝑆𝐵(Γ)= 𝑆𝐵(𝑀(Γ)) . Generally, differences in Boltzmann entropy are what we refer to as differences of thermodynamic entropy. To relate the Boltzmann entropy to time reversal breaking, consider the following scenario. For every microscopic trajectory that starts at 𝑀0and ends at 𝑀𝑛, the following relation holds log(𝑃(𝛾=(Γ,𝑓Γ,…)|𝑀(Γ)=𝑀0) 𝑃(𝜃𝛾=(Γ,𝑓Γ,…)|𝑀(Γ)=𝜋𝑀𝑛))=𝑆𝐵(𝑀𝑛)−𝑆𝐵(𝑀0)(1.54) 1.3. Time reversal and entropy production 15 Figure 1.2.: Illustration of how to connect entropy production with path probabilities. Given that a microscopic path 𝛾 starts in 𝑀0 , its probability is given by 𝑃(𝛾|𝛾(0)=𝑀0)=1/|𝑀0| . If a path is connecting 𝑀0 and 𝑀𝑛 , then the reversed path 𝜃𝛾 is connecting 𝜋𝑀𝑛 and 𝜋𝑀0 . By the same construction, the probability of 𝜃𝛾 is 𝑃(𝜃𝛾|𝜃𝛾(0)=𝜋𝑀𝑛)=1/|𝜋𝑀𝑛|=1/|𝑀𝑛| . With Eq. (1.53) one arrives at Eq. (1.54). where 𝜃𝛾 is the time reversed trajectory. This result follows from the fact that the underlying dynamics are deterministic, and so the path probability is only defined by the probabilities of the initial points, which are determined by the phase space volume. This construction is illustrated in Fig. 1.2 We can see from Eq. (1.54) that the time reversal can indeed pick up changes in Boltzmann entropy. However, usually we have no access to the microscopic degrees and therefore need a relation for the macroscopic states 𝑀 . For the coarse grained dynamics we can, however, obtain a very similar relation. If the trajectory starts at 𝑀0 and ends in 𝑀𝑛 , then for the trajectories 𝜔=(𝑀0,…,𝑀𝑛)and 𝜃𝜔=(𝜋𝑀𝑛,…,𝜋𝑀0) log(𝑃[𝜔] 𝑃[𝜃𝜔])=[𝑆𝐵(𝑀𝑛)−𝑆𝐵(𝑀0)]+[−log(𝜇𝑡(𝑀𝑛)+log(𝜇0(𝑀0)](1.55) for any initial and final probability distribution 𝜇0,𝜇𝑡 on Ω . What is generally meant as entropy production in statistical mechanics is encoded in the difference of Boltzmann entropies. The term (−log(𝜇(𝑀𝑛)+log(𝜇(𝑀0)) arises due to the stochasticity of the coarse grained dynamics and has a priori no relation to entropy production [53]. On the other hand, including this term gives rise to beneficial mathematical properties, as we will see in section 1.3.3, and may also be physically informed. The whole right hand side of Eq. (1.55) is called stochastic entropy production. We also want to emphasize that to make the connection to the Boltzmann entropy for every macro observable 𝑀 , the underlying microstates Γ with 𝑀=𝑀(Γ) have to be distributed by the microcanonical distribution. Therefore, the system must be at least locally in equilibrium. Eq. (1.55) is strictly speaking only valid for closed systems, however, it can also be generalized to open systems. If the system is coupled to multiple (equilibrated) heat baths, then one 2. Mean Back Relaxation This chapter is largely identical to the publication [4]. We introduce here the mean back relaxation (MBR) as the central observable of this dissertation. •Section 2.1 was taken from [4] with small changes to integrate Section 2.2. • Section 2.2 was written by me and is not included in [4] as is the case for Fig. 2.1. • The remaining sections are taken from [4]. The word "cut off" was exchanged with "length parameter" for overall consistency with the remaining thesis. Matthias Krüger introduced me to the concept of MBR. The mathematical definition of MBR was finalized by me and Matthias Krüger. I performed the mathematical calculations and numerical simulations. I wrote the original draft of the paper under the supervision of Matthias Krüger. Some of the words might be his. 2.1. Introduction The analysis of particle trajectories is a fundamental building block of modern statistical physics, both in simulation and experiment. Especially in the latter, nowadays advanced techniques allow particle tracking with high precision and time resolution, leading to an extensive use of statistical analysis and the determination of microrheological and non-equilibrium properties of complex media [42, 44, 47, 70, 71]. Correlation functions obtained from these trajectories, like the mean squared displacement (MSD), are in equilibrium related to powerful theorems like the fluctuation dissipation theorem (FDT) or Green-Kubo relation [35, 38, 72–75]. Breakage of time reversal symmetry is quantified by entropy production, for which powerful theorems exist [3, 33, 61, 63, 65, 76–78]. Multi-point correlation functions, that depend on multiple time points, have the potential to be sensitive to time reversal breakage and can therefore serve as intriguing observables in non-equilibrium systems. Conditioning trajectories, for example by selecting trajectories that start at certain positions [79–81] has lead to interesting insights. In the following, we will analyze a multi-point observable that has been termed mean back relaxation (MBR) [2, 82]. It was shown that MBR is a marker for broken detailed balance in confinement [2]. In this chapter we expand on the properties of this quantity and analyze it in more depth. We also extend it towards microscopic densities, which yields a marker for broken detailed balance in confinement and for bulk systems. 24 2. Mean Back Relaxation The chapter is organized as follows. In sections 2.2 and 2.3, we motivate and define MBR. We discuss its properties in confined systems in section 2.4, where we reproduce the proof that the deviation of the long time value of MBR from 12 marks broken detailed balance in confinement and a fluctuation theorem. We probe these findings on an active Brownian particle trapped in a potential. In section 2.5 we derive conditioned correlations in a path integral formalism. In section 2.6, we apply this to find a general relation between MBR and mean squared displacement in Gaussian systems. At last in section 2.7 we introduce a new version of MBR based on densities, showing that its deviation from 12marks broken detailed balance in bulk systems and confinement. 2.2. Mean Back Relaxation: Motivation The central idea of MBR lies in the following question: If a particle moved, for example, to the right in a previous time interval, what is it going to do in the future? Generally, one would expect that if there is no overall drift in the system, the particle would move to the left. But by how much? As a dimensionless quantitative measure, we define the back relaxation as the negative ratio between the displacement 𝑑 in the intervall [−𝜏,0]and the displacement 𝑏between [0,𝑡](ignoring division by 0) ≈−𝑏𝑑.(2.1) A more rigorous mathematical definition can be found in section 2.3. The mean of Eq. (2.1) is the mean back relaxation. The hope is that how the particle responds to displacements 𝑑 is different in equilibrium and non-equilibrium settings, and thus MBR might reveal differences in time forward and time reversed dynamics. MBR can be understood as a measure of persistence. If 𝑏 and 𝑑 have the same sign, the back relaxation is negative, while if they have opposite signs, it is positive. Negative MBR, therefore, means that the particle tends to move in the same direction, while positive MBRs indicate that the particle moves in the opposite direction. This connection is visualized in Fig. 2.1. The most typical observable investigated for colloidal particles is the MSD. However, it is not particularly useful to investigate time reversal symmetry, because Δ𝑥2(𝑡)=∫d𝑥𝑡d𝑥0(𝑥𝑡−𝑥0)2𝑊2(𝑥𝑡,𝑡;𝑥0,0) (2.2) =∫d𝑥𝑡d𝑥0(𝑥𝑡−𝑥0)2𝑊2(𝑥0,𝑡;𝑥𝑡,0) (2.3) it is insensitive to replacing the joint two-point probability 𝑊2 with its time reversed. Therefore, MSD is not able to detect breaking of time reversal symmetry. Other options usually involve currents, which are primarily used in the context of Markov networks, which can detect breaking of time reversal symmetry if one can construct one for the particular system. The most famous example is the entropy production itself. That MBR might be a possible marker is motivated by another conditioned correlation, the mean 2.3. Mean Back Relaxation: Conditioned correlations 25 Figure 2.1.: Illustrates the meaning of basic values of MBR. If the particle moves on average in the same direction between −𝜏 and 0 and 0 and 𝑡 , MBR is negative. If future displacement is not influenced by the past displacement, MBR vanishes. For positive values, the particle typically moves in the opposite direction in the respective time intervals. The value of 12 is the expected long time 𝑡value in equilibrium under the conditions of Eq. (2.8) & (2.9) . conditional displacement (MCD). Here, one fixes the initial point of the trajectory 𝑥0 and then looks at the mean conditioned displacement ⟨𝑥(𝑡)−𝑥0⟩|𝑥0 . With this quantity, one can derive conditions on monotonicity in equilibrium. It has been used to show that a driven particle in an optical tweezers suspended in a viscoelastic background is out of equilibrium [80, 83]. However, the MCD is not easily applicable to a bulk system because it is not possible to sample unique 𝑥0positions. 2.3. Mean Back Relaxation: Conditioned correlations The mean back relaxation (MBR) [2] is the average future displacement of a particle divided by the displacement in the past. For particle position coordinate 𝑥 , we define it in terms of two time periods 𝑡and 𝜏and a length parameter 𝑙as, (𝜏,𝑡,𝑙)=⟨−𝑥𝑡−𝑥0 𝑥0−𝑥−𝜏𝜗𝑙(|𝑥0−𝑥−𝜏|)⟩(2.4) =∫d𝑥𝑡d𝑥0d𝑥−𝜏(−𝑥𝑡−𝑥0 𝑥0−𝑥−𝜏)×(2.5) ×𝜗𝑙(|𝑥0−𝑥−𝜏|)𝑊3(𝑥𝑡,𝑡;𝑥0,0;𝑥−𝜏,−𝜏). 𝑊3 is the three point probability function, i.e., the probability of finding the particle at the positions 𝑥𝑡,𝑥0,𝑥−𝜏 at the corresponding times. MBR is thus the ratio of the displacement 𝑏=𝑥𝑡−𝑥0 in future, i.e., between 0 and 𝑡 and the displacement in the past, 𝑑=𝑥0−𝑥−𝜏 , i.e., between times −𝜏 and 0. One observes that the particle typically travels in the opposite direction or "relaxes back" compared to its earlier movement, 26 2. Mean Back Relaxation in which case 𝑏 and 𝑑 have opposite signs [2]. We therefore include by convention a minus sign, so that MBR is positive in these cases. In contrast to the definition provided in Ref. [2], we have made here explicit that |𝑑|<𝑙 is excluded, with length parameter 𝑙>0 . We use in Eq. (2.5) a Heaviside step function 𝜃(𝑥)in the following factor 𝜗𝑙(|𝑥0−𝑥−𝜏|)=𝜃(|𝑥0−𝑥−𝜏|−𝑙) ⟨𝜃(|𝑥0−𝑥−𝜏|−𝑙)⟩.(2.6) As mentioned, the factor 𝜗𝑙(|𝑥𝜏−𝑥0|) excludes the region of vanishing denominator in Eq. (2.5) . The denominator in Eq. (2.6) is introduced for normalization 1 . This definition of MBR in Eq. (2.5) has a practical implementation in data from experiments or simulations; instances with |𝑥𝜏−𝑥0|<𝑙 are excluded from trajectories. In other words, the object ⟨𝜃(|𝑥0−𝑥−𝜏|−𝑙)⟩ needs not to be evaluated explicitly. A sample code for evaluation of MBR of a Brownian particle in a harmonic potential can be found at SI. In Ref. [2], MBR was evaluated in the limits of small 𝜏 and small 𝑙 . The existence of these limits cannot be expected a priori. We continue here by analyzing MBR for finite 𝑙and finite 𝜏. To illustrate the meaning of the condition, we give the following equivalent formulation (𝜏,𝑡,𝑙)=∫d𝑑d𝑏(−𝑏𝑑)𝜗𝑙(|𝑑|)𝑝(𝑏,𝑑) =∫d𝑑(−⟨𝑏⟩|𝑑 𝑑)𝜗𝑙(|𝑑|)𝑝(𝑑).(2.7) Here ⟨𝑏⟩|𝑑=⟨𝑥𝑡−𝑥0⟩|𝑑=∫d𝑏[𝑏𝑝(𝑏|𝑑)] is the conditional average of the displacement 𝑏=𝑥𝑡−𝑥0 given 𝑑 . Eq. (2.7) shows that MBR probes how conditions affect the movement of the particle. Division by 𝑑 , despite the need of a length parameter 𝑙 , is advantageous, as it renders MBR dimensionless. This gives MBR a quantitative meaning, which we expect to allow useful quantitative comparison of different systems. 2.4. MBR in stationarity 2.4.1. Long time limit of MBR and time reversal In this subsection, we reproduce for completeness some of the results given in Ref. [2]. We will in this subsection discuss stationary systems in confinement. We thus assume that a stationary distribution 𝑊1(𝑥) with a finite mean ⟨𝑥⟩ exists. We also assume that 1⟨𝜃(|𝑥0−𝑥−𝜏|−𝑙)⟩is nonzero if displacements |𝑑|≥𝑙occur. 2.4. MBR in stationarity 27 the joint three-point distribution function can be separated for 𝑡→∞, i.e., ⟨𝑥⟩=∫d𝑥𝑥𝑊1(𝑥)with ⟨𝑥⟩finite,(2.8a) lim 𝑡→∞𝑊3(𝑥𝑡,𝑡;𝑥0,0;𝑥−𝜏,−𝜏)(2.8b) =𝑊1(𝑥𝑡)𝑊2(𝑥0,0;𝑥−𝜏,−𝜏). We introduced the joint two-point probability 𝑊2 [26]. In this subsection, we assume the following symmetry of 𝑊2 𝑊2(𝑥𝑡,𝑡;𝑥0,0)=𝑊2(𝑥0,𝑡;𝑥𝑡,0),(2.9) which is the usual condition for detailed balance for the joint distribution [26, 40]. We insert Eq. (2.8) in the definition of MBR, Eq. (2.5), and obtain lim 𝑡→∞(𝜏,𝑡,𝑙)=∫d𝑥𝑡d𝑥0d𝑥−𝜏(−𝑥𝑡−𝑥0 𝑥0−𝑥−𝜏)×(2.10) ×𝜗𝑙(|𝑥0−𝑥−𝜏|)𝑊1(𝑥𝑡)𝑊2(𝑥0,0;𝑥−𝜏,−𝜏) =∫d𝑥0d𝑥−𝜏(−⟨𝑥⟩−𝑥0 𝑥0−𝑥−𝜏)×(2.11) ×𝜗𝑙(|𝑥0−𝑥−𝜏|)𝑊2(𝑥0,0;𝑥−𝜏,−𝜏). Evaluating half of Eq. (2.11) by a change of variables, 𝑥0↔𝑥−𝜏 yields a form in terms of the difference of 𝑊2with arguments exchanged, lim 𝑡→∞(𝜏,𝑡,𝑙)=12+ 12∫d𝑥0d𝑥−𝜏(−⟨𝑥⟩−𝑥0 𝑥0−𝑥−𝜏)𝜗𝑙(|𝑥0−𝑥−𝜏|)× ×[𝑊2(𝑥0,0;𝑥−𝜏,−𝜏)−𝑊2(𝑥−𝜏,0;𝑥0,−𝜏)],(2.12) The term in the second line vanishes when inserting Eq. (2.9) . The long time value of MBR is thus lim 𝑡→∞(𝜏,𝑡,𝑙)(2.8)&(2.9) =12.(2.13) We conclude that, if Eq. (2.13) is violated and Eq. (2.8) holds, Eq. (2.9) , i.e., detailed balance, must be broken. Notably, the result in Eq. (2.13) is independent of the time period 𝜏, and it is independent of the length parameter 𝑙. Eq. (2.13) is not restricted to particle position 𝑥 . It is generally valid for observables A with 𝜋𝐴=𝜖𝐴,𝜖∈{−1,1} with 𝜋 the kinematic reversal operator and the corresponding detailed balance condition 𝑊2(𝐴𝑡,𝑡;𝐴0,0)=𝑊2(𝜖𝐴0,𝑡;𝜖𝐴𝑡,0) . We provide the corresponding general derivations in appendix A.1. 28 2. Mean Back Relaxation 2.4.2. Fluctuation theorem In this subsection we assume Eq. (2.8) to be given, without assuming symmetry of 𝑊2 . We introduce path probability 𝑝[𝑥(𝑡)] , so that, with detailed balance, 𝑝[𝑥(𝑡)]=𝑝[𝜃𝑥(𝑡)] for any path, with bracket […] indicating the functional dependence on 𝑥(𝑡) . The path probabilities of 𝑥(𝑡) and 𝜃𝑥(𝑡) yield a fluctuation theorem for MBR. The stochastic total change of entropy for the stationary process is defined as [33, 53, 54, 79] 𝑠[𝑥(𝑡)]=log𝑝[𝑥(𝑡)] 𝑝[𝜃𝑥(𝑡)].(2.14) Stochastic total change of entropy therefore is log ratio of the probability of a given path and the probability of its reversed. We rewrite Eq. (2.12) by change of integration variables lim 𝑡→∞(𝜏,𝑡,𝑙) =12+12⟨𝑥0+𝑥−𝜏−2⟨𝑥⟩ 𝑥0−𝑥−𝜏𝜗𝑙(|𝑥0−𝑥−𝜏|)⟩.(2.15) Eq. (2.15) is reformulated in the fluctuation theorem [3] ⟨(−⟨𝑥⟩−𝑥0 𝑥0−𝑥−𝜏)𝜗𝑙(|𝑥0−𝑥−𝜏|)(1+𝑒−𝑠)⟩=1.(2.16) Note that for any observable 𝑂(𝑥(𝑡)) the average of the time reversed observable can be rewritten as ⟨𝑂[𝜃𝑥(𝑡)]⟩=∫𝑥(𝑡)𝑂[𝑥(𝑡)]𝑝[𝜃𝑥(𝑡)] 𝑝[𝑥(𝑡)]𝑝[𝑥(𝑡)] (2.17) =⟨𝑂[𝑥(𝑡)]𝑒−𝑠⟩(2.18) using entropy production as defined in Eq. (2.14) and 𝑥(𝑡) denoting a path integral. The fluctuation theorem Eq. (2.16) therefore states that the long time value of MBR evaluated from original and reversed dynamics add up to unity lim 𝑡→∞((𝜏,𝑡,𝑙)+𝑟(𝜏,𝑡,𝑙)) =⟨−⟨𝑥⟩−𝑥0 𝑥0−𝑥−𝜏𝜗𝑙(|𝑥0−𝑥−𝜏|) −⟨𝑥⟩−𝑥−𝜏 𝑥−𝜏−𝑥0𝜗𝑙(|𝑥−𝜏−𝑥0|)⟩=1,(2.19) where we introduced 𝑟(𝜏,𝑡,𝑙) , the MBR evaluated from time backward trajectories. Notably, the result in Eq. (2.19) is independent of the time period 𝜏 , and it is independent of the length parameter 𝑙 . In equilibrium, original and reversed dynamics are statistically indistinguishable, which is equivalent to 𝑠=0 for all paths, hence the long time value of MBR equals 12as stated previously. 2.4. MBR in stationarity 29 0 5 10 15 20 25 30 t/γ k 0.0 0.2 0.4 0.6 0.8 1.0 M(τ, t, l) original reversed org. + rev. active passive Figure 2.2.: MBR as a function of time 𝑡 , obtained from simulations of an active or passive Brownian particle, Eq. (2.20) , for 𝜏=2.0𝛾𝑘,𝑙=0.2√𝛾𝐷 𝑘 and 𝛾𝐷𝜑 𝑘= 0.2 . Blue line shows a passive particle with 𝑣0=0 , i.e., a system with detailed balance, and MBR approaches the long time value 12 . The solid red line shows an active Brownian particle with 𝑣0/(√𝑘 𝛾𝐷𝐷)=7 . The long time value deviates from 12 , indicating a non-equilibrium system. The red dotted line is for the same system, evaluated for the time reversed trajectories. In this case, for 𝑡→∞ , MBR deviates from 12 in opposite direction. Their sum is shown as a dark red dotted line, which approaches unity in the long time limit, as predicted by the fluctuation theorem Eq. (2.16) . Reprinted from Ref. [4]. To sum up, in a system with Eq. (2.8) fulfilled, Eq. (2.16) holds, and the long time values of MBR evaluated from original and reversed dynamics add up to unity, Eq. (2.19) . With Eqs. (2.8) and (2.9) fulfilled, the long time value of MBR equals 12 . If Eq. (2.8) is fulfilled, a deviation of the long time value from 12 thus marks the invalidity of Eq. (2.9) , i.e., breakage of detailed balance. These results are independent of time period 𝜏 and length parameter 𝑙. 2.4.3. Example: Active Brownian particle To demonstrate the results of the previous subsection, we probe a two-dimensional active Brownian particle confined in a harmonic trap of stiffness 𝑘 , described by the 30 2. Mean Back Relaxation 0246810 τ/γ k 0.50 0.55 0.60 0.65 0.70 M(τ, t → ∞, l) active passive 0.01 0.02 0.03 D v2 0/γ k 0.2 0.4 0.6 τmax/γ k k Dϕγ=0.67 k Dϕγ=5.0 Figure 2.3.: Long time limit of MBR of ABP as function of 𝜏 with 𝑙=0.2√𝛾𝐷 𝑘 and 𝛾𝐷𝜑 𝑘=0.2 . In the passive case 𝑣0=0 (blue line) the MBR is 12 independent of 𝜏 . In the active case with 𝑣0/(√𝑘 𝛾𝐷𝐷)=7 (red line), MBR depends on 𝜏 , and the deviation from 12 shows a maximum, denoted 𝜏m𝑎𝑥 . The inset shows 𝜏max as a function of 𝐷𝑣20 , for different 1 𝐷𝜑 , with all times in units of 𝛾𝑘 . Different line colors indicate different 1 𝐷𝜑 , equidistant between 𝑘 𝐷𝜑𝛾=0.67 and 𝑘 𝐷𝜑𝛾=5.0 (corresponding to a spacing of ≈0.394 between lines). Reprinted from Ref. [4]. Langevin equations[16, 84] 𝐱=−𝑘𝛾𝐱+𝑣0𝐞(𝜑)+𝐟(2.20a) 𝜑=𝑓𝜑(2.20b) with 𝐞=(cos(𝜑),sin(𝜑))⊺ and white noise random forces ⟨𝑓(𝑖)(𝑡)𝑓(𝑗)(𝑡′)⟩=2𝐷𝛿(𝑡− 𝑡′)𝛿𝑖𝑗 and ⟨𝑓𝜑(𝑡)𝑓𝜑(𝑡′)⟩=2𝐷𝜑𝛿(𝑡−𝑡′) . 𝐷 and 𝐷𝜑 are the translational and rotational diffusion coefficients, respectively. Simulation results are shown in Fig. 2.2, where we use one of the coordinates to evaluate MBR. Simulation details can be found in appendix A.2. If the active propulsion 𝑣0=0 vanishes, the system is in equilibrium. The blue line in Fig. 2.2 shows this case, for 𝜏=2.0𝛾𝑘 and 𝑙=0.2√𝛾𝐷 𝑘 , and indeed MBR approaches the equilibrium 12 value for large times 𝑡 . The red lines in Fig 2.2 are for finite 𝑣0 , using the same values of 𝜏 and 𝑙 as for the passive case. In this case the long time MBR deviates from 12 . This result indicates the breakage of detailed balance. Evaluation of MBR for the reversed trajectories results in a deviation from 12 of opposite sign for 𝑡→∞ , i.e., they add up to one as stated by the fluctuation theorem Eq. (2.16) . The 2.4. MBR in stationarity 31 10−210−1100 l/ 1 √βk 0.5 0.6 0.7 0.8 0.9 1.0 M(τ, t → ∞, l) active passive norm. + rev. Figure 2.4.: The long time limit of MBR of ABP as function of 𝑙 with 𝜏=2.0𝛾𝑘 and 𝛾𝐷𝜑 𝑘=0.2 . In the passive case 𝑣0=0 (blue line) there is no dependence, as expected from Eq. (2.13) . In the active case 𝑣0/(√𝑘 𝛾𝐷𝐷)=7 (red line) MBR depends on 𝑙 , with the limit 𝑙→0 apparently existing. The fluctuation theorem Eq. (2.16) does not depend on 𝑙 shown by the red dotted line for 𝑣0/(√𝑘 𝛾𝐷𝐷)=7. Reprinted from Ref. [4]. active Brownian particle therefore demonstrates that a deviation of MBR from 12 detects the breakage of time reversal symmetry, and this breakage is accurately described by the corresponding fluctuation theorem. Figure 2.3 shows the long time value of MBR as a function of 𝜏 . For 𝑣0=0 , this long time value is indeed independent of 𝜏 , as stated by Eq. (2.13) . For finite 𝑣0 , i.e., with detailed balance broken, the long time value does depend on 𝜏 . For 𝜏→∞ , the long time MBR seems to approach 12 , and the deviation from 12 is most pronounced for an intermediate value of 𝜏 , denoted 𝜏max . The model in Eq. (2.20) encodes three time scales, 1 𝐷𝜑 , 𝐷𝑣20 and 𝛾𝑘 , resembling the time scales of rotational diffusion, the time scale where active and passive motion are comparable, and the time scale of relaxation in the harmonic potential, respectively. The inset in Fig. 2.3 shows the dependence of 𝜏max on these, by varying 1 𝐷𝜑 and 𝐷𝑣20 and by scaling the axes in units of 𝛾𝑘 . 𝜏m𝑎𝑥 increases with 1 𝐷𝜑 and with 𝐷𝑣20 , indicating that breakage of detailed balance is sensitive to 𝐷𝜑 and 𝑣0 : As regards the dependence on propulsion velocity 𝑣0 , we interpret that for smaller 𝑣0 , it takes a longer time for activity to become noticeable, and hence 𝜏m𝑎𝑥 is larger. Decreasing rotational diffusion 𝐷𝜑 makes the motion more persistent and thereby increases the timescales of motion. This seems to also increase the timescale 𝜏m𝑎𝑥. Figure 2.4 shows the long time value as a function of the length parameter 𝑙 . As expected, for 𝑣0=0 , this value is independent of 𝑙 . Also, for finite 𝑣0 , the sum of forward and 38 2. Mean Back Relaxation particle interactions [86], so that ⟨𝜌𝑞⟩=∫d𝑥⟨𝜌(𝑥)⟩𝑒𝑖𝑞𝑥 (2.51) is finite as well. For example, for homogeneous systems, the mean density is constant in space ⟨𝜌(𝑥)⟩=𝜌0, and ⟨𝜌𝑞⟩=𝜌0𝛿(𝑞)vanishes for any finite 𝑞[86]. We thus introduce the density mean back relaxation dMBR, 𝜌(𝜏,𝑡,𝑙,𝑞)=⟨−𝜌𝑞(𝑡)−𝜌𝑞(0) 𝜌𝑞(0)−𝜌𝑞(−𝜏)𝜗𝑙(|𝜌𝑞(0)−𝜌𝑞(−𝜏)|)⟩.(2.52) dMBR additionally depends on the wave vector 𝑞 compared to MBR. A global displacement 𝑥𝑖+𝑎 yields a factor 𝜌𝑞(𝑡)→𝑒𝑖𝑞𝑎𝜌𝑞(𝑡) , which cancels in Eq. (2.52) . dMBR does thus not depend on the choice of coordinate origin. Most important, we can thus state lim 𝑡→∞𝜌(𝜏,𝑡,𝑙,𝑞)(A.1)&(A.2) =12,(2.53) so that a deviation of dMBR from 12 marks broken detailed balance. Furthermore, the sum of long time values of dMBR for original and reversed trajectories adds up to unity, analogously to Eq. (2.19), lim 𝑡→∞𝜌(𝜏,𝑡,𝑙,𝑞)+lim 𝑡→∞𝜌𝑟(𝜏,𝑡,𝑙,𝑞)=1.(2.54) To demonstrate these properties, we revisit the two coupled particles of Eq. (2.39) , see Fig. 2.6. Indeed, for 𝑇1=𝑇2 , dMBR approaches 12 , while it deviates from 12 for 𝑇1≠𝑇2 , with Eq. (2.54) fulfilled. As expected, the deviation from 12 depends on 𝑞 . We expect that the dependence on 𝑞yields information on the length scales of the system. 2.8. Summary We discussed several properties of the recently introduced mean back relaxation, and extended the definition by including a length parameter 𝑙 , so that MBR depends, additionaly to observation time 𝑡 , on the parameters of time period 𝜏 and length 𝑙 . In stationary confined systems, deviation of the long time value of MBR from 12 marks breakage of detailed balance [2]. With or without detailed balance, MBR from forward and reversed paths sum up to unity. These statements are true for any values of 𝜏 and 𝑙. We exemplify this for the case of a trapped active Brownian particle. We point out that Eq. (2.13) is generally valid for observables that are even or odd under time reversal. Future work will investigate MBR with particle velocity, which is odd under time reversal. Using a path integral approach we find a general solution for MBR in terms of multipoint density correlations. Evaluating this for Gaussian systems yields a relation between mean squared displacement (MSD) and MBR, from which additional properties of MBR 2.8. Summary 39 0.0 0.5 1.0 1.5 2.0 q/qk kBT 0.6 0.8 1.0 1.2 1.4 |Mρ(τ, t → ∞, l, q)| 0 25 50 t/γ1 k 0.00 0.25 0.50 |Mρ(τ, t, q)| original reversed ∆T T= 3.0 ∆T T= 0.0 Figure 2.6.: Density MBR, Eq. (2.52) , for the model of two coupled particles of Eq. (2.39) , with 𝜏=0.1𝛾1 𝑘 , 𝛾1 𝛾2=1,𝑘′ 𝑘=0.003 and 𝑙=0.01 ( 𝑙 is dimensionless for dMBR). Main plot shows the long time limit as a function of 𝑞 , displaying the value of 12 for equal temperatures, and a deviation for different temperatures. The deviation from 12 is maximal for 𝑞max ≈0.11√𝑘 𝑘𝐵𝑇 for this set of parameters. Inset shows the behavior for 𝑞=0.8√𝑘 𝑘𝐵𝑇 as a function of time, where we note that the numerical data obey Eq. (2.54). Reprinted from Ref. [4]. in Gaussian systems can be inferred. In Gaussian overdamped equilibrium systems, MBR is monotonic in time, and takes values between 0 and 12 . We exemplified the relation between MBR and MSD via the example model of two trapped coupled Brownian particles. Studying the regime of parameters for which the relaxation times are well separated allows for additional insights: For intermediate times, the MSD is diffusive, and MBR takes (nearly) time independent values, which show a strong dependence on the chosen parameters. For even longer times, MSD approaches a constant, and MBR approaches 12 for any set of parameters. We point out that Eq. (2.13) is valid for the latter time regime only, and the value of 12 is found because 1d stationary Gaussian systems obey detailed balance. We define a density MBR, dMBR, using the Fourier transform of microscopic density as observable. dMBR enjoys the same properties as MBR: It marks breakage of detailed balance, and sums up to unity from normal and reversed paths. We illustrate its advantages using the model of coupled Brownian particles: i) The final value of MBR is typically approached faster compared to position MBR; its relaxation time remains finite when the trapping potential vanishes, in contrast to that of position MBR. ii) Density MBR uses the positions of both particles as an input, and it detects broken detailed balance for the case of different temperatures, in contrast to position MBR. 40 2. Mean Back Relaxation Future work will investigate how the relaxation time of density MBR can be estimated a priori. It remains open whether monotonicity and positivity of MBR in equilibrium can be proven more generally. For dMBR, more examples need to be studied, by testing it in many-body systems [92, 93]. 3. Variance of Back Relaxation This chapter is based on sections V.A and V.B of Ref. [5], which correspond to sections 3.1, 3.2 and 3.3 which were slightly modified to fit the structure of the remaining thesis. This part of the original draft was written by me under the supervision of (MK). Some of the words might be his. Section 3.4 is new and written by me. 3.1. Introduction In chapter 2 we have extensively analyzed the mean of the back relaxation. For practical purposes, it is important to estimate the statistical error of MBR. In this chapter, we address this by defining the variance of back relaxation (VBR). We will also find the relation between VBR and MSD for a Gaussian process and give guidance on how to select the length parameter 𝑙. 3.2. Formal expression VBR is given by the variance of the observable averaged in Eq. (2.5), VBR(𝜏,𝑡,𝑙)=⟨(−𝑥(𝑡)−𝑥(0) 𝑥(0)−𝑥(−𝜏)𝜗𝑙(𝑥(0)−𝑥(−𝜏)))2⟩ −⟨(−𝑥(𝑡)−𝑥(0) 𝑥(0)−𝑥(−𝜏)𝜗𝑙(𝑥(0)−𝑥(−𝜏)))⟩2.(3.1) 3.3. Gaussian process: Dependence on 𝑡,𝜏and 𝑙 For a Gaussian process, VBR is found in terms of MSD and MBR (see appendix B for details), VBR(𝜏,𝑡,𝑙)=(𝜏,𝑡)2[(𝛼(𝜏,𝑡)−1)𝑔(𝜂)+ℎ(𝜂)].(3.2) 42 3. Variance of Back Relaxation Figure 3.1.: Variance of back relaxation (VBR) versus the length parameter 𝜂=𝑙 √2Δ𝑥2(𝜏) , from Eq. (3.2) , for various values of 𝛼=4(1−Δ𝑥2(𝑡+𝜏)−Δ𝑥2(𝑡) Δ𝑥2(𝜏))−2Δ𝑥2(𝑡) Δ𝑥2(𝜏) . Gray vertical line gives 𝜂(𝑔) min , the position of the minimum of VBR approached for large 𝛼. Reprinted from Ref. [5]. We introduced the functions 𝑔(𝜂)= 1 erfc(𝜂)2(1 √𝜋𝑒−𝜂2 𝜂−erfc(𝜂)),(3.3) ℎ(𝜂)=(1 erfc(𝜂)−1).(3.4) with the complementary error function erfc(𝑥) . We also introduced a dimensionless number 𝛼(𝜏,𝑡)= 1 (𝜏,𝑡)2Δ𝑥2(𝑡) Δ𝑥2(𝜏).(3.5) Notably, for Gaussian systems 𝛼≥1. We introduced the dimensionless length parameter 𝜂=𝑙 √2Δ𝑥2(𝜏) . Notably, for the Gaussian process, MBR is independent of 𝑙 , but VBR in Eq. (3.2) depends on it. Note that MBR in Eq. (3.2) can be replaced in terms of MSD via Eq. (2.38) , so that VBR may equivalently be expressed purely in terms of MSD. For any 𝜏 and 𝑡 , VBR in Eq. (3.2) has a minimum as a function of 𝜂 : 𝑔(𝜂→∞)→√𝜋 2𝑒𝜂2 𝜂 , and ℎ(𝜂→∞)→√𝜋𝜂𝑒𝜂2 , i.e., VBR diverges as 𝜂→∞ as long as MBR is nonzero. For 𝜂→0 , 𝑔 diverges as 𝑔(𝜂→0)→1 √𝜋1𝜂 while ℎ(𝜂→0)→0 stays finite. VBR thus diverges as a power law for 𝜂→0, except for special cases MBR =0or 𝛼=1. 3.3. Gaussian process: Dependence on 𝑡,𝜏and 𝑙43 The divergence of VBR for large 𝜂 is understood from the observation that ⟨𝜃(|𝑥0−𝑥−𝜏|−𝑙)⟩ goes to zero in that limit. In other words, for 𝜂→∞ , there are less and less occurrences of displacements |𝑥0−𝑥−𝜏|larger than 𝑙, yielding worse and worse statistics. For small 𝜂 , events with smaller and smaller |𝑥0−𝑥−𝜏| are taken into account in Eq. (2.5) , yielding a larger variance due to the smaller and smaller denominator in that equation. The minimum of VBR as a function of 𝑡 , 𝜏 and 𝑙 is of interest, as one may expect the smallest statistical error of MBR at that minimum. We will continue by discussing this minimum as a function of 𝜂 , starting with 𝑡 and 𝜏 fixed. Notably, 𝑔(𝜂) shows a global minimum at 𝜂(𝑔) min =0.654334 , corresponding to a length 𝑙 of 𝑙≈0.92537√Δ𝑥2(𝜏) , i.e., 𝑙 comparable to the square root of MSD at time 𝜏 . In contrast, ℎ(𝜂) in Eq. (3.4) is a monotonically growing function of 𝜂 , and the minimum of VBR, 𝜂min , thus depends on 𝛼 in Eq. (3.2) , and is smaller than 𝜂(𝑔) min , 𝜂min ≤𝜂(𝑔) min . In practical situations, MBR ∼(1) and if Δ𝑥2(𝑡)≫Δ𝑥2(𝜏) , 𝛼−1≫1 . We expand around this case, i.e., perform an asymptotic expansion around 𝛼−1infinite, finding, for the minimum of VBR, 𝜂min(𝛼)=𝜂(𝑔) min −ℎ′(𝜂(𝑔) min) 𝑔′′(𝜂(𝑔) min)1 𝛼−1+((𝛼−1)−2),(3.6) with expansion coefficients ℎ′(𝜂(𝑔) min)=5.84259 and 𝑔′′(𝜂(𝑔) min)=6.71207 . For 𝛼→∞ , 𝜂min approaches 𝜂(𝑔) min, and is smaller for finite 𝛼. Figure 3.1 shows VBR as a function of 𝜂 for several values of 𝛼 . For all curves shown, VBR has a single minimum. Both the value of 𝜂min as well as the minimal value of VBR decrease with 𝛼 . For large 𝛼 , the position 𝜂min is seen to become independent of 𝛼 (Eq. (3.6) ), while VBR, at that minimum, grows linearly in 𝛼 , as expected from Eq. (3.2) . As is evident in Eq. (3.2) , 𝜂min depends only on 𝛼 , and Fig. 3.2 shows that dependence. For large 𝛼 , 𝜂min approaches 𝜂(𝑔) min with a power law, where the red dashed line shows the two terms given in Eq. (3.6) . For 𝛼→1 , 𝜂min goes to zero as 𝜂min ∝(𝛼−1)12 . Recall that 𝛼>1for a Gaussian process. The inset of Fig. 3.2 shows VBR (𝜂min)/MBR2 , as a function of 𝛼 . We see that this ratio increases linearly with 𝛼 for large 𝛼 , as expected from Eq. (3.2) , and asymptotically reaches zero for 𝛼→1with a power law of (𝛼−1)12. How does 𝛼(𝜏,𝑡) depend on 𝜏 and 𝑡 ? To investigate this, we rewrite Eq. (3.5) purely in terms of MSD 𝛼(𝜏,𝑡)= 4Δ𝑥2(𝑡)Δ𝑥2(𝜏) (Δ𝑥2(𝑡+𝜏)−Δ𝑥2(𝑡)−Δ𝑥2(𝜏))2 =𝛼(𝑡,𝜏),(3.7) i.e., 𝛼 is symmetric in its arguments. In case of diffusive behavior, i.e., Δ𝑥2(𝑡)≃2𝐷𝑡 for large 𝑡,𝛼(𝜏,𝑡)grows without bound with increasing time. If MSD reaches a finite 44 3. Variance of Back Relaxation Figure 3.2.: Position of minimum of VBR, 𝜂min , as a function of 𝛼 . Solid blue line shows the numerically exact solution, and the dotted grey and red lines show the zeroth and first order terms of Eq. (3.6) , respectively. Inset shows VBR, evaluated at 𝜂min , as a function of 𝛼−1 , displaying the two power laws with a cross over at 𝛼−1≈0.1. Reprinted from Ref. [5]. value, i.e., lim 𝑡→∞Δ𝑥2(𝑡)=𝐴, we obtain lim 𝑡→∞𝛼(𝜏,𝑡)= 4𝐴 Δ𝑥2(𝜏).(3.8) If MSD grows with time, there is thus a regime where 𝛼 decreases with increasing 𝜏 . With the symmetry 𝛼(𝜏,𝑡)=𝛼(𝑡,𝜏), the statement hols true for 𝜏and 𝑡exchanged. Summarizing, 𝜂min takes values between zero and 0.65 , so that, especially, for large 𝛼 , 𝜂∼(1) is a useful rule of thumb, i.e., the length parameter that minimizes VBR may be estimated as of order √Δ𝑥2(𝜏) . VBR( 𝜂min )/MBR 2 grows as a function of 𝛼 , which itself depends on 𝜏and 𝑡. 3.4. Summary We have calculated the variance of back relaxation to get an idea of the statistics of MBR. We give an explicit expression for a Gaussian process and find that, while MBR is independent of the length parameter 𝑙 , VBR does depend on it. VBR diverges for small and large 𝑙 , with a minimum in between. This minimum can serve as an estimate for an ideal length parameter. Generally, this minimum depends on the MSD, 𝜏 and 𝑡 . As a rule of thumb, we provide that 𝑙 should be chosen as of order 𝑙≈√Δ𝑥2(𝜏) . This is the main result in this section. In the first study of MBR in [2], we tried to 3.4. Summary 45 minimize the length parameter to increase the statistics. However, in the final analysis, the length parameter was estimated by a similar principle as presented in this chapter to avoid detector noise at very small length scales. The results here provide theoretical considerations on how to choose the length parameter. It also provides guidelines for temporal and spatial resolution for experiments. For any given 𝜏 , the spatial resolution should be below √Δ𝑥2(𝜏)to apply MBR. 4. MBR,VBR and effective Energy in cells and model This chapter is an adapted reprint of the publication [5] and is a continuation of chapter 3. The different sections have been reordered for better readability. The experimental data was obtained by Till Moritz Muenker (TMM) and was originally presented in [2]. The original manuscript was written by me and TMM under the supervision of MK and Timo Betz (TB). Some of the words may be theirs. This chapter contains • Section 4.1 is an abreviated version of section I of [5] to give a context to the work originally performed and extended upon in [2] by the same authors. • Section 4.2 is identical to section III of [5]. The analysis of the experimental data has been done by me, in collaboration with TMM. The model was designed by me and MK, all mathematical calculations were performed by me. • Section 4.3 is identical to section IV of [5]. The analysis of the experimental data has been performed by TMM. The mathematical derivations have been performed by me. • Section 4.4 is identical to sections V.C and V.D of [5]. The analysis of the experimental data has been performed by me. Mathematical derivations and simulations have been performed by me. •Section 4.5 is identical to section VI of [5]. In formulas, "MBR" was replaced by for overall consistency in the thesis. 4.1. Introduction Over the past decades, studies have shown that the mechanical properties of cells play a key role in many essential cellular functions like migration, proliferation, and development, and often these properties are dysregulated in severe diseases like cancer or Covid-19 [94–101]. A plethora of studies has demonstrated this by probing the viscoelastic material properties of cells from the outside using techniques like atomic force microscopy [102], flow cytometry [103], or micropipette aspiration [104]. However, even though recent studies emphasize that also intracellular mechanics are relevant for many cellular processes [100, 105, 106], they have been less extensively studied. 54 4. MBR,VBR and effective Energy in cells and model Figure 4.4.: Regimes of qualitatively different shapes of MBR (𝑡,𝜏) of the RHC model of Eq. (4.3) . Green regime: MBR grows monotonically as a function of time 𝑡 , see inset (a). In the red area, MBR shrinks monotonically as a function of time 𝑡 fir 𝜏→0 , see inset (c). In the blue area, MBR is non-monotonic as a function of time, for 𝜏→0 , see inset (b). The blue area with light red stripes corresponds to the regime, where, as a function of 𝜏 , MBR turns from non-monotonic to monotonic in 𝑡 , i.e., the red solid line moves to the left with increasing 𝜏 . A red dashed dotted line indicates the boundary between non-monotonic and monotonic for a specific 𝜏 as labeled. Long time value of MBR changes sign at 𝐷𝑞/𝐷1=1 shown as a gray dashed line. Reprinted from Ref. [5]. thus shown as a dark green line. The upper bound of Eq. (4.11) is shown as a light red curve in Fig. 4.4. On the right hand side of this curve, i.e., for −𝜆21𝑓(𝜆1,𝜆2)+𝜆22𝑓(𝜆2,𝜆1) 𝜆21𝑔(𝜆1,𝜆2)+𝜆22𝑔(𝜆2,𝜆1)<𝐷𝑞 𝐷1(4.12) fulfilled, MBR is negative and decreases monotonically, i.e., Eq. (4.7) is a sufficient but not necessary condition for such behavior. Notably, as is the case for the other boundaries, Eqs. (4.11) and (4.12) only depend on the ratios 𝛾1 𝛾2and 𝑘1 𝑘2. Also, the long time value of MBR takes a simple form which is not influenced by the presence or absence of the particle 𝑥2 in Eq. (4.3) , i.e., the long time value is identical to the one found for the case of 𝑘1=0studied in Ref. [2], (𝜏→0,𝑡→∞)=12(1−𝐷𝑞 𝐷1).(4.13) The order of limits is not important in Eq. (4.13) , i.e., Eq. (4.13) is found by either order. 4.2. Mean Back Relaxation: Model and experiment 55 Figure 4.5.: Illustration of MBR as resulting from MSD of RHC, at 𝑡=1𝛾1 𝑘2 , for different 𝜏 . Arrows indicate the MSD at different 𝜏 values, and insets at the beginning of the arrows show construction of the ratio 𝛽 (with (𝜏,𝑡)=12(1−𝛽) , see main text) at the corresponding value of 𝜏 . In the insets, the lengths of the red lines equal Δ𝑥2(𝜏) and the length of the green lines equal Δ𝑥2(𝑡+ 𝜏)−Δ𝑥2(𝜏) . 𝛽=Δ𝑥2(𝑡+𝜏)−Δ𝑥2(𝑡) Δ𝑥2(𝜏) is thus the ratio between the length of green and red lines. For small 𝜏 the red line is longer then the green one, MBR is positive. For increasing 𝜏 , the green line grows relatively larger compared to the red one, eventually leading to a negative MBR at intermediate 𝜏 . For large 𝜏 , the lengths of the two lines approach each other, and 𝛽→ 1. MBR vanishes. Reprinted from Ref. [5]. The long time value of MBR thus solely depends on the ratio 𝐷𝑞/𝐷1 , changing sign at 𝐷𝑞/𝐷1=1, which is shown as a dashed line in Fig. 4.4. In the following, we focus on parameters for which MBR shows a maximum as a function of 𝑡 , as this is the behavior observed in the cell data in Fig. 4.1(a). See the top curve in Fig. 4.1(b), which corresponds to 𝜏→0 . Remarkably, for the parameters chosen, it looks very similar to the top curve in Fig. 4.1(a). As the simpler model of Ref. [2] shows no maximum as a function of time, we see that the particle 𝑥2 , i.e., memory, is crucial in obtaining qualitative agreements with cell data. Dependence on 𝜏 We continue with analyzing the dependence of MBR on time 𝜏 . Eq. (4.12) above, i.e., the boundary between non-monotonic and monotonic dependence on time 𝑡 can be 56 4. MBR,VBR and effective Energy in cells and model extended for finite 𝜏. For finite 𝜏, curves are negative and monotonic in 𝑡, if −𝜆1𝑓(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+𝜆2𝑓(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏) 𝜆1𝑔(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+𝜆2𝑔(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏) <𝐷𝑞 𝐷1.(4.14) Notably, 𝜏 enters this expression exponentially in the combinations 𝜏𝜆𝑖 . With increasing 𝜏 , the left hand side of Eq. (4.14) decreases, i.e., the curve separating blue and red areas in Fig. 4.4 moves to the left. For 𝜏≫1𝜆𝑖, this line approaches 𝐷𝑞 𝐷1=1. The green line in Fig. 4.4, given by Eq. (4.6) , i.e., the "left" border of non-monotonic behavior, is independent of 𝜏 . The area of nonmonotonic MBR behavior thus stays of finite size for any 𝜏. In Fig. 4.4, we marked the area 1<𝐷𝑞 𝐷1<−𝜆21𝑓(𝜆1,𝜆2)+𝜆22𝑓(𝜆2,𝜆1) 𝜆21𝑔(𝜆1,𝜆2)+𝜆22𝑔(𝜆2,𝜆1)(4.15) by red stripes. In this area, MBR changes from non-monotonic for small 𝜏 , to monotonic negative behavior for larger 𝜏 . On the other hand, parameters in the blue area left of 𝐷𝑞 𝐷1=1are non-monotonic for all values of 𝜏. Figure 4.1(b) shows curves for parameters in the latter case, i.e., in the blue area, as a function of 𝑡 for various values of 𝜏 . For small 𝜏 , 𝜏≪𝜆−1 𝑖 , the curve does not depend on 𝜏 , and it is non-monotonic by construction, and it ends with a positive value for 𝑡→∞, as stated by Eq. (4.13). For 𝜏 comparable to 𝜆−1 1 and 𝜆−1 2 , MBR decreases and the long time value eventually turns negative (Eq. (4.13) holds only for 𝜏→0 ). For large 𝜏 , 𝜏≫𝜆−1 𝑖 , it asymptotically vanishes with a power law in 𝜏 , which will be discussed in more detail below. The astonishing similarity with curves obtained from cells, shown in panel Fig. 4.1(a) suggests that the experimental system is located in the blue regime of Fig. 4.4. The mentioned behavior is further visualized via Eq. (2.38) above, using the MSD of this model, in Fig. 4.5. Eq. (2.38) can be written (𝜏,𝑡)=12(1−𝛽(𝜏,𝑡)) , with the ratio 𝛽(𝜏,𝑡)=Δ𝑥2(𝑡+𝜏)−Δ𝑥2(𝑡) Δ𝑥2(𝜏) . The numerator of 𝛽 is shown as a green line in the insets of Fig. 4.5, while the denominator is shown as a red line. For small 𝜏 , MSD grows faster between 0 and 𝜏 compared to the interval 𝑡 to 𝑡+𝜏 , hence the ratio 𝛽 is smaller then one: MBR is positive. With 𝜏 increasing 𝛽 increases as well, hence MBR gets smaller. When 𝛽 exceeds unity, MBR is negative. For large 𝜏 , long time diffusion dominates, the green and red lines approach each other in length, i.e., 𝛽→1 and MBR →0 . MBR thus vanishes for large 𝜏. Limit of 𝑡→∞as a function of 𝜏 The dependence on 𝜏 can be investigated easier in the limit of 𝑡→∞ , in which case we find, 4.2. Mean Back Relaxation: Model and experiment 57 (𝜏,𝑡→∞)= 12(1+ 2𝐷𝑞 𝐷1𝑘2 𝛾1𝜏 Ψ(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+Ψ(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏))−1 (4.16) For 𝜏→0 , this expression approaches Eq. (4.13) . Notably, in this expression, 𝜏 appears in the product 𝜏𝜆𝑖 , but also in a power law form. The latter dominates in the limit of 𝜏≫𝜆−1 𝑖, (𝜏→∞,𝑡→∞)=14Ψ(𝜆1,𝜆2)+Ψ(𝜆2,𝜆1) 𝐷𝑞 𝐷𝑞𝑘2 𝛾1𝜏.(4.17) For large 𝜏 , the MBR vanishes as a power law, and not, as might be naively expected, exponentially. Figure 4.2(b) shows the limit of 𝑡→∞ as a function of 𝜏 for various values of 𝐷𝑞/𝐷1 . The curves start at 𝜏=0 , with the value given by Eq. (4.13) . For large 𝜏 , MBR approaches zero with the power law of Eq. (4.17) . The power law changes sign when Ψ(𝜆1,𝜆2)+ Ψ(𝜆2,𝜆1)=0, which corresponds to 𝐷𝑞 𝐷1=−𝑓(𝜆1,𝜆2)+𝑓(𝜆2,𝜆1) 𝑔(𝜆1,𝜆2)+𝑔(𝜆2,𝜆1)(4.18) Notably, at this value of 𝐷𝑞/𝐷1 , the power law vanishes, and MBR approaches zero exponentially with 𝜏 . This critical activity as a function of 𝛾1 𝛾2 is also shown as the blue dotted line in Fig. 4.4. Eq. (4.18) and Eq. (4.13) thus give rise to a range of activity values for which the large 𝑡 limit, as a function of time 𝜏, changes sign form positive to negative. This occurs for −𝑓(𝜆1,𝜆2)+𝑓(𝜆2,𝜆1) 𝑔(𝜆1,𝜆2)+𝑔(𝜆2,𝜆1)<𝐷𝑞 𝐷1<1(4.19) For very large activity, 𝐷𝑞/𝐷1→∞, MBR(𝜏,𝑡→∞)approaches a limiting line, given by min(𝜏,𝑡→∞) =12𝑔(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+𝑔(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏) 2𝜏+𝑔(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+𝑔(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏).(4.20) In the RHC model of Eq. (4.3) , MBR cannot fall below this line. This line is shown in Fig. 4.2 as a thick black line. For 𝜏→∞ , it approaches zero as 𝑔(𝜆1,𝜆2)+𝑔(𝜆2,𝜆1) 2𝜏 . For 𝜏→0 , 58 4. MBR,VBR and effective Energy in cells and model it diverges as 𝑔(𝜆1,𝜆2)𝜆21+𝑔(𝜆2,𝜆1)𝜆22 2𝜏 , in agreement with the finding of Eq. (4.13) that, in the limit 𝜏→0, MBR can reach arbitrary negative values. The surprising agreement of the curves shown in Fig. 4.2(a) and (b) is a further support that the simple model captures the system in a cell in many aspects qualitatively. Notably, the experimental curve in Fig. 4.2(a) crosses from positive to negative, i.e., indicating to be in the range of Eq. (4.19). 4.3. MBR and effective energy As discussed in the introduction, we observed in Ref. [2] a surprising phenomenological relation between MBR and effective energy (see Eq. (4.21) below for its definition). While in Ref. [2], we restricted to small values of 𝜏 , we will here consider a range of 𝜏 values. 4.3.1. Effective energy and FDT The effective energy 𝐸eff(𝜔) is defined as the ratio of correlation and response function at frequency 𝜔,𝐸eff(𝜔)=𝜔𝐶(𝜔) 2𝜒′′(𝜔).(4.21) For systems obeying detailed balance, FDT is recovered with 𝐸eff =𝑘𝐵𝑇 . Studying 𝐸eff is thus one way of quantifying the violation of FDT and of detailed balance. 4.3.2. Effective energy and MBR in RHC model system For the RHC model of Eq. (4.3) , response function and correlation in Eq. (4.2) can be found analytically, see appendix C for the detailed calculation [11, 14, 123]. Notably, the linear response function 𝜒 is independent of the activity parameter 𝐷𝑞 , so that the dependence of the ratio of 𝜒 and 𝐶 on 𝐷𝑞 results purely from the correlation 𝐶 . 𝐸Eff can thus be found as the ratio of correlation taken at 𝐷𝑞and at 𝐷𝑞=0, 𝐸Eff 𝑘𝐵𝑇=⟨𝑥(𝜔)𝑥(−𝜔)⟩ ⟨𝑥(𝜔)𝑥(−𝜔)⟩𝐷𝑞=0 (4.22) =1+𝐷𝑞 𝐷1𝑘22 𝛾211𝜔2𝑘21 𝛾22+𝜔2 𝑘21 𝛾22+𝜔2+𝑘21 𝛾1𝛾2(4.23) 𝜔→0 =1+𝐷𝑞 𝐷1𝑘22 𝛾211𝜔21 1+𝛾2 𝛾1(4.24) =1+𝐸0 𝑘𝐵𝑇𝜔2𝑐 𝜔2.(4.25) 4.3. MBR and effective energy 59 In the third line, we took the limit 𝜔→0to identify the asymptotic behavior, and we introduced the "corner" frequency 𝜔2𝑐=𝑘22 𝛾21. We extract the amplitude 𝐸0=𝑘𝐵𝑇𝐷𝑞 𝐷11 1+𝛾2 𝛾1.(4.26) 𝐸0 depends on activity 𝐷𝑞 𝐷1 and on the ratio 𝛾2 𝛾1 of the friction of bath and tracer particles. For 𝛾2≪𝛾1 , we recover the case studied in Ref. [2], where particle 𝑥2 is absent, and 𝐸0=𝑘𝐵𝑇𝐷𝑞/𝐷1. Eq. (4.16) can be solved for 𝐷𝑞/𝐷1 in terms of MBR (we use for brevity, in the following equations, notation 𝑀≡(𝜏,𝑡→∞)), 𝐷𝑞 𝐷=(1−2𝑀)(𝐹12+𝐹21) 4𝑀𝑘2 𝛾1𝜏−(1−2𝑀)(𝐺12+𝐺21)(4.27) with abbreviations 𝐹𝑖𝑗=𝑓(𝜆𝑖,𝜆𝑗)(1−𝑒−𝜆𝑖𝜏),(4.28) 𝐺𝑖𝑗=𝑔(𝜆𝑖,𝜆𝑗)(1−𝑒−𝜆𝑖𝜏).(4.29) With Eq. (4.26), this yields a relation between 𝐸0and 𝑀, 𝐸0 𝑘𝐵𝑇=1 1+𝛾2 𝛾1(1−2𝑀)(𝐹12+𝐹21) 4𝑀𝑘2 𝛾1𝜏−(1−2𝑀)(𝐺12+𝐺21).(4.30) Figure 4.6(b) shows 𝐸0 as a function of 𝑀 from Eq. (4.30) for various values of 𝜏 . In the limit 𝜏→0 , Eq. (4.30) simplifies to the linear relation 𝐸0 𝑘𝐵𝑇=1 1+𝛾2 𝛾1(1−2𝑀) , which was displayed in Ref. [2] (for 𝛾2≪𝛾1 ). Such relation is also observed for cell data. For finite 𝜏 , the relation is curved, as seen in Fig. 4.6(b). Even in these cases, for moderate deviations of MBR from 12, a linear relation is found 𝐸0 𝑘𝐵𝑇=1 1+𝛾2 𝛾1× ×𝑒−𝜆1𝜏(𝜆2−𝑘2 𝛾1)−𝜆2−(𝑒−𝜆2𝜏(𝜆1−𝑘2 𝛾1)−𝜆1) 𝑘2 𝛾1(𝜆1−𝜆2)𝜏× ×(1−2𝑀)+((1−2𝑀)2) (4.31) =𝜆(𝜏) 𝑘𝐵𝑇(𝑀−12)+((𝑀−12)2).(4.32) The slope of this linear relation, 𝜆(𝜏), decreases with 𝜏, as is visible in Fig. 4.6(b). Figure 4.7(b) shows the slope 𝜆(𝜏) as a function of 𝜏 , for various ratios of 𝛾1/𝛾2 , showing the mentioned decrease. For large 𝜏 , 𝜏≫{𝜆−1 1,𝜆−1 2} , the slope decreases with a power 60 4. MBR,VBR and effective Energy in cells and model Figure 4.6.: (a) Amplitude 𝐸0 of effective energy against the long time limit of MBR for multiple 𝜏 between 5×10−5s and 0.01s , for different cell types as labeled. For each 𝜏 value the values from different cells fall on a straight line, which is fitted and also shown. For increasing 𝜏 the slope of that line decreases. (b) Effective energy amplitude 𝐸0 against the long time limit of MBR for 𝛾1 𝛾2=0.2 and 𝑘1 𝑘2=1.0 in the RHC model of Eq. (4.3) . Color indicates the respective 𝜏 . For small 𝜏 the relation between 𝐸0 and MBR is linear, and it is curved for larger 𝜏 . Close to MBR =1/2 , it is linear for any 𝜏 , see main text. Reprinted from Ref. [5]. law 𝜏−1, i.e., in this limit, 𝐸0 𝑘𝐵𝑇=− 2 1+𝛾2 𝛾1𝛾1 𝑘2𝜏(𝑀−12)+((𝑀−12)2).(4.33) Interestingly, the prefactor of the power law in Eq. (4.33) is very similar to the result of 𝜆 for 𝜏→0 found above. What is the range of validity of the linear relation in Eq. (4.32) ? It must break if  approaches its minimum as a function of activity min given in Eq. (4.20). In this case, increasing activity 𝐷𝑞 𝐷 does no longer decrease 𝑀 , as seen in Fig. 4.2(b). As 𝐸0 in Eq. (4.26) increases with 𝐷𝑞 , independent of 𝜏 , the linear relation can than no longer hold. This lower bound for MBR depends on 𝜏 and diverges in the limit 𝜏→0 . A smaller 𝜏 , therefore, increases the range where the long time value of MBR is sensitive to an increase of the driving 𝐷𝑞 𝐷. For larger 𝜏 , 𝑀 comes closer to its minimum, and the relation between 𝑀 and 𝐸0 must become less precise. We will thus, in our experiments, restrict to the range where 𝑀 decreases as a function of 𝜏(see Fig. 4.2(a)). 4.3.3. Experiment: Effective energy and MBR relation for various 𝜏 In a previous study [2], we determined the effective energy and the viscoelastic properties of 8 different cell types and cell conditions using optical tweezers based active and 4.3. MBR and effective energy 61 Figure 4.7.: (a) Slope 𝜆(𝑡) of the linear relation between 𝐸0 and long time limit of MBR, extracted from Fig. 4.6(a), against 𝜏 . The magnitude of the slope decreases with increasing 𝜏 . The grey dotted line shows MBR as a function of time 𝑡 of Fig. 4.1a) for 𝜏=1.82ms , shifted and rescaled along the 𝑦 -axis, i.e., MBR in the blue area in the inset, which shows original MBR of Fig. 4.1a). (b) Slope 𝜆(𝜏) of the linear expansion in Eq. (4.32) for 𝑘1 𝑘2=1 and various 𝛾1 𝛾2 as labeled. For large 𝜏 , 𝜆 decays as a power law, 𝜆≃−2𝜏−1/(1+𝛾2/𝛾1) and for 𝜏→0it approaches the constant −2/(1+𝛾2/𝛾1). Reprinted from Ref. [5]. passive microrheology (Material and Methods in [5]). In particular, we investigated HeLa cells, passivated HeLa cells, A549 cells, C2C12 cells, CT26 cells, HoxB8 cells, and MDCK cells (Material and Methods in [5]). This study yielded, for all cells investigated, good agreement with the phenomenological relation, 𝐸Eff =𝐸0(𝜔0 𝜔)𝜈+𝑘𝐵𝑇, (4.34) with 𝜈≈1 for all cell types. We chose 𝜔0=1Hz , which defines and yields, for each cell type, a value for amplitude 𝐸0 . In Ref. [2], we chose a small 𝜏 -value of 0.6ms for evaluation of MBR, and observed a striking linear dependence between effective energy amplitude 𝐸0 and MBR long-time value. Here we investigate this relation for different 𝜏 values, limiting ourselves to 𝜏≤0.01s ; In Fig. 4.2(a), we see that, in this regime, MBR decreases with 𝜏 , see also the discussion at the end of Sec. 4.3.2. Strikingly, when we compare the effective energy amplitude 𝐸0 to MBR longtime value for different values of 𝜏, we observe linear dependencies for all values of 𝜏, as can be seen in Fig. 4.6(a). For each 𝜏 -value, the relation between 𝐸0 and MBR longtime value can be accurately fit by a linear function (see Supplementary table in [5] for fitting uncertainties) 𝐸0=𝜆(𝜏)(MBR(𝑡→∞,𝜏)−12).(4.35) This result suggests that knowledge of the 𝜏 -specific slope 𝜆(𝜏) , as shown in Fig. 4.7(a) is sufficient to determine the effective energy amplitude 𝐸0 from an MBR measurement in these cases. As Fig. 4.7(a) shows, the magnitude of the slope decreases with increasing 𝜏 , similarly as in the model system discussed in Sec. 4.3.2 and shown in Fig. 4.7(b). 62 4. MBR,VBR and effective Energy in cells and model Figure 4.7(a) also shows, as a comparison, the time 𝑡 dependent (𝑡) of Fig. 4.1 a) (grey dotted line) shifted and rescaled along the 𝑦 -axis. This shows that the time dependence of 𝜆(𝜏) for small 𝜏 is comparable to the time dependence of (𝑡) for small 𝑡 . This allows to estimate, from MBR, on what time scales 𝜆(𝜏) is expected to vary. Generally MBR can therefore also be used to scan relevant time scales in the system. Returning to the initial questions, we thus extend the relation between MBR and effective energy to finite values of 𝜏, compared to Ref. [2]. 4.4. VBR in model and cell data 4.4.1. Variance: RHC model The RHC model of Eq. (4.3) is Gaussian, so that VBR is given in Eq. (3.2) . Evaluating VBR for RHC thus allows testing of Eq. (3.2) in numerical simulations [124] as well as estimating VBR for our experiments. Figure 4.8(b) shows VBR as a function of the length parameter 𝑙 , for fixed 𝑡 and 𝜏 , confirming Eq. (3.2) using numerical simulations of RHC. We observe the mentioned divergence for small and large 𝑙 . The amplitude of VBR is to a good estimate given by the ratio of MSD evaluated at the times 𝑡 and 𝜏 . Thus, increasing the time 𝜏 with keeping 𝑡 fixed leads to a reduction of VBR (This overdamped process has a monotonic MSD). Because the MSD increases linearly for large 𝑡 , VBR, evaluated at 𝜂min has no upper bound in this model. Therefore, the statistical error of MBR increases with 𝑡 . Increasing 𝜏 on the other hand reduces VBR. Notably the curves are rather flat around the minimum, which softens the requirement of a precise choice of the length parameter 𝑙. 4.4.2. Variance: Experiment For the data obtained in the mentioned cell, we find that VBR has a similar functional behavior as the Gaussian reference model. The dots show the variance directly obtained from the cell data, while the line shows the prediction from MSD of this data using Eqs. (2.38) and (3.2) . There is a surprising agreement between the two for small 𝜏 . However, one should keep in mind that the experimental resolution is about 2nm, so that what is seen at smaller 𝑙 may be measurement noise. For larger values, the cell VBR grows and lies above the Gaussian prediction. From this, we conclude that the data obtained from cells are notably non-Gaussian. However, since the curve is rather flat in the region of the minimum, the exact choice of the length parameter is not that important and the Gaussian estimate is still reasonable. Therefore, using the Gaussian estimate for the length parameter is also a good starting point for cell data. 4.5. Summary 63 Figure 4.8.: (a) VBR from A549 cells (red dots) compared to the Gaussian prediction using the particle’s MSD via Eq. (3.2) (grey line) for 𝜏=1ms and 𝑡=1s . Both curves show a similar qualitative behavior with a divergence for 𝑙→0 and 𝑙→∞ . The two disagree, especially for larger 𝜏 , demonstrating a non-Gaussianity of the process. (b) VBR of the RHC model of Eq. (4.3) against the length parameter 𝑙 for 𝑘1 𝑘2=1.0,𝛾1 𝛾2=0.2,𝐷𝑞 𝐷1=0.5 and 𝑡=1.0𝛾1 𝑘2 for two values of 𝜏 as labeled. The lines show the analytical prediction of Eq. (3.2) and crosses give numerical results from simulation. One can see the growth for low and high 𝑙 . We note that the curve is rather flat around the minimum, which means that, especially for larger 𝜏 , VBR is rather insensitive to the precise choice of 𝑙. Reprinted from Ref. [5]. 4.5. Summary Mean back relaxation depends on two time parameters 𝜏 and 𝑡 , and one length parameter 𝑙 . Here we investigated in detail the dependence of MBR on 𝜏 and 𝑡 , both in a linear driven RHC model system as well as for data obtained from cells. The RHC model is found to qualitatively reproduce the characteristic shapes of MBR as a function of 𝑡 found in cells, namely a non-monotonic behavior with MBR first rising and then decaying with 𝑡 . This shape as a function of 𝑡 is found in the investigated cells for any 𝜏 , with however the height of the maximum as well as the final value strongly dependent on 𝜏 . This behavior is also found in the RHC system for a certain regime of parameters. We find a strong dependence of MBR on 𝜏 , which emphasizes the importance of this parameter. It is thus not only the future motion that is of interest (i.e., the dependence on 𝑡 ), but also the motion in the past, i.e., dependence on 𝜏 . There are other parameter regimes where MBR, as a function of 𝑡 is monotonically decreasing or monotonicaly increasing. Notably, non-monotonic shapes cannot be found in the simpler model of Ref. [2]. The RHC model used here introduces one more (bath) particle, which yields memory, thus indicating that non-monotonic MBR behavior is a sign of memory. For large 𝜏 , MBR decays to zero, found both in cell data, as well in the RHC model system. In the latter, it does so with a power law. Depending on the parameters, the power law can approach zero from above or below. The large 𝑡 limit of MBR shows, in cells, a minimum as a function of 𝜏 . Such behavior is 70 5. Entropy bound for time reversal markers Notably, when adding Eq. (5.10) for ⟨𝑂𝑠⟩ and ⟨𝑂∗𝑠⟩ , the linear term in Eq. (5.9) drops out, so that Eq. (5.11) does in general not saturate for small 𝑠 . As will be discussed below, it saturates for a binary process for short times, for any value of 𝑠. A straight forward way to extract a bound for ⟨𝑠⟩ from Eq. (5.11) is by considering 𝑂+=1 , i.e., path independent. In order for 𝑂 to be positive, the antisymmetric part, 2𝑂−[𝜔]=𝑂[𝜔]−𝑂[𝜃𝜔], must be normalized to |𝑂−[𝜔]|≤1. This yields ⟨𝑠⟩≥⟨𝑂−⟩log(1+⟨𝑂−⟩ 1−⟨𝑂−⟩)≥0.(5.12) Eq. (5.12) is a main result of this section, a bound for entropy production ⟨𝑠⟩ in terms of the average of antisymmetric observable 𝑂− . This relation is thus fundamentally different from uncertainty relations, which bound entropy production in terms of mean and variance [69]. The condition of |𝑂−[𝜔]|≤1 may seem to be a strong restriction of validity of Eq. (5.12) . However, a bound between ⟨𝑠⟩ and ⟨𝑂−⟩ can only be useful if 𝑂− is normalizable, i.e., if a maximum value of max 𝜔|𝑂−|<∞ exists. Whenever this maximum exists, 𝑂− can be normalized to fulfill |𝑂−[𝜔]|≤1 . Eq. (5.12) is thus applicable for any normalizable antisymmetric observable. We also note that the right hand side of Eq. (5.12) is nonnegative, so that any nonzero ⟨𝑂−⟩yields a positive bound for ⟨𝑠⟩. Eq. (5.12) can be read in two ways: (i) A given ⟨𝑠⟩ yields a bound for how far the mean of (any) 𝑂− can deviate from zero. Using, e.g., a time interval from −𝑡0 to 𝑡0 , 𝑂− can be the time-moment at which a certain event occurs, which is then bound by ⟨𝑠⟩ via Eq. (5.12) . This will be investigated in future work. (ii) A given non-vanishing value of ⟨𝑂−⟩yields a lower bound for entropy production. We will analyze this below. The form of Eq. (5.12) is illustrated in Fig. 5.1. For small ⟨𝑂−⟩ , the bound grows quadratically in ⟨𝑂−⟩while it diverges logarithmically for ⟨𝑂−⟩→1. 5.4. Optimal observable: Signum of entropy Eq. (5.12) , as mentioned, is valid for any normalizable antisymmetric observable, and, naturally, the observable that maximizes the right hand side of it, yields the best estimate for ⟨𝑠⟩ . Which observable is it? Answering this important question has been found non-trivial for entropy bounds [65, 158], while it has a clear answer for Eq. (5.12) . To see this, rewrite 3 ⟨𝑂−⟩=∑ 𝜔𝑂−[𝜔]𝑝[𝜔]=12∑ 𝜔𝑂−[𝜔](𝑝[𝜔]−𝑝[𝜃𝜔]) ≤⟨sign(𝑠)⟩.(5.13) 3Eq. (5.13) holds also for −𝑂−and thus for |⟨𝑂−⟩|. 5.4. Optimal observable: Signum of entropy 71 Figure 5.1.: Illustration of Eq. (5.12) in terms of anti-symmetric observable ⟨𝑂−⟩ , with accessible area marked in green. For ⟨𝑂−⟩→±1 , the bound diverges logarithmically. Reprinted from Ref. [3]. In the second step, we used the anti-symmetry of 𝑂− . The inequality in the last step of Eq. (5.13) follows by noting that the sum is maximized if 𝑂−[𝜔]=1 for 𝑝[𝜔]>𝑝[𝜃𝜔] and 𝑂−[𝜔]=−1for 𝑝[𝜔]<𝑝[𝜃𝜔]. This is the definition of 𝑂−=sign(𝑠)4. As the right hand side of Eq. (5.12) is a monotonically growing function of |⟨𝑂−⟩| (compare Fig. 5.1), 𝑂−=sign(𝑠) yields the optimal bound for ⟨𝑠⟩ from Eq. (5.12) . To emphasize this, we write explicitly ⟨𝑠⟩≥⟨sign(𝑠)⟩log(1+⟨sign(𝑠)⟩ 1−⟨sign(𝑠)⟩) ≥⟨𝑂−⟩log(1+⟨𝑂−⟩ 1−⟨𝑂−⟩).(5.14) The first inequality of Eq. (5.14) bounds ⟨𝑠⟩ by ⟨sign(𝑠)⟩ . Writing ⟨sign(𝑠)⟩=12∑𝜔sign(𝑝[𝜔]− 𝑝[𝜃𝜔])(𝑝[𝜔]−𝑝[𝜃𝜔]) shows that ⟨sign(𝑠)⟩≥0 , and that ⟨sign(𝑠)⟩=0 only if ⟨𝑠⟩=0 , i.e., Eq. (5.14) yields a finite bound for any finite ⟨𝑠⟩ . The second inequality of Eq. (5.14) restates that 𝑂−=sign(𝑠) yields the optimal bound, so that any other 𝑂− lies below it. 4 The terms with 𝑝[𝜔]=𝑝[𝜃𝜔] cancel in the sum in Eq. (5.13) due to 𝑂−[𝜔]=−𝑂−[𝜃𝜔] , and 𝑂−[𝜔] can be chosen arbitrarily in these cases. 72 5. Entropy bound for time reversal markers 5.5. Coarse graining A bound of ⟨𝑠⟩ in terms of ⟨sign(𝑠)⟩ is fundamentally interesting, and it is also useful, as, e.g., ⟨sign(𝑠)⟩ has beneficial properties under coarse graining. Therefore consider coarse grained paths Ω with probabilities 𝑃(Ω)=∑𝜔∈Ω𝑝(𝜔) , and coarse grained entropy production 𝑆=log𝑃[Ω] 𝑃[𝜃Ω] . Naturally, 𝑂−=sign(𝑆) fulfills Eq. (5.13) , so that, for any grouping of paths 0≤⟨sign(𝑆)⟩≤⟨sign(𝑠)⟩.(5.15) Coarse graining thus leads, in general, to a decrease of ⟨sign(𝑠)⟩ , reminiscent of the finding that ⟨𝑠⟩ also decreases under coarse graining [66]. Notably, grouping paths according to the sign of 𝑠, i.e., with sign(𝑠[𝜔])=sign(𝑆[Ω𝑜])conserves ⟨sign(𝑠)⟩, ⟨sign(𝑠)⟩=12∑ Ω𝑜sign(𝑃[Ω𝑜]−𝑃[𝜃Ω𝑜])× ×∑ 𝜔∈Ω𝑜(𝑝[𝜔]−𝑝[𝜃𝜔]) =⟨sign(𝑆𝑜)⟩(5.16) Under this "optimal" (index o) coarse graining, the bound provided by sign(𝑠) is invariant, so that the macroscopic ⟨sign(𝑆𝑜)⟩ yields the same bound as the microscopic ⟨sign(𝑠)⟩ . Furthermore, as the bound must hold for 𝑠 and 𝑆𝑜 alike, coarse grained entropy production 𝑆𝑜 never falls below the original, microscopic bound. This algorithm thus provides a controlled coarse graining of entropy production, which is especially useful if the bound from sign(𝑠)is close to 𝑠. 5.6. Example: Network on a ring To display this in an example, consider a network on a ring, where every state is connected to two neighbors (see inset sketch of Fig. 5.2). In every discrete time step, a particle jumps to the left (right) with probability 𝑝 ( 𝑞 ). For 𝑞≠𝑝 , the system violates detailed balance and shows a directed flow. After 𝑁 steps, the probability of finding a specific path with 𝑛𝐿 steps to the left, is given by the binomial distribution 𝑝[𝜔]=1𝐿𝑝𝑛𝐿𝑞𝑁−𝑛𝐿 , with 𝐿 the number of states in the network. With it, entropy production after 𝑁steps is given by, ⟨𝑠⟩=𝑁(𝑝−𝑞)log(𝑝𝑞).(5.17) Optimal coarse graining can be performed here in straight forward manner: Because 𝑠=𝑑log(𝑝/𝑞) , the sign of entropy production equals the sign of 𝑑=𝑛𝐿−𝑛𝑅 (for 𝑝>𝑞 ), with 𝑛𝑅 the number of steps to the right, i.e., sign(𝑑[𝜔])=sign(𝑠[𝜔]) . This 5.6. Example: Network on a ring 73 Figure 5.2.: Network on a ring model: Entropy production ⟨𝑠⟩ as a function of 𝑁 , Eq. (5.17) , for 𝑝=0.57 , versus the bound obtained from Eq. (5.12) for various observables as sign(𝑠),sign(𝑑) and tanh(𝑑2) using numerical simulations. Gray dotted line is the asymtotic limit of large 𝑁 . Graph also shows optimally coarse grained entropy ⟨𝑆𝑜⟩. Reprinted from Ref. [3]. system thus allows coarse graining towards measurement of the net displacement 𝑑 , under preservation of the bound. We may expect that 𝑑is easier to measure than 𝑠. Having established that Eq. (5.12) is maximal for 𝑂−=sign(𝑑[𝜔]) , we can test the quality of the estimate for ⟨𝑠⟩provided by it. For 𝑁=1,⟨sign(𝑑)⟩=𝑝−𝑞and ⟨𝑠⟩𝑁=1 =⟨sign(𝑑)⟩log(1+⟨sign(𝑑)⟩ 1−⟨sign(𝑑)⟩).(5.18) For 𝑁=1 , the bound of Eq. (5.12) thus meets entropy production exactly, for any 𝑝 and 𝑞 , i.e., arbitrarily far from equilibrium. This is the above mentioned case of binary process, where a particle either jumps right or left. Figure 5.2 shows ⟨𝑠⟩ and the bound of Eq. (5.12) as a function of 𝑁 . For 𝑁>1 , the bound grows sublinear in 𝑁 for an intermediate range, and thus falls below the value of 𝑠 . For 𝑁≫1 , it approaches a linear asymptote, which can be found via a large deviation principle. We find for 𝑝>12and 𝑁→∞, [159] ⟨sign(𝑠)⟩=⟨sign(𝑑)⟩∼1− 2 1−𝑞𝑝(1 4𝑝𝑞)−𝑁2 √𝜋2𝑁,(5.19) i.e., ⟨sign(𝑑)⟩ approaches unity exponentially fast with 𝑁 . Because of this, the bound for 74 5. Entropy bound for time reversal markers Figure 5.3.: Network on a ring model, with identical parameters as in Fig. 5.2, here focusing on the comparison to FTUR. FTUR is used with two different observables as indicated, sign(𝑠) or 𝑠 . For large 𝑁 , Eq (5.12) and FTUR for sign(𝑠) scale linearly in 𝑁 , while FTUR for 𝑠 scales with log(𝑁) . Reprinted from Ref. [3]. 𝑠 of Eq. (5.12) grows linear in 𝑁 , and plugging (5.19) into Eq. (5.12) yields 12log(1 4𝑝𝑞)𝑁 , shown as a gray line in the graph. The ratio between this large 𝑁 asymptote and ⟨𝑠⟩ of Eq. (5.17) varies between 12for 𝑝→1and 14for 𝑝→12. The coarse graining groups paths according to their displacement, i.e., Ω for 𝑑>0 and 𝜃Ω for 𝑑<0 . This way, the coarse grained entropy ⟨𝑆𝑜⟩ can be determined, which is also shown in Fig. 5.2. The curve demonstrates that it, as expected, stays above the bound. As only two coarse grained paths with finite 𝑆𝑜exist, it is here found from ⟨𝑆𝑜⟩=log(𝑃[Ω𝑜] 𝑃[𝜃Ω𝑜])𝑃[Ω𝑜] +log(𝑃[𝜃Ω𝑜] 𝑃[Ω𝑜])𝑃[𝜃Ω𝑜].(5.20) 𝑁odd =⟨sign(𝑑)⟩log(1+⟨sign(𝑑)⟩ 1−⟨sign(𝑑)⟩).(5.21) Notably, the bound Eq. (12) is saturated with respect to 𝑆𝑜 for odd 𝑁 as indicated. For 𝑁 even, paths with zero entropy production exist, and the second equality Eq. (5.21) is not valid, and Eq. (12) lies below ⟨𝑆𝑜⟩ . For large 𝑁 , these differences vanish, so that the bound of Eq. (5.12) and ⟨𝑆𝑜⟩ share the same asymptote. In this system, entropy production may thus be coarse grained by a maximal loss of a factor between 2 and 4 , depending on 𝑝, using the optimal algorithm. 5.7. Summary 75 According to Eq. (5.13) , any other normalizable antisymmetric observable should yield a lower bound, which we examplify by using 𝑂−=tanh(𝑑/2) . Indeed, it lies lower, but approaches the optimal bound for large 𝑁 , because then, a typical trajectory shows 𝑑≫1so that tanh(𝑑/2)becomes equivalent to sign(𝑑). While the optimal coarse graining is possible in an exact manner in this model, we expect approximate preservation of sign(𝑠) to be possible in more complicated systems, which will be investigated in future work. Can we compare to other relations such as TUR? The original TUR is not applicable to time discrete dynamics as used in this example. We thus compare to the so called FTUR [69] as shown in Fig. 5.3. Not knowing the optimal observable for FTUR, we try sign(𝑠) and 𝑠 , analytically computing the required variances for these two. Interestingly, for each of the three curves shown in the figure, there exists a regime of 𝑁 where it provides the highest estimate. It is also remarkable that FTUR used with sign(𝑠) can provide a better estimate compared to using 𝑠 . This shows the advantage of Eq. (5.17) , for which the optimal observable is known, leading to the coarse graining scheme. It also shows that the comparison between these relations is rich and nontrivial and needs to be studied in future work. 5.7. Summary Entropy production is bound by the mean of normalizable antisymmetric observables. The optimal observable is identified to be the signum of entropy production, so that we determine a bound between entropy production and its sign, sign(𝑠) . The latter may often be estimated from simple observables, like here, the displacement on a ring. The network example shows that, measuring sign(𝑠) (did the particle move left or right?) is expected to require a lower experimental resolution compared to measuring 𝑠 (where did the particle move when?). One can also estimate ⟨sign(𝑠)⟩ by use of Eq, (5.13) , i.e., by testing various observables 𝑂− and finding the maximum deviation from zero. For the investigated network, ⟨sign(𝑠)⟩ approaches unity exponentially fast with number of steps, so that the bound grows with the expected linear dependence. Grouping paths according to sign(𝑠) yields a coarse graining algorithm that preserves sign(𝑠) and the bound. The presented analysis is not restricted to specific dynamics. Due to this, it is, additionally to the here discussed discrete system, also valid for fluids and biological systems [2]. Future work may as well investigate applications to Langevin systems, like active Brownian particles, or quantum mechanics [16, 160–162]. 6. Time reversal breaking and entropy production bound in cells This chapter is entirely written by me and is not based on a publication. The experimental data was obtained by TMM and originally presented in [2], here with additional data for drugged HeLa cells. The data analysis was performed by me. 6.1. Time reversal symmetry breaking in cells In chapter 4, we analyzed MBR for cell data and saw that we can mimic its characteristic qualitatively with an unconfined Gaussian model. However, we have not spoken about the time reversal breaking. A key condition to prove for MBR to approach 12 in the long limit was lim 𝑡→∞𝑊3(𝑥𝑡,𝑡;𝑥0,0;𝑥−𝜏,−𝜏)=𝑊1(𝑥𝑡)𝑊2(𝑥0,0;𝑥−𝜏,−𝜏).(6.1) However, in cellular systems, at least on the observable time frame, Eq. (6.1) is typically not valid. The plateau at around 1s in Fig. (4.1) is presumably comparable to the plateau one gets from a diffusive regime on intermediate times, as discussed in section 2.6.2. However, at the same time, we have already seen from VBR that the cell process is not Gaussian. So strictly speaking, the MBR-MSD relation, which predicts a plateau for a diffusive regime, is also not applicable. However, to determine if the cell process breaks time reversal symmetry, this is not a major drawback. Suppose MBR evaluated from the forward trajectories compared to MBR evaluated from the backward differ for any combination of 𝜏 and 𝑙 . Then, time reversal symmetry is broken [118]. Figure 6.1 shows the MBR for multiple different cell types, similar to Fig. 4.1 in chapter 4. However, this time we show the MBR, once evaluated from forward trajectories (solid line), and evaluated from the backward trajectories (dashed line). We find that for multiple cell types, forward and backward MBR clearly differ, indicating the breakage of time reversal symmetry. Notably, the drugged cells, which are a modified version of the HeLa wild type show a smaller signature than the wild type cells. For those, the actin network and microtubules have been destroyed and, hence, the activity of motor proteins is reduced. This is a strong hint that the observed breaking of time reversal symmetry is actually part of the cellular process and not an experimental error. It is also 78 6. Time reversal breaking and entropy production bound in cells Figure 6.1.: (a) MBR for multiple different cell types for 𝜏=0.047s and 𝑙=0.002µm . Solid line shows MBR evaluated for time forward trajectories and dotted line for time backward trajectories. For the wild type cells the solid and dotted line disagree, indicating breakage of time reversal symmetry. For the drugged HeLa cells, both lines are nearly identical. (b) Shows the antisymmetric part of MBR as the difference between the lines shown in (a). The faces show the standard deviation of a bootstrap (4000 samples). Given the bootstrap distribution, MBR breaks time reversal symmetry for all cells except the drugged ones with over 2𝜎certainty. obvious that solid and dashed lines do not add up to one within 𝑡 to 1s , which means the true MBR long time value has not been reached yet. That means Eq. (6.1) is not fulfilled in the observable time frame as we discussed earlier, and thus the MBR does not need to approach 12 in equilibrium. It, however, is remarkable that the drugged cells actually get very close to 12 . In the spirit of the Gaussian model, this would mean that diffusivity on intermediate timescales has to be significantly smaller for drugged cells compared to the wild type case. This could hint that maybe an increased diffusivity in the wild type cell is the result of active cellular processes. It has also been reported by other experiments that the destruction of microtubules, a part of the cytoskeleton, reduces the movement of the colloidal particles inside the cytoplasm [163]. Figure 6.1(b) shows the mean of the anti-symmetric part of MBR, which is given by half of the difference between the time forward and reversed curves in (a). We can see that it increases with 𝑡. Generally, one has to ask with which certainty MBR detects the time reversal breaking in the data. The two primary contributions here are the amount of statistics and the heterogeneity of the individual cells. To give a measure, we performed a bootstrap on the original data and computed the standard deviation of the bootstraped distribution [164]. The result is shown as the colored area in Fig. 6.1(b). For all cells, apart from the drugged ones, we have over 2𝜎 certainty from the boostrapped distribution that MBR detects broken time reversal symmetry. With that, we can say with relatively high certainty that time reversal symmetry is actually broken for micrometer sized colloidal particles in cells. 6.2. Time reversal breaking of MBR and dependence on length and timescales 79 6.2. Time reversal breaking of MBR and dependence on length and timescales Before we continue with the analysis of the cell data, we want to spend a few lines on which properties of 𝑊3 are ideal for MBR to detect broken detailed balance. We will also analyze how MBR depends on the length and timescales. 6.2.1. When does MBR detect broken time reversal breaking: Insights from the probability density Assume that we have a translational invariant system i.e. all probabilities are invariant under shifting 𝑥→𝑥+𝑎 , and that the system is inversion symmetric i.e. all probabilities are invariant under 𝑥→−𝑥 . Under these assumptions, we can make the first observation that any two point observable can not detect the breakage of time reversal symmetry, because 𝑊2(𝑥𝑡,𝑡,𝑥𝜏,𝜏)=𝑊2(𝑥𝑡−𝑥𝜏,𝑡;0,𝜏)(6.2) =𝑊2(𝑥𝜏−𝑥𝑡,𝑡;0,𝜏)(6.3) =𝑊2(𝑥𝜏,𝑡;𝑥𝑡,𝜏)(6.4) is always symmetric. MBR, as a three point observable, therefore, has an advantage over simpler two point observables under these conditions. At next, we look at the joint distribution 𝑝(𝑏,𝑡;𝑑,𝜏) of a displacement 𝑑 in time 𝜏 and a following displacement of 𝑏 in time 𝑡 afterwards. We can express this distribution in terms of the three point probability 𝑝(𝑏,𝑡;𝑑,𝜏)=∫d𝑥𝑊3(𝑥+𝑑+𝑏,𝑡+𝜏;𝑥+𝑑,𝜏;𝑥,0) (6.5) Under time reversal we get 𝜃𝑝(𝑏,𝑡;𝑑,𝜏)=∫d𝑥𝑊3(𝑥,𝑡+𝜏;𝑥+𝑑,𝑡;𝑥+𝑑+𝑏,0) (6.6) =∫d𝑥𝑊3(𝑥−𝑏−𝑑,𝑡+𝜏;𝑥−𝑏,𝑡;𝑥,0) (6.7) =𝑝(−𝑑,𝜏;−𝑏,𝑡).(6.8) where we shifted the integral via 𝑥=𝑥+𝑏+𝑑 . This means the time reserved distribution 𝜃𝑝(𝑏,𝑡;𝑑,𝜏) is the same as the original one, however, we execute the steps 𝑏,𝑑 in opposite order and opposite direction. We have not assumed inversion symmetry. Time reversal breakage can therefore be expressed as the following quantity Δ𝑝(𝑏,𝑡;𝑑,𝜏)=𝑝(𝑏,𝑡;𝑑,𝜏)−𝑝(−𝑑,𝜏;−𝑏,𝑡)(6.9) 86 6. Time reversal breaking and entropy production bound in cells Figure 6.7.: (a) The probability density 𝑝(𝑑,𝜏,𝑏,𝜏) sampled numerically for the RHCDW of Eq. (6.15) for 𝜏=𝑡=2.19𝐿2 𝐷,Δ𝑉 𝑘𝐵𝑇=2.0 and 𝐷𝑞 𝐷=0.1 . The probability density is peaked at 𝑏=𝑑=0 . (b) The anti-symmetric probability density Δ𝑝(𝑏,𝜏;𝑑,𝜏) from (a). The probability density shows an octapole structure, where the peaks have alternating signs. For very small and large 𝑏,𝑑 there is no time reversal breakage in the probability distribution. The grey lines show were Δ𝑝(𝑏,𝜏;𝑑,𝜏) has to vanish due to inversion symmetry. Figure 6.8.: Illustration of how MBR and observable Eq. (6.17) interact with Δ𝑝(𝑏,𝜏;,𝑑,𝜏) in Fig. 6.7(b). (a) Shows the back relaxation multiplied with Δ𝑝(𝑏,𝜏;𝑑,𝜏) with normalization computed from the density shown in Fig. 6.7(a). MBR ( 𝑙=0.1𝐿) projects out the poles at 𝑑=0.5 and suppresses the peaks for higher 𝑑 , leading to an overall non-vanishing mean antisymmetric part of MBR. (b) The observable Eq. (6.17) for 𝑛=4 , 𝑅=2𝐿 and 𝜖=0.1𝐿 flips all poles such that they all contribute positively to the mean of the anti-symmetric part of the observable. 6.2. Time reversal breaking of MBR and dependence on length and timescales 87 Alternative observable The structure of Δ𝑝(𝑏,𝑡;𝑑,𝜏)suggests to define a different type of observable, which reflects the symmetries of the probability distribution. Because of the observed pole like shape, one might construct observables of the following form 𝑂(𝑏,𝑑)=Ψ(𝜑)𝜌(𝑟)(6.17) with the polar coordinates 𝜑=arctan(𝑏/𝑑) and 𝑟=√𝑑2+𝑏2 . The major hypothesis of Eq. (6.17) is that the contributions of 𝜑 and 𝑟 decouple, which seems to be a good approximation in the case of RHCDW. As basis functions for the angular dependency, we choose the circular harmonics, which are essential combinations of sine and cosine functions Ψ𝑛,𝛿(𝜑)=−sin(𝑛𝜑+𝛿)/√𝜋(6.18) with a frequency parameter 𝑛=0,1,2,… and a phase 𝛿∈[0,2𝜋] . We normalize such that 1𝜋∫2𝜋 0sin(𝑛𝜑+𝛿)2=1.(6.19) For the radial component, we have no a priori guess for a good candidate. Therefore, we simply choose 𝜌𝑅,𝜖(𝑟)=𝜒[𝑅−𝜖,𝑅+𝜖](𝑟)/√𝑅𝜖 (6.20) with the step function 𝜒[𝑎,𝑏](𝑥)={1,𝑥∈[𝑎,𝑏] 0,otherwise (6.21) which means we filter radii between 𝑅−𝜖 and 𝑅+𝜖 . In this setup 𝑅 functions as a length scale parameter, whereas 𝜖 is a tool to control the statistics. The function is again normalized such that ∫∞ 0d𝑟𝜌(𝑟)2𝑟=1.(6.22) In total, the observable is given by 𝑂(𝑏,𝑑)=− 1 √𝜋𝑅𝜖sin(𝑛𝜙+𝛿)𝜒[𝑅−𝜖,𝑅+𝜖](𝑟)(6.23) and normalized such that ∫∫d𝑏d𝑑𝑂𝑛,𝛿,𝐿,𝜖(𝑏,𝑑)2=1 (6.24) and the different observables are orthonormal in 𝑛. 88 6. Time reversal breaking and entropy production bound in cells Figure 6.9.: The anti-symmetric part of observable Eq. (6.17) for 𝑛=4 and 𝜖=0.05𝐿 for different 𝑅 and 𝜏 for 𝐷𝑞 𝐷1=0.1 and 𝛿=0 . (a) Δ𝑉 𝑘𝐵𝑇=2.0 for small and very large 𝜏 the anti-symmetric part vanishes, for intermediate 𝜏 it is finite. The same is true for the radius 𝑅 . With increasing 𝜏 , the R values for which the anti-symmetric part is finite also increase. The observable peaks at 𝑅≈2𝐿 , the distance between the wells, and 𝜏≈2𝐿2 𝐷 .(b) Δ𝑉 𝑘𝐵𝑇=4.0 has a higher barrier. The overall shape seams to be similar to (a), however, we can now clearly see two peaks. The second one is similar to the peak in (a), the first one is for smaller 𝐿≈0.5𝐿and 𝜏≈0.8𝐿2 𝐷. We now look at the new observables for the RHCDW. Figure 6.8(b) shows for 𝑛=4 and 𝑅=2 how the observable flips the signs of Δ𝑝 in such a way that (mostly) positive contributions remain. For this plot and in the following, we set 𝛿=0 . Crucially, we can now use 𝑅 very precisely to scan for different length scales. Figure 6.9 shows how the observable mean changes with 𝑅 and 𝜏 . (a) is for a moderate barrier height Δ𝑉 𝑘𝐵𝑇=2.0 and low activity 𝐷𝑞 𝐷=0.1 . One can see that for small 𝜏 the anti-symmetric part vanishes, as for very large 𝜏 . There is an intermediate range where the breakage of time reversal symmetry is visible. With increasing 𝜏 , the required 𝑅 also increases. In (a), one can only see a maximum at 𝑅≈2.0𝐿 and 𝜏≈1.5𝐿2 𝐷 . This length scale would correspond to the distance between the two minima. In (b), we see a high barrier height of Δ𝑉 𝑘𝐵𝑇=4.0 , where we can now clearly see two peaks. The smaller one is similar to the one in (a), the higher one is at rather small values of 𝑅≈0.5𝐿 and 𝜏≈0.06𝐿2 𝐷 . It is possible that this peak also exists in (a), but is rather blurred. The length scale here is not so obvious. It could correspond to the width of the potential. All these considerations were for 𝑛=4 . We can also test for different 𝑛 and see if the octopole structure is universal. Figure 6.10 shows different 𝑛 values, in (a) for different 𝑅 and (b) for different 𝜏 . The most prominent contribution is from 𝑛=4 , as expected, however, 𝑛=8 also has a non-vanishing mean. This 𝑛=8 parameter would correspond to 16 poles, which is mainly relevant for large 𝑅 compared to 𝜏 . This shows that the structure of Δ𝑝(𝑏,𝑑) is a bit more complex than the simple octapole structure that is obvious in Fig. 6.7(b). To summarize the study of the probability density and RHCDW, we have seen that MBR prefers a quadrupole structure in Δ𝑝(𝑏,𝜏,;𝑑,𝜏) close to the value of the length parameter |𝑑|≈𝑙 to show time reversal breaking. We could demonstrate that these 6.2. Time reversal breaking of MBR and dependence on length and timescales 89 Figure 6.10.: The anti-symmetric part of observable Eq. (6.17) for 𝜖=0.05𝐿,Δ𝑉 𝑘𝐵𝑇=2.0 and 𝐷𝑞 𝐷=0.1 . (a) The parameter 𝑛 versus the radius 𝑅 for 𝜏=1.38𝐿2 𝐷 . One can see that the strongest signal can be seen for 𝑛=4 , however, there are also contributions from 𝑛=8 , which corresponds to 16 poles in Δ𝑝(𝑏,𝑑) . (b) shows the same for 𝑅=2.03𝐿 and different 𝜏 . Again, there are primarily contributions from 𝑛=4 and 𝑛=8 . In (a) 𝑛=8 peaks at higher 𝑅 and in (b) for smaller 𝜏 , which indicates that the 𝑛=8 contributions arise from more extreme displacements 𝑏,𝑑than 𝑛=4. poles are present in the RHCDW model. The change in sign of the anti-symmetric part of MBR can be explained by first measuring the inner peaks for small 𝑙 , and later for the outer peaks for large 𝑙 . For this pole like structure, we have also looked at observables that incorporate the symmetries of Δ𝑝(𝑏,𝜏;𝑑𝜏) . We found that the RHCDW TRBP is dominated by an octapole, however, higher moment poles also exist. For this observables, we could also see that there is a clear relation between length and timescales for which time reversal symmetry is broken. We also noted with MBR, that asymmetric 𝑡≠𝜏seems to give a stronger signal to detect time reversal breaking. 6.2.3. Cells: Length and timescales of long time MBR time reversal breakage We now have a more detailed look at the time reversal breakage of MBR in cells. While Fig. 6.1 already shows that time reversal symmetry of colloidal particles in the cytoplasm is broken, we look at the 𝜏 and 𝑙 dependence of the long time value in Fig. 6.11. (a) shows the long time value of MBR color coded for HeLa cells for different 𝑙 and 𝜏 . We obseve that for very small and large 𝜏 MBR detects no breakage in time reversal symmetry for the long time value at 𝑡=1s . For an intermediate regime of 2400ms on the other hand, we can see a non-vanishing mean of the anti-symmetric part of MBR. For small 𝑙 the anti-symmetric part is positive, while for larger ones it gets negative. The length parameter in between, for which the anti-symmetric part is zero, seems to increase with 𝜏 . A typical approach in biology and biophysics is to knock out some cellular components to investigate how they influence the overall cellular properties. In (b), we show the results of cells that are treated with Cytochalasin-B (CytoB), which 90 6. Time reversal breaking and entropy production bound in cells Figure 6.11.: Shows the anti-symmetric part of long time MBR 𝑡=1s for various 𝜏 and 𝑙 for different types of HeLa cells. (a) The wild type HeLa cell. For small and large 𝜏 there is no time reversal breaking. For an intermediate range from 2400ms we see a non-vanishing anti-symmetric part. For small 𝑙 the anti-symmetric part is negative and gets positive for large 𝑙 . The white area shows the region for which no event in 5% of trajectories was recorded. (b) HeLa cell drugged with CytoB. Similar to (a), with a smaller amplitude and time reversal breaking is less pronounced for larger 𝜏. (c) HeLa cell drugged with Nocoda. The anti-symmetric part vanishes, so MBR shows no time reversal breaking. (d) HeLa with CytoB and Nocoda (drugged) also shows no time reversal breaking. inhibits the formation of actin. This stops certain motors that sit in the actin network, like myosin, and so suppresses active fluctuations transmitted by the actin network. In this case, we observe a similar overall structure of the anti-symmetric part of long time MBR as in the wild type case (a). However, here the time reversal breaking is only visible between 2200ms . The deviation between forward and backward MBR seems to be a bit smaller than in the wild type case, but the length scales remain similar. In (c) the cells are instead drugged with Nocodazole (Nocoda), which destroys the microtubules inside the cells, so all effects transmitted by the microtubules are prohibited. In this case, MBR does not show obvious time reversal breakage for any parameter combinations of 𝜏 and 𝑙 . (d) shows both drugs combined, which previously were always referred to as drugged (HeLa) cells. In this case, we also see no time reversal breakage of MBR. Overall, 6.2. Time reversal breaking of MBR and dependence on length and timescales 91 Figure 6.12.: Derivative of the anti-symmetric long time MBR for 𝑡=1s as in Fig. 6.11 in 𝑙 direction. (a) for HeLa wild type and (b) Hela drugged with CytoB. One can see that the highest change of MBR is in the region of ca. 220nm and 2-100ms. this investigation suggests that the time reversal breaking observed in the trajectory data is somehow related to processes on microtubules and less so in the actin network. However, a more in depth analysis would be needed to say which processes exactly, because there are multiple candidates, as molecular motors attached to microtubuli or the (de-)polyramization process of the microtubuli themselves. It is also possible, since destroying the microtubules is a rather invasive process, that there is some secondary effect which inhibits relevant time reversal breaking fluctuations. However, it would be very interesting to, for example, directly target motor proteins that operate on the microtubules to see if they are the main driving force of the time reversal breaking. To get an idea of the relevant length scales, we can again look at the derivative of MBR in the 𝑙 direction, which is shown in Fig. 6.12, in (a) for wild type HeLa cells and in (b) for the CytoB drugged ones. The length scale for which MBR changes the most seems to lie in the region of roughly 220nm with 𝜏 between 2100ms . The upper bound in 𝑙 may be higher because we lack statistics for larger length 𝑙 . These numbers fit roughly with the typical length and timescales of molecular motors, which have a typical center of mass step motion of 8nm with a few 10ms cycle time, depending on load and ATP concentration [167]. However, this is only a very crude estimate and only serves as one possible candidate. By no means is this a clear indication that molecular motors are responsible for the observed effects. Up to now, we have focused on the long time MBR. What is with the equal 𝑡=𝜏 MBR that we have analyzed for the RHCDW? Figure 6.13 shows the anti-symmetric MBR in (a) for HeLa cells and (b) for A549 cells. Surprisingly, there is no huge time reversal breakage detected by MBR. This suggests that the asymmetry between 𝑡 and 𝜏 is an important feature for MBR in cells. We have also seen this for RHCDW, but for cells, the effect seems to be even more pronounced. To sum up, we have seen that MBR shows time reversal breakage of trajectories of colloidal particles in living cells. In HeLa cells this time reversal breakage vanishes if the microtubules are inhibited, which suggests that there is some underlying connection 92 6. Time reversal breaking and entropy production bound in cells Figure 6.13.: Anti-symmetric of MBR for 𝜏=𝑡 for different 𝜏 and length parameters 𝑙 . (a) HeLa cells and (b) A549 cells. Both show no time reversal breaking, in contrast to long time MBR in Fig. 6.11. This means the asymmetry between 𝑡and 𝜏is important for MBR to detect time reversal breaking. to the active processes. Also, the asymmetry between 𝑡 and 𝜏 seems to be important to detect the time reversal brekage with MBR in cells. 6.3. Entropy production bound in cells Now, as we have found observables that detect the break time reversal symmetry in the cell data, we can apply our results from chapter 5 to get a lower bound on entropy production. To use Eq. (5.12) , we need an anti-symmetric observable whose absolute value is bounded by 1 . Because we know that the optimal observable is given by the sign(𝑠), we also apply a sign function to the anti-symmetric part of MBR 𝑂=sign (−12𝑥(𝑡+𝜏)−𝑥(𝜏) 𝑥(𝜏)−𝑥(0) 𝜗𝑙(𝑥(𝜏)−𝑥(0)) +12𝑥(0)−𝑥(𝑡) 𝑥(𝑡)−𝑥(𝑡+𝜏)𝜗𝑙(𝑥(𝑡)−𝑥(𝑡+𝜏))).(6.25) Remember the normalization factor in 𝜗𝑙(𝑥)=𝜃(|𝑥|−𝑙) ⟨𝜃(|𝑥|−𝑙)⟩ . However, we can expect that in a stationary system ⟨𝜃(|𝑥(𝜏)−𝑥(0)|−𝑙)⟩=⟨𝜃(|𝑥(𝑡+𝜏)−𝑥(𝑡)|−𝑙)⟩.(6.26) So we do not have to explicitly calculate this factor. Therefore, the observable simplifies to 𝑂=sign (−(𝑥(𝑡+𝜏)−𝑥(𝑡))𝜃(|𝑥(𝜏)−𝑥(0)|−𝑙) 𝑥(𝜏)−𝑥(0) +(𝑥(0)−𝑥(𝑡))𝜃(|𝑥(𝑡+𝜏)−𝑥(𝑡)|−𝑙) 𝑥(𝑡)−𝑥(𝑡+𝜏))(6.27) 6.3. Entropy production bound in cells 93 Figure 6.14.: The mean of the sign of the anti-symmetric part of MBR in Eq. (6.27) for different cell types for 𝜏=0.17s and 𝑙=0.002µm . Standard error shown by the areas which was obtained by the bootstrap method (2000 samples). For this we calculate the ⟨𝑂⟩𝑛 for every individual trajectory and use as final mean ⟨𝑂⟩=1𝑁∑⟨𝑂𝑛⟩ . (a) For A549 cells and a reference measurement in Agarose, the latter is a purely passive system. One can see that the measurement for A549 deviates from 0 above the standard error, indicating the breaking of time reversal symmetry. The Agarose measurement stays close to zero. (b) The same curves for HeLa cells and drugged HeLa cells. The wild type clearly deviates from zero with high certainty, while the drugged cells fluctuate around it. for the sign case. Figure 6.14 shows the mean of the observable Eq. (6.27) . In (a) we see the result for A549 cells and for a reference measurement in Agarose, with 𝜏=0.17ms and 𝑙=0.002µm . The measurement in Agarose should not show time reversal breakage, since it is a passive system. The areas show the standard error obtained from a bootstrap of the recorded data. For Agarose, ⟨𝑂⟩ is close to zero, as we expect. For A549 the mean is positive, indicating that time reversal is broken. In (b) we have the same observable, however, for HeLa cells and drugged HeLa cells. For the normal HeLa cells, time reversal symmetry is broken with at least 2 𝜎 certainty, while for the drugged cells, this is at least not clearly the case, compared to the standard error. Because Eq. (6.27) is a bounded anti-symmetric observable with |𝑂|=1 , we can apply Eq. (5.12) to get a lower bound on entropy production ⟨𝑠⟩ . The last aspect we need to clarify is, how the entropy production depends on the times 𝑡 and 𝜏 . To evaluate the observable 𝑂 in Eq. (6.27) , one needs at least a trajectory of length 𝑡+𝜏 . Therefore, the entropy production is a lower bound on the entropy production ⟨𝑠(𝑡+𝜏)⟩ . However, many possible combinations of 𝑡 and 𝜏 lead to one specific 𝑡+𝜏 , which means we get several lower bounds from observables 𝑂(𝜏,𝑡) for ⟨𝑠(𝑡+𝜏)⟩ . Figure 6.15 shows different bounds for the entropy production based on ⟨𝑂(𝜏,𝑡)⟩ for HeLa cells. The 𝑥 -axis shows the time 𝑡+𝜏 and 𝜏 is color coded. All combinations of 𝑡,𝜏 that result in the same 𝑡+𝜏 are on a vertical line. The highest lower bound for ⟨𝑠⟩ is given by the 94 6. Time reversal breaking and entropy production bound in cells 0123 t+τ[s] 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 hsibound σ·(t+τ) 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] Figure 6.15.: The entropy production bound Eq. (5.12) applied to the observable means of Fig. 6.14 for HeLa cells with 𝑙=0.002µm . The dots show different combinations of 𝑡 and 𝜏 that add up to 𝑡+𝜏 on the 𝑥 -axis, with 𝜏 color coded. The gray dotted line shows the straight line, starting from zero, with the highest slope crossing the entropy bound. The slope then serves as a lower bound for the entropy production rate 𝜎 . In this graph 𝜎≈0.00141s . For the other cell types, this plot can be found in appendix D. largest deviation from zero. From this, we can get an optimal bound by ⟨𝑠(𝑡)⟩≥max 𝑡,𝜏with 𝑡+𝜏=𝑡⟨𝑂(𝜏,𝑡)⟩log(1+⟨𝑂(𝜏,𝑡)⟩ 1−⟨𝑂(𝜏,𝑡)⟩).(6.28) Generally, one expects in a steady state that the entropy production scales linearly in time ⟨𝑠(𝑡)⟩=𝜎𝑡 , with an entropy production rate 𝜎 . We can get a lower bound for 𝜎 from Eq. (6.28) by maximizing 𝜎≥max 𝑡,𝜏 1 𝑡+𝜏⟨𝑂(𝜏,𝑡)⟩log(1+⟨𝑂(𝜏,𝑡)⟩ 1−⟨𝑂(𝜏,𝑡)⟩).(6.29) In Fig. 6.14, this is illustrated by the gray dotted line. Getting the lower bound for the rate is equivalent to finding the straight line with the highest slope that connects the origin with the entropy bound dots. With the entropy production rate, we can reduce the analysis down to one number. Figure 6.16 shows on the 𝑦− axis the lower bound for entropy production rate 𝜎 for different cells. The error bars represent the 95% confidence interval of a bias corrected and accelerated bootstrap (BCa) [164]. For the bootstrap we calculate 𝜎 for a set of trajectories and then repeat for multiple randomly sampled sets. We can see that the entropy production rate is roughly of the order of 𝜎≈0.00151s . Generally, we would expect the overall entropy production inside 6.3. Entropy production bound in cells 95 0 5 10 15 20 25 30 E0[kBT] 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 σBound [1/s] A549 C2C12 CT26 Drugged HeLa HoxB8 MDCK Agarose Figure 6.16.: The lower bound for entropy production rate 𝜎 for different cell types and Agarose. Error bars obtained via bias corrected and accelerated bootstrap (4000 samples) with confidence interval of 95% and 𝑙=0.002µm . The 𝑥 -axis shows the effective energy amplitude, which was discussed in Section 4.3, and quantifies the FDT violation. One can see that a higher FDT violation correlates with a higher amount of time reversal breakage i.e. a higher bound on the entropy production rate. the cell to be much higher. ATP consumption is in the order of 109 per second with approximately 12𝑘𝐵𝑇 per hydrolysis [168, 169]. However, we only have access to the entropy production that is responsible for the dynamics of the one colloidal particle that we observe. This highly projected dynamics will have a much smaller entropy production by default. Secondly, we do not know how tight the bound is. Generally, we can expect to be off by multiple orders of magnitude. All in all, the absolute value of the entropy production rate does not tell us too much on its own. However, it is interesting to compare it between different cells. The 𝜎 -bound is a way to quantify the time reversal breaking in the stochastic trajectories. We can, for example, see that the wild type HeLa cells have a much higher 𝜎 -bound than the drugged HeLa cells. At the same time, the drugged cells seem to have a higher 𝜎− bound than the Agarose reference measurement. This could be a hint that even in the drugged cells, still active processes are present. It could also be related to a decaying process due to the damage caused by the drugs. The 𝑥 -axis of Fig. 6.16 shows the effective energy amplitude, which we encountered in chapter 4. We can see here that a larger 𝜎 -bound correlates with a larger effective energy amplitude. This means that a more severe time reversal breaking could also be related to a more severe FDT violation. However, the error bars are indeed very large, which is probably also related to the large heterogeneity of the cells. So it is hard to say how strong these correlations are, and if one can make any quantitative 102 7. Summary, Discussion and Outlook In this work, we have primarily focused on MBR of the position of one colloidal particle in one dimension. While it is surprising that MBR for positions shows some quite remarkable results, there are different directions to explore further. For scalar observables, we already introduced the density MBR. This quantity has the neat feature that we combine multiple particles and dimensions into one stochastic quantity. Also for bulk systems, we recover the MBR 12 theorem. In our example, we have looked primarily at the Fourier transform of the density. The reason is that it is easier to determine in Fourier space for individual particle trajectories. In real space, one has to deal with delta functions or bins, which is not preferable if the system is not dense. However, it might be interesting to look at the density of actual camera data. The pixels of the camera or detector would already serve as bins. MBR could therefore also be applied to the plain camera data, or more abstractly to a real space field. The absolute square of the Fourier transformed density is the structure factor [85]. This is one of the primary quantities obtained in scattering experiments. It would be very interesting if one could use the dynamic structure factor of an active system in MBR. Because the density MBR already shows time reversal breaking, one could hope for this to be the case. The only downside of the structure factor is that it loses the phase information of the Fourier transformed density, so it contains less information. It is therefore possible that relevant information to detect time reversal breaking with MBR can be lost. Initial studies by Laila Henkes in her Master Thesis, however, suggest that it is possible to use the structure factor, at least in some systems [171]. Technically, the structure factor does not necessarily have to be determined from scattering experiments. One could also use camera recordings and Fourier transform the resulting image, which is a technique already used by experimenters [172]. Considering the desired resolution and equipment, the latter approach might be beneficial in certain situations. This approach would also give rise to the missing phase of the density. Another direction one can explore is to generalize MBR for vector valued quantities. We have taken some trials in this direction, however, we have not come to a final result. The easiest way, which keeps the MBR a scalar without introducing new parameters, is to rewrite MBR as (𝜏,𝑡,𝑙)=⟨−𝐛𝐝 𝐝2𝜗𝑙(|𝐝|)⟩(7.1) where 𝐛 is the vector displacement from time 𝑡 to 0, and 𝐝 the displacement from −𝜏 to 0 . In this definition, we compare the future displacement from 0 to 𝑡 projected on the direction of the displacement from −𝜏 to 0. This definition has the nice feature that it collapses in the one-dimensional case to the original definition Eq. (2.5) . Another possibility is to only look at the projection of the trajectory in one direction 𝑒 . MBR in this case can be direction dependent. In two dimensions, this means that (𝜏,𝑡,𝑙,𝜙) generally depends on an angle 𝜙 . This might be a useful approach to detect anisotropies in the system. Referring back to the usage in cells, it’s not a priori clear that the cytoplasm is a homogeneous material. Generally, one would expect quite the opposite. If cells have a principal access, this might be visible in MBR. There have been initial studies by Sarah Lädke that show exactly this angular dependence of MBR in cells 7.2. Discussion and outlook 103 using darkfield microscopy [173]. The last possibility we briefly discuss is to define a MBR matrix. In this case, we compare the 𝑏 displacements, for example, in 𝑥 -direction to the 𝑑displacements in 𝑦-direction. For general components 𝑥𝑖,𝑥𝑗 𝑖𝑗(𝜏,𝑡,𝑙)=⟨−𝑥𝑖(𝑡)−𝑥𝑖(0) 𝑥𝑗(0)−𝑥𝑗(−𝜏)𝜗𝑙(||𝑥𝑗(0)−𝑥𝑗(−𝜏)||⟩.(7.2) With 𝑖=𝑗 we get the ordinary MBR, with the MBR 12 theorem. For 𝑖≠𝑗 the properties of this quantity are not well understood. If the 𝑥𝑖 and 𝑥𝑗 directions are uncorrelated, then the MBR matrix element vanishes. If they are correlated, they can have a non-vanishing value and detect time reversal breakage. Systems for which the off-diagonal terms are interesting are rotating or spinning particles, where a coupling between multiple spatial directions occurs. Further, it will depend on the chosen basis 𝑥𝑖 and 𝑥𝑗 . There have been preliminary studies for this quantity, as for the other ones. However, until now, there have not been any major results, which is more related to a lack of a focused study. Future investigation might find some interesting results of MBR for different spatial directions. Entropy bound Apart from MBR the second major result was concerning the derived entropy production bound. It has a relatively high validity, because it only requires that the entropy production can be written as the log ratio of path probabilities. In that sense, it is superior to the first iterations of TUR, which additionally require an overdamped time continuous dynamics. However, its range of validity is comparable to that of the FTUR relation. The main difference is that the observables it can be used for are a little different. The primary applications are signum observables of antisymmetric quantities, as the anti-symmetric part of MBR or a current. While one could also have continuous observables, limiting oneself to the signum type is definitely reasonable since we know that the signum of entropy production is the most ideal observable. The challenge is to find an antisymmetric observable that always has the same sign as the stochastic entropy production. The advantage of these types of observables is that they are relatively easy to construct, and one only needs to measure their mean. On the other hand, one can not assume that these observables track the full extent of the dynamics. Missing in the earlier discussion is an interpretation of the bound. The TUR is usually related to precision, for example, how precisely a clock runs. The signum observables that can be constructed using the bound derived in this thesis have a yes-no structure. For example, was entropy production positive? Did the clock run forward? The only important part is that if the answer in the forward trajectory is yes, the answer in the backward trajectory is no. The relevant antisymmetric observables are then constructed as returning −1 and 1 (we ignore here that the answer can also be neither of the two given choices, which would correspond to zero). In simple terms, the entropy bound 104 7. Summary, Discussion and Outlook Eq. (5.12) measures how much entropy production is at least required to make yes more likely than no. Or in the case of the clock, how much entropy production does one need at least to make the clock run forward more likely than backward? These types of questions are encoded in the signum observables. This is a little different from asking for the precision. However, the concepts have, of course, similarities. If a clock is significantly more likely to run forward in a time frame, then it’s also reasonable that it is more precise. For real world applications, the question of the tightness of the bound remains an important question if one tries to infer the entropy production. Generally, one has to say, one can not expect Eq. (5.12) to give an accurate estimate of entropy production, and usually will underestimate it by several orders of magnitude. However, the issue can also occur with other bounds, such as TUR. The bound can still be useful by comparing different systems, as we have done for the various cell types. The only important part is that the time reversal breaking, which is essentially quantified by the bound, scales here with the scale of the activity in the system. The comparison with the results from the FDT violations suggests that there might be a connection. Going beyond MBR At last, we may ask what other types of observables are also possible. In chapter 6, we constructed for the RHCDW model a new type of observable that tries to align with the structure of the time reversal breaking part of the displacement probability distribution TRBP. Because these observables are similar to determining coefficients of basis functions i.e. we try to decompose the time reversal breaking part into basis functions, they may be a very interesting way to learn about time reversal breaking length and timescales in the system. The major challenge here is to find a suitable set of basis functions. This will depend on which typical shapes TRBP displays, like the poles we saw for RHCDW. This direction may also be a very interesting pathway to explore in the future. 7.3. Final Remarks The thesis has primarily focused on the development and application of MBR. A main avenue of future theoretical investigations will focus on the density MBR and similar quantities to enable new experimental workflows, like scattering experiments and multi-particle tracking. The investigation of length and time scales of biophysical systems will be a promising application of this work. One primary challenge to solve to date is the many parameters MBR depends on, like 𝑡 , 𝜏 , 𝑙 , 𝑞 , and so on. It is not always clear how these parameters relate to each other. As a general curiosity, it would also be interesting to look at completely different systems, for example, from nonlinear dynamics, quantum mechanics, or financial markets, and how MBR relates to typical quantities used in those respective fields. While we have a well developed understanding of Gaussian systems. What we lack is a deep understanding of how to 7.3. Final Remarks 105 tackle non-Gaussian systems analytically and efficiently compute corrections to MBR from the Gaussian case. Such a derivation could reveal how MBR responds to various interparticle interactions and driving. Independent of which pathways are chosen, there are many interesting roads to explore. As the Klingons say - Dajqu’ tuch. Qapla’! A. Mean Back Relaxation A.1. Eq. (2.13) for general observable We state the assumptions of Eqs. (2.8) and (2.9) for a general phase space observable 𝐴 , ⟨𝐴⟩=∫d𝐴𝐴𝑊1(𝐴)with ⟨𝐴⟩finite (A.1a) lim 𝑡→∞𝑊3(𝐴𝑡,𝑡;𝐴0,0;𝐴−𝜏,−𝜏)(A.1b) =𝑊1(𝐴𝑡)𝑊2(𝐴0,0;𝐴−𝜏,−𝜏) We also repeat the symmetry condition for 𝑊2[40] 𝑊2(𝐴𝑡,𝑡;𝐴0,0)=𝑊2(𝜖𝐴0,𝑡;𝜖𝐴𝑡,0),(A.2) denoted as the usual condition for detailed balance for the joint distribution in Ref. [26]. We assume that 𝐴 is either symmetric ( 𝜖=1 ) or antisymmetric ( 𝜖=−1 ) under path reversal. Assuming Eq. (A.2) and stationarity, one can use that 𝑊1(𝐴)=∫d𝐴𝑡𝑊2(𝐴𝑡,𝑡;𝐴,0)= ∫d𝐴0𝑊2(𝐴,𝑡;𝐴0,0)to find ⟨𝐴⟩=∫d𝐴𝑡d𝐴0𝐴0𝑊2(𝐴𝑡,𝑡;𝐴0,0) =∫d𝐴𝑡d𝐴0𝜖𝐴𝑡𝑊2(𝐴𝑡,𝑡;𝐴0,0) =𝜖⟨𝐴⟩.(A.3) We proceed as in the main text, we start by the definition of the MBR and insert Eq. (A.1) for the long time limit lim 𝑡→∞2𝐴(𝜏,𝑡,𝑙)(A.1) =∫d𝐴0d𝐴−𝜏(−⟨𝐴⟩−𝐴0 𝐴0−𝐴−𝜏) ×𝜗𝑙(|𝐴0−𝐴−𝜏|)𝑊2(𝐴0,0;𝐴−𝜏,−𝜏)(A.4) +∫d𝐴0d𝐴−𝜏(−⟨𝐴⟩−𝜖𝐴−𝜏 𝜖𝐴−𝜏−𝜖𝐴0) ×𝜗𝑙(|𝜖𝐴0−𝜖𝐴−𝜏|)𝑊2(𝜖𝐴−𝜏,0;𝜖𝐴0,−𝜏). 108 A. Mean Back Relaxation We used the coordinate transform 𝐴0→𝜖𝐴−𝜏 and 𝐴−𝜏→𝜖𝐴0 in the second summand. By adding a zero in the counter of the second summand and algebraic transformations one gets, using Eq. (A.3) lim 𝑡→∞𝐴(𝜏,𝑡,𝑙)=12∫d𝐴0d𝐴−𝜏𝐴0−𝐴−𝜏 𝐴0−𝐴−𝜏(A.5) ×𝜗𝑙(|𝐴0−𝐴−𝜏|)𝑊2(𝜖𝐴−𝜏,0;𝜖𝐴0,−𝜏) +12∫d𝐴0𝐴−𝜏(−⟨𝐴⟩−𝐴0 𝐴0−𝐴−𝜏)𝜗𝑙(|𝐴0−𝐴−𝜏|) ×[𝑊2(𝐴0,0;𝐴−𝜏,−𝜏)−𝑊2(𝜖𝐴−𝜏,0;𝜖𝐴0,−𝜏)] The first term is evaluated as 12 because of the normalization ⟨𝜗𝑙(|𝐴0−𝐴−𝜏|)⟩=1 . Using Eq. (A.2) we find that the second term vanishes and obtain lim 𝑡→∞𝐴(𝜏,𝑡,𝑙)(A.1)&(A.2) =12.(A.6) A.2. Simulations Simulations are performed with the Julia "Differential Equations" package with a RungeKutta-Milstein method (RKMil) without adaptive time stepping. Internal time step is 𝑑𝑡=10−3 and the particle position is saved at Δ𝑡=10−2 in units of the simulation time scale. The simulations are performed in dimensionless space and time. For the active Brownian particle length is given in units of √𝛾𝐷 𝑘 and time in units of 𝛾𝑘 . Therefore the following dimensionless SDEs were simulated (compare Eq. (2.20)) 𝐱=−𝐱+𝑣0 √𝑘 𝛾𝐷𝐷𝐞(𝜑)+𝐟(A.7a) 𝜑=𝑓𝜑(A.7b) with ⟨𝑓(𝑖)(𝑡)𝑓(𝑗)(𝑡′)⟩=2𝛿(𝑡−𝑡′)𝛿𝑖𝑗 and ⟨𝑓𝜑(𝑡)𝑓𝜑(𝑡′)⟩=2𝐷𝜑𝛾 𝑘𝛿(𝑡−𝑡′) . These equations thus involve two dimensionless parameters, 𝑣0/(√𝑘 𝛾𝐷𝐷)and 𝐷𝜑𝛾 𝑘. For the coupled particles of Eq. (2.39) , we scale length in terms of √𝑘𝐵𝑇 𝑘 and time in terms of 𝛾1 𝑘. The dimensionless equations are (compare Eq. (2.39)) 𝑥1=−𝑘′ 𝑘𝑥1−(𝑥1−𝑥2)+𝑓1(𝑡)(A.8a) 𝑥2=−𝑘′ 𝑘𝛾1 𝛾2𝑥2−𝛾1 𝛾2(𝑥2−𝑥1)+𝑓2(𝑡)(A.8b) with ⟨𝑓1(𝑡)𝑓1(𝑡′)⟩=2𝛿(𝑡−𝑡′) and ⟨𝑓2(𝑡)𝑓2(𝑡′)⟩=2𝛾1 𝛾2𝑇+Δ𝑇 𝑇𝛿(𝑡−𝑡′) which depend on the dimensionless parameters 𝛾2 𝛾1,𝑇+Δ𝑇 𝑇and 𝑘′ 𝑘. A.3. Two coupled particles in harmonic potential 109 Figures 1-3 have a trajectory length of 𝑡=105 in units of the simulation time scale with 80 trajectories, inset Figure 2 trajectory length of 𝑡=104with 1400 trajectories. Long time MBR in Figure 2 and 3 are evaluated at 𝑡=40 . Figure 5, main, has a trajectory length of 𝑡=104 and 80 trajectories, and inset trajectory length of 𝑡=400 and 800 trajectories. Figure 5 long time MBR evaluated at 𝑡=1000. A.3. Two coupled particles in harmonic potential The Langevin equations of the two linearly coupled particles, discussed in section 2.6.2, are given by 𝑥1=−𝑘′ 𝛾1𝑥1−𝑘𝛾1(𝑥1−𝑥2)+𝑓1(A.9a) 𝑥2=−𝑘′ 𝛾2−𝑘𝛾2(𝑥2−𝑥1)+𝑓2(A.9b) which is rewritten into the matrix equation 𝐱=−𝐌⋅𝐱+𝐟(A.10) with 𝐌=(𝑘′+𝑘 𝛾1−𝑘𝛾1 −𝑘𝛾2𝑘′+𝑘 𝛾2)(A.11) which has the general solution 𝐱(𝑡)=∫𝑡 −∞𝑒−𝐌(𝑡−𝑠)⋅𝐟(𝑠).(A.12) To solve this equation we diagnoalize 𝐌with the eigenvalues 𝜆1/2=(𝑘′+𝑘)(𝛾1+𝛾2) 2𝛾1𝛾2 ±√((𝑘′+𝑘)(𝛾1+𝛾2) 2𝛾1𝛾2)2−𝑘′2+2𝑘𝑘′ 𝛾1𝛾2 (A.13) and corresponding eigenvectors (1,𝑘′ 𝑘+1−𝛾1 𝑘𝜆𝑖)⊺=(1,𝑎(𝜆𝑖))⊺ . After basis transform one gets the following expression 𝑥1(𝑡)= 1 𝑎(𝜆2)−𝑎(𝜆1)(A.14) ∫𝑡 −∞[(𝑎(𝜆2)𝑒−𝜆1(𝑡−𝑠)−𝑎(𝜆1)𝑒−𝜆2(𝑡−𝑠))𝑓1(𝑠) +(−𝑒𝜆1(𝑡−𝑠)+𝑒𝜆2(𝑡−𝑠))𝑓2(𝑠)]d𝑠. 110 A. Mean Back Relaxation From here we can evaluate the correlation function ⟨𝑥1(𝑡)𝑥1(0)⟩= 𝑘𝐵𝑇 𝑘[(𝜙(𝜆1,𝜆2)+(1+Δ𝑇 𝑇)𝜓(𝜆1,𝜆2))𝑒−𝜆1𝑡 +(𝜙(𝜆2,𝜆1)+(1+Δ𝑇 𝑇)𝜓(𝜆2,𝜆1))𝑒−𝜆2𝑡](A.15) with dimensionless 𝜙(𝜆1,𝜆2)=𝑘3 𝛾311 (𝜆2−𝜆1)2(𝑎(𝜆2)2 𝜆1−2𝑎(𝜆1)𝑎(𝜆2) 𝜆1+𝜆2)(A.16) 𝜓(𝜆1,𝜆2)=𝑘3 𝛾31𝛾1 𝛾21 (𝜆2−𝜆1)2(1𝜆1−2 𝜆1+𝜆2)(A.17) The mean squared displacement is obtained by the identity Δ𝑥2(𝑡)=⟨(𝑥(𝑡)−𝑥(0))2⟩ =2⟨𝑥2⟩−2⟨𝑥(𝑡)𝑥(0)⟩(A.18) and we get Δ𝑥21(𝑡)=2𝑘𝐵𝑇 𝑘[Ψ(𝜆1,𝜆2)(1−𝑒−𝜆1𝑡) +Ψ(𝜆2,𝜆1)(1−𝑒−𝜆2𝑡)](A.19) with Ψ(𝜆1,𝜆2)=𝜙(𝜆1,𝜆2)+(1+Δ𝑇 𝑇)𝜓(𝜆1,𝜆2) . The relation is rewritten into the timescales 𝜏𝑖=1𝜆𝑖 . In the main text we investigate the case 𝑘′≪𝑘 . In this case the first time scale 𝜏1=1𝜆1approaches lim 𝑘′/𝑘→01𝜆1=1𝑘𝛾1𝛾2 𝛾1+𝛾2(A.20) However in the same limit lim 𝑘′/𝑘→0𝜆2=0 (A.21) so the second time scale diverges as one would expect. It diverges with a power law lim 𝑘′/𝑘→0𝑘′ 𝑘𝜏2=lim 𝑘′/𝑘→0𝑘′ 𝑘 𝜆2=𝛾1+𝛾2 2𝑘(A.22) which means 𝜏2≈𝛾1+𝛾2 2𝑘′ . On time scales 𝜏1≪𝑡≪𝜏2 we approach a diffusive regime A.3. Two coupled particles in harmonic potential 111 for which we can find a diffuson coefficient by expanding in terms of 𝜆2𝑡 Δ𝑥21(𝑡)=2𝑘𝐵𝑇 𝑘[Ψ(𝜆1,𝜆2)(1−𝑒−𝜆1𝑡) +Ψ(𝜆2,𝜆1)𝜆2𝑡+(𝜆22𝑡2)](A.23) where we identify 𝐷𝐿=𝑘𝐵𝑇 𝑘𝜆2Ψ(𝜆2,𝜆1) as the diffusion coefficient. In the limit 𝑘′ 𝑘→0 one gets 𝐷≈𝑘𝐵𝑇 𝛾1+𝛾2+𝑘𝐵Δ𝑇𝛾2 (𝛾1+𝛾2)2(A.24) 118 C. Random horse and cart model 𝛽𝑘2Δ𝑥21(𝑡)=2𝐷𝑞 𝐷1𝑘2 𝛾1𝑡+(𝑓(𝜆1,𝜆2)+𝑔(𝜆1,𝜆2)𝐷𝑞 𝐷1)(1−𝑒−𝜆1𝑡) +(𝑓(𝜆2,𝜆1)+𝑔(𝜆2,𝜆1)𝐷𝑞 𝐷1)(1−𝑒−𝜆2𝑡)(C.5) with time scales 𝜆1,2=𝑘2 𝛾112[1+𝑘1 𝑘2+𝛾1 𝛾2𝑘1 𝑘2 ±√(1+𝑘1 𝑘2+𝛾1 𝛾2𝑘1 𝑘2)2−4𝛾1 𝛾2𝑘1 𝑘2](C.6) and constants 𝑓(𝜆1,𝜆2)=2𝑘2 𝛾1−𝜆2 𝜆1−𝜆2(C.7) 𝑔(𝜆1,𝜆2)=2𝑘2 𝛾1(𝜆2−𝑘2 𝛾1)(𝜆2+𝑘2 𝛾1) (𝜆1+𝜆2)(𝜆1−𝜆2)𝜆1(C.8) By introducing the coefficients Ψ(𝜆1,𝜆2)=𝑓(𝜆1,𝜆2)+𝑔(𝜆1,𝜆2)𝐷𝑞 𝐷1 (𝜏,𝑡)=12Ψ(𝜆1,𝜆2)(1−𝑒−𝜆1𝑡)(1−𝑒−𝜆1𝜏)+Ψ(𝜆2,𝜆1)(1−𝑒−𝜆2𝑡)(1−𝑒−𝜆2𝜏) 2𝐷𝑞 𝐷1𝑘2 𝛾1𝜏+Ψ(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+Ψ(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏).(C.9) In the limit 𝜏→0we obtain (𝜏→0,𝑡)=12𝜆1Ψ(𝜆1,𝜆2)(1−𝑒−𝜆1𝑡)+𝜆2Ψ(𝜆2,𝜆1)(1−𝑒−𝜆2𝑡) 2𝐷𝑞 𝐷1𝑘2 𝛾1+𝜆1Ψ(𝜆1,𝜆2)+𝜆2Ψ(𝜆2,𝜆1)(C.10) and in the long time limit 𝑡→∞this expression simplifies to (𝜏→0,𝑡→∞)=12(1−𝐷𝑞 𝐷1)(C.11) C.2. Maximum of MBR The MBR has for certain parmeters 𝛾1 𝛾2,𝑘1 𝑘2 a maximum in 𝑡 which one can obtain by C.3. Critical activity 119 setting the derivative of Eq.(C.9) to zero. The result is given by 𝑡max =1 𝜆1−𝜆2log(−𝜆1Ψ(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏) 𝜆2Ψ(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏)).(C.12) However, not for every time scales 𝜆1,𝜆2 there is a finite time solution as is illustrated in Fig. 4.4 because the expression in the log can be negative or smaller than 1. For activities smaller than 𝐷𝑞 𝐷1<(𝜆1+𝜆2)𝜆2 𝑘2 𝛾1+𝜆1(C.13) there is no maximum, which is remarkably independent of 𝜏 . Also for too high activities there is no maximum, which we find by setting lim 𝑡→0d d𝑡(𝜏,𝑡)!=0 𝐷𝑞 𝐷1>−𝜆1𝑓(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+𝜆2𝑓(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏) 𝜆1𝑔(𝜆1,𝜆2)(1−𝑒−𝜆1𝜏)+𝜆2𝑔(𝜆2,𝜆1)(1−𝑒−𝜆2𝜏),(C.14) however, this bound is 𝜏dependent. In the limit 𝜏→0we obtain Eq. (4.8). C.3. Critical activity The critical activity is given by the ration of active to passive diffusion 𝐷𝑞 𝐷 , for which the long time MBR gets negative and has a local minimum. This property depends on the timescales of the system 𝜆1and 𝜆2. The long time MBR for 𝜏≫𝜆1,𝜆2is given by (𝜏,𝑡→∞)=12Ψ(𝜆1,𝜆2)+Ψ(𝜆2,𝜆1) 2𝐷𝑞 𝐷1𝜏+Ψ(𝜆1,𝜆2)+Ψ(𝜆2,𝜆1).(C.15) Because 𝐷𝑞 𝐷1>0 the sign of the long time MBR is determined by the sign of Ψ(𝜆1,𝜆2)+ Ψ(𝜆2,𝜆1) . Using Ψ(𝜆1,𝜆2)=𝑓(𝜆1,𝜆2)+𝐷𝑞 𝐷1𝑔(𝜆1,𝜆2) we find for the critical activity Ψ(𝜆1,𝜆2)+Ψ(𝜆2,𝜆1)=0 𝐷𝑞 𝐷=−𝑓(𝜆1,𝜆2)+𝑓(𝜆2,𝜆1) 𝑔(𝜆1,𝜆2)+𝑔(𝜆2,𝜆1)(C.16) C.4. Effective energy The diffusing potential with memory can be described by the following Langevin equations 120 C. Random horse and cart model 𝑥1=−𝑘1 𝛾1(𝑥1−𝑥2)−𝑘2 𝛾1(𝑥1−𝑞)+𝜉1(𝑡)(C.17a) 𝑥2=−𝑘1 𝛾2(𝑥2−𝑥1)+𝜉2(𝑡)(C.17b) 𝑞=𝜉𝑞(C.17c) with 𝑥1 the tracer particle, 𝑥2 the bath particle that mimics the viscoelastic environment and 𝑞 the trap position. The system is projected onto 𝑥1 coordinate and by integrating out the bath particle a generalized Langevin equation can be found by −𝑘2𝑥(𝑡)+∫𝑡 −∞Γ(𝑡−𝑠)𝑥(𝑠)=𝑘2𝑞(𝑡)+𝑓(𝑡)(C.18) with a Gaussian random force ⟨𝑓(𝑡)𝑓(𝑡′)⟩=𝑘𝐵𝑇Γ(|𝑡−𝑡′|) . The correlation function in Fourier space is therefore given by ⟨𝑥(𝜔)𝑥(−𝜔)⟩=𝑘22⟨𝑞(𝜔)𝑞(−𝜔)+⟨𝑓(𝜔)𝑓(−𝜔)⟩⟩ 𝜔2Γ(𝜔)Γ(−𝜔)+𝑘22+𝑖𝑘2𝜔(Γ(𝜔)−Γ(−𝜔)) (C.19) To get the effective energy we compare the correlation function with the active driving 𝑞to the correlation function without the active driving and obtain 𝐸eff(𝜔) 𝑘𝐵𝑇=1+𝑘22⟨𝑞(𝜔)𝑞(−𝜔)⟩ ⟨𝑓(𝜔)𝑓(−𝜔)⟩(C.20) =1+𝑘22⟨𝑞(𝜔)𝑞(−𝜔)⟩ 𝑘𝐵𝑇Γ(𝜔)(C.21) =1+𝐷𝑞 𝐷𝑘22 𝛾11𝜔21 Γ(𝜔)(C.22) with Γ(𝜔) the Fourier transform of Γ(|𝑡|) and the Fourier transform of the free diffusion ⟨𝑞(𝜔)𝑞(−𝜔)⟩=𝐷𝑞 𝜔2 with 𝐷=𝑘𝐵𝑇 𝛾1 . For the model system the memory kernel is given by Γ(𝑡)=[2𝛾1𝛿(𝑡)+𝑘1𝑒−𝑘1 𝛾2𝑡]𝜃(𝑡)(C.23) with the Fourier transform Γ(𝜔)=𝛾1+𝑘21 𝛾21 𝑘21 𝛾22+𝜔2 we get the final expression for the effective energy in the main text that scales with 𝐷𝑞 𝐷. D. Entropy bound in cells - Supplementary plots The experimental details can be found in Ref. [2]. Multiple trajectories for different cells and media have been recorded A549 ( 𝑁=179 ), C2C12 ( 𝑁=177 ), CT26 ( 𝑁=173 ), Drugged ( 𝑁=75 ), HeLa ( 𝑁=231 ), HoxB8 ( 𝑁=170 ), MDCK ( 𝑁=183 ), Agarose ( 𝑁=27 ), HeLa CytoB ( 𝑁=60 ) and HeLa Nocoda ( 𝑁=63 ) with a sampling rate of 1.57×10−5s . For A549, the trajectory lengths lie between 3s - 10s with an average of 8.7s . For HoxB8, the trajectory lengths lie between 5s - 10s with an average of 5.6s . For all other cells and media, the trajectory lengths are exactly 10s. 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound A549, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound HoxB8, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound HeLa, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound Drugged, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] 122 D. Entropy bound in cells - Supplementary plots 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound MDCK, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound CT26, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound C2C12, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] 123 t+τ[s] 0.000 0.002 0.004 0.006 0.008 hsibound Agarose, l=0.002 0.5 1.0 1.5 2.0 2.5 3.0 τ[s] Bibliography [1] G. Knotz, U. Parlitz, and S. Klumpp, “Synchronization of a genetic oscillator with the cell division cycle”, New Journal of Physics 24, 033050 (2022). [2] T. M. Muenker, G. Knotz, M. Krüger, and T. Betz, “Accessing activity and viscoelastic properties of artificial and living systems from passive measurement”, Nature Materials 23, 1283–1291 (2024). [3] G. Knotz, T. M. Muenker, T. Betz, and M. Krüger, “Entropy bound for time reversal markers”, Frontiers in Physics 11, Publisher: Frontiers, 1331835 (2024). [4] G. Knotz and M. Krüger, “Mean back relaxation for position and densities”, Physical Review E 110, 044137 (2024). [5] G. Knotz, T. M. Muenker, T. Betz, and M. Krüger, Evaluating non-equilibrium trajectories via mean back relaxation: Dependence on length and time scales, arXiv:2507.05912 [cond-mat], July 2025. [6] R. Brown, “A Brief Account of Microscopical Observations made in the Months of June, July, and August, 1827, on the Particles contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies”, in The Miscellaneous Botanical Works of Robert Brown, Vol. 1, edited by J. J. Bennett, Cambridge Library Collection - Botany and Horticulture (Cambridge University Press, 2015), pp. 463–486. [7] A. Einstein, “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen”, Annalen der Physik 322, 549–560 (1905). [8] M. Von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen”, Annalen der Physik 326, 756–780 (1906). [9] J. Perrin, Les atomes (CNRS éditions, 2014). [10] G. Gallavotti, Statistical Mechanics (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999). [11] R. Zwanzig, Nonequilibrium statistical mechanics (Oxford University Press, Oxford ; New York, 2001). [12] H. Mori, “Transport, Collective Motion, and Brownian Motion”, Progress of Theoretical Physics 33, 423–455 (1965). [13] G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation”, Physical Review A 37, 4419–4428 (1988). [14] C. Ayaz, L. Scalfi, B. A. Dalton, and R. R. Netz, “Generalized Langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme”, Physical Review E 105, 054138 (2022). 124 Bibliography [15] H. Gest, “The discovery of microorganisms by Robert Hooke and Antoni van Leeuwenhoek, Fellows of The Royal Society”, Notes and Records of the Royal Society of London 58, 187–201 (2004). [16] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, “Active Particles in Complex and Crowded Environments”, Reviews of Modern Physics 88, 045006 (2016). [17] F. Schmidt, H. Šípová-Jungová, M. Käll, A. Würger, and G. Volpe, “Nonequilibrium properties of an active nanoparticle in a harmonic potential”, Nature Communications 12, 1902 (2021). [18] M. E. Cates and J. Tailleur, “Motility-Induced Phase Separation”, Annual Review of Condensed Matter Physics 6, 219–244 (2015). [19] A. P. Solon, Y. Fily, A. Baskaran, M. E. Cates, Y. Kafri, M. Kardar, and J. Tailleur, “Pressure is not a state function for generic active fluids”, Nature Physics 11, 673–678 (2015). [20] J. Shea, G. Jung, and F. Schmid, “Passive probe particle in an active bath: can we tell it is out of equilibrium?”, Soft Matter 18, 6965–6973 (2022). [21] P. Nelson, Biological Physics: Energy, Information, Life (WH Freeman New York, 2004). [22] K. C. Neuman and S. M. Block, “Optical trapping”, Review of Scientific Instruments 75, 2787–2809 (2004). [23] C. J. Bustamante, Y. R. Chemla, S. Liu, and M. D. Wang, “Optical tweezers in single-molecule biophysics”, Nature Reviews Methods Primers 1, 25 (2021). [24] W. Ebeling and I. M. Sokolov, Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems (World Scientific, 2005). [25] S. P. Hardy and W. Kőhler, “Investigating bacillary dysentery: The role of laboratory, technique and people”, International Journal of Medical Microbiology 296, 171–178 (2006). [26] H. Risken, The Fokker-Planck equation: methods of solution and applications, 2nd ed, Springer series in synergetics v. 18 (Springer-Verlag, New York, 1996). [27] D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146 , 530–533 (1908)]”, American Journal of Physics 65, 1079–1081 (1997). [28] R. Durrett, Probability:Theory and Examples, 5th ed., Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge, England, Apr. 2019). [29] C. Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences. 3rd ed., Vol. 13, Springer Ser. Synergetics, ISSN: 0172-7389 (Berlin: Springer, 2004). [30] N. G. v. Kampen, Stochastic processes in physics and chemistry, 3rd ed, NorthHolland personal library, OCLC: ocm81453662 (Elsevier, Amsterdam ; Boston, 2007). [31] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, Heidelberg, 1992). [32] A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, 2010). Bibliography 125 [33] U. Seifert, “Stochastic thermodynamics, fluctuation theorems and molecular machines”, Reports on Progress in Physics 75, Publisher: IOP Publishing, 126001 (2012). [34] J. A. Hertz, Y. Roudi, and P. Sollich, “Path integral methods for the dynamics of stochastic and disordered systems”, Journal of Physics A: Mathematical and Theoretical 50, Publisher: IOP Publishing, 033001 (2016). [35] R. Kubo, “The fluctuation-dissipation theorem”, Reports on Progress in Physics 29, 255 (1966). [36] L. Onsager, “Reciprocal Relations in Irreversible Processes. I.”, Physical Review 37, 405–426 (1931). [37] L. Onsager, “Reciprocal Relations in Irreversible Processes. II.”, Physical Review 38, 2265–2279 (1931). [38] M. Baiesi and C. Maes, “An update on the nonequilibrium linear response”, New Journal of Physics 15, 013004 (2013). [39] J. Caspers and M. Krüger, “Identities for nonlinear memory kernels”, Physical Review E 112, 024124 (2025). [40] S. R. de Groot and P. Mazur, Non-equilibrium thermodynamics, Series in physics (North-Holland Publ. Co., Amsterdam, 1962). [41] B. Y. - K. Hu, “Kramers–Kronig in two lines”, American Journal of Physics 57, 821–821 (1989). [42] T. M. Squires and T. G. Mason, “Fluid Mechanics of Microrheology”, Annual Review of Fluid Mechanics 42, 413–438 (2010). [43] I. De Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems”, Reviews of Modern Physics 89, 015001 (2017). [44] P. Martin, A. J. Hudspeth, and F. Jülicher, “Comparison of a hair bundle’s spontaneous oscillations with its response to mechanical stimulation reveals the underlying active process”, Proceedings of the National Academy of Sciences 98, 14380–14385 (2001). [45] T. Betz, M. Lenz, J.-F. Joanny, and C. Sykes, “ATP-dependent mechanics of red blood cells”, Proceedings of the National Academy of Sciences 106, 15320–15325 (2009). [46] M. Guo, A. J. Ehrlicher, M. H. Jensen, M. Renz, J. R. Moore, R. D. Goldman, J. Lippincott-Schwartz, F. C. Mackintosh, and D. A. Weitz, “Probing the Stochastic, Motor-Driven Properties of the Cytoplasm Using Force Spectrum Microscopy”, Cell 158, 822–832 (2014). [47] W. W. Ahmed, É. Fodor, M. Almonacid, M. Bussonnier, M. - H. Verlhac, N. Gov, P. Visco, F. v. Wijland, and T. Betz, “Active Mechanics Reveal Molecular-Scale Force Kinetics in Living Oocytes”, Biophysical Journal 114, 1667–1679 (2018). [48] F. Schwabl, Statistische Mechanik: mit 26 Tabellen und 186 Aufgaben, 3., aktualisierte Aufl, Springer-Lehrbuch (Springer, Berlin Heidelberg, 2006). [49] C. Maes and M. H. van Wieren, “A Markov Model for Kinesin”, Journal of Statistical Physics 112, 329–355 (2003). [50] G. Szabó, T. Tomé, and I. Borsos, “Probability currents and entropy production in nonequilibrium lattice systems”, Physical Review E 82, 011105 (2010). 126 Bibliography [51] R. Clausius, Die mechanische Wärmetheorie, 3rd ed. (Friedrich Vieweg und Sohn, Braunschweig, 1887). [52] L. Boltzmann, “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”, in Kinetische Theorie II: Irreversible Prozesse Einführung und Originaltexte (Vieweg+Teubner Verlag, Wiesbaden, 1970), pp. 115–225. [53] C. Maes and K. Netočný, “Time-reversal and entropy”, Journal of Statistical Physics 110, 269–310 (2003). [54] U. Seifert, “Entropy Production along a Stochastic Trajectory and an Integral Fluctuation Theorem”, Physical Review Letters 95, 040602 (2005). [55] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, “Probability of second law violations in shearing steady states”, Physical Review Letters 71, 2401–2404 (1993). [56] D. M. Carberry, S. R. Williams, G. M. Wang, E. M. Sevick, and D. J. Evans, “The Kawasaki identity and the Fluctuation Theorem”, The Journal of Chemical Physics 121, 8179 (2004). [57] G. E. Crooks, “Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences”, Phys. Rev. E 60, 2721–2726 (1999). [58] C. Jarzynski, “Nonequilibrium Equality for Free Energy Differences”, Physical Review Letters 78, 2690–2693 (1997). [59] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski’s Equality”, Science 296, 1832–1835 (2002). [60] F. Schlögl, “Thermodynamic uncertainty relation”, Journal of Physics and Chemistry of Solids 49, 679–683 (1988). [61] A. C. Barato and U. Seifert, “Thermodynamic Uncertainty Relation for Biomolecular Processes”, Physical Review Letters 114, 158101 (2015). [62] A. Dechant and S. - i. Sasa, “Current fluctuations and transport efficiency for general Langevin systems”, Journal of Statistical Mechanics: Theory and Experiment 2018, 063209 (2018). [63] J. M. Horowitz and T. R. Gingrich, “Thermodynamic uncertainty relations constrain non-equilibrium fluctuations”, Nature Physics 16, 15–20 (2020). [64] P. Pietzonka, “Classical Pendulum Clocks Break the Thermodynamic Uncertainty Relation”, Physical Review Letters 128, 130606 (2022). [65] C. Dieball and A. Godec, “Direct Route to Thermodynamic Uncertainty Relations and Their Saturation”, Physical Review Letters 130, 087101 (2023). [66] U. Seifert, “From Stochastic Thermodynamics to Thermodynamic Inference”, Annual Review of Condensed Matter Physics 10, 171–192 (2019). [67] T. Van Vu, V. T. Vo, and Y. Hasegawa, “Entropy production estimation with optimal current”, Physical Review E 101, 042138 (2020). [68] K. Proesmans and C. Van Den Broeck, “Discrete-time thermodynamic uncertainty relation”, EPL (Europhysics Letters) 119, 20001 (2017). [69] Y. Hasegawa and T. Van Vu, “Fluctuation Theorem Uncertainty Relation”, Physical Review Letters 123, 110602 (2019). Bibliography 127 [70] D. Krapf, E. Marinari, R. Metzler, G. Oshanin, X. Xu, and A. Squarcini, “Power spectral density of a single Brownian trajectory: what one can and cannot learn from it”, New Journal of Physics 20, 023029 (2018). [71] K. Hubicka and J. Janczura, “Time-dependent classification of protein diffusion types: A statistical detection of mean-squared-displacement exponent transitions”, Physical Review E 101, 022107 (2020). [72] G. S. Agarwal, “Fluctuation-dissipation theorems for systems in non-thermal equilibrium and applications”, Zeitschrift für Physik A Hadrons and nuclei 252, 25–38 (1972). [73] T. Harada and S. - i. Sasa, “Equality Connecting Energy Dissipation with a Violation of the Fluctuation-Response Relation”, Physical Review Letters 95, 130602 (2005). [74] T. Speck and U. Seifert, “Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state”, Europhysics Letters (EPL) 74, 391–396 (2006). [75] M. Baiesi, C. Maes, and B. Wynants, “Fluctuations and response of nonequilibrium states”, Physical Review Letters 103, 010602 (2009). [76] R. K. P. Zia and B. Schmittmann, “Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states”, Journal of Statistical Mechanics: Theory and Experiment 2007, P07012–P07012 (2007). [77] A. Ghanta, J. C. Neu, and S. Teitsworth, “Fluctuation loops in noise-driven linear dynamical systems”, Physical Review E 95, 032128 (2017). [78] S. Thapa, D. Zaretzky, R. Vatash, G. Gradziuk, C. Broedersz, Y. Shokef, and Y. Roichman, “Nonequilibrium probability currents in optically-driven colloidal suspensions”, SciPost Physics 17, 096 (2024). [79] É. Roldá’n, I. Neri, M. Dörpinghaus, H. Meyr, and F. Jülicher, “Decision Making in the Arrow of Time”, Physical Review Letters 115, 250602 (2015). [80] J. Berner, B. Müller, J. R. Gomez-Solano, M. Krüger, and C. Bechinger, “Oscillating modes of driven colloids in overdamped systems”, Nature Communications 9, 999 (2018). [81] F. Mori, S. N. Majumdar, and G. Schehr, “Time to reach the maximum for a stationary stochastic process”, Physical Review E 106, 054110 (2022). [82] P. Ronceray, “Two steps forward – and one step back?– Measuring fluctuationdissipation breakdown from fluctuations only”, Journal Club for Condensed Matter Physics, 10.36471/JCCM_July_2023_02 (2023). [83] I. Santra and M. Krüger, “Forces from coarse-graining non-equilibrium degrees of freedom: exact results”, Journal of Statistical Mechanics: Theory and Experiment 2024, 123203 (2024). [84] O. Dauchot and V. Démery, “Dynamics of a Self-Propelled Particle in a Harmonic Trap”, Physical Review Letters 122, 068002 (2019). [85] J. K. G. Dhont, “Studies in Interface Science”, in An Introduction to Dynamics of Colloids, Vol. 2 (Elsevier, 1996). [86] J. Hansen and I. McDonald, Theory of Simple Liquids: with Applications to Soft Matter (Elsevier Science, 2013). [87] T. J. Doerries, S. A. M. Loos, and S. H. L. Klapp, “Correlation functions of nonMarkovian systems out of equilibrium: analytical expressions beyond single- 134 Bibliography Acknowledgments I want to first of all thank Matthias Krüger for supervising and supporting me in the last few years. Your enthusiasm was incredibly encouraging, and you always knew how to help me focus on what really mattered and sharpen my arguments with your experience and expertise. At the same time, it was also pleasant to have free space to try different things and approaches, even though I did not tell you about all the explorations that did not pan out. I also want to thank Peter Sollich and Timo Betz for their participation on my Thesis Advisory Committee, and I am really grateful for their insightful advice and guidance in the past years. Furthermore, I want to thank Timo Betz and Till Münker for the great collaboration. Ever since I stumbled into the "MBR meeting", your excitement was often a huge motivation booster, especially when we, the "Theoreticians", were disappointed by some intermediate setbacks, and helped me to maintain perspective during the odyssey we encountered. Generally, I want to thank every current and past participant of the MBR meeting for the stimulating discussions. Also, I would like to thank all current and former group members for creating a welcoming and enjoyable atmosphere. Especially Juliana Caspers and Ion Santra for many insightful discussions. I am grateful to Rupayan Saha, Laila Henkes, and Manuel Buriks for their careful proofreading of the thesis. I want to thank my friends and family, especially my parents, for supporting me from the very beginning, through my studies, and up to this point. Finally, I want to thank Alina for your love and support, including your help with proofreading during the last weeks, even as you were dealing with your own challenges. Bibliography 135 Declaration on the use of ChatGPT and comparable tools In this thesis, I have used the GWDG Chat AI service (Gemma 3) for brainstorming, proofreading, and optimization of self-written text passages. Further, I used Grammarly for proofreading and optimization of self-written text passages. I have also used the GWDG CoCo AI Service (Qwen 2.5 coder) for the development and optimization of software code. I hereby declare that I have stated all uses completely.