Resistive dissipative magnetohydrodynamics from the Boltzmann-Vlasov equation

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Resistive dissipative magnetohydrodynamics from the Boltzmann-Vlasov equation © 2019 American Physical Society Published version Denicol, Gabriel S.; Molnár, Etele; Niemi, Harri; Rischke, Dirk H. Denicol, G. S., Molnár, E., Niemi, H., & Rischke, D. H. (2019). Resistive dissipative magnetohydrodynamics from the Boltzmann-Vlasov equation. Physical Review D, 99(5), Article 056017. https://doi.org/10.1103/PhysRevD.99.056017 2019 Resistive dissipative magnetohydrodynamics from the Boltzmann-Vlasov equation Gabriel S. Denicol,1Etele Molnár,2,3,* Harri Niemi,2,4,5 and Dirk H. Rischke2,6 1Instituto de Física, Universidade Federal Fluminense, UFF, Niterói, 24210-346, RJ, Brazil 2Institut für Theoretische Physik, Johann Wolfgang Goethe–Universität, Max-von-Laue-Street 1, D–60438 Frankfurt am Main, Germany 3Institute of Physics and Technology, University of Bergen, Allegaten 55, 5007 Bergen, Norway 4Department of Physics, University of Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Finland 5Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland 6Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Received 5 February 2019; published 28 March 2019) We derive the equations of motion of relativistic, resistive, second-order dissipative magnetohydrodynamics from the Boltzmann-Vlasov equation using the method of moments. We thus extend our previous work [Phys. Rev. D 98, 076009 (2018)], where we only considered the nonresistive limit, to the case of finite electric conductivity. This requires keeping terms proportional to the electric field Eμin the equations of motions and leads to new transport coefficients due to the coupling of the electric field to dissipative quantities. We also show that the Navier-Stokes limit of the charge-diffusion current corresponds to Ohm’s law, while the coefficients of electrical conductivity and charge diffusion are related by a type of Wiedemann-Franz law. DOI: 10.1103/PhysRevD.99.056017 I. INTRODUCTION Second-order theories of relativistic dissipative fluid dynamics play an essential role in understanding the dynamics of ultrarelativistic heavy-ion collisions [1]. Moreover, strong electromagnetic fields are created in noncentral heavy-ion collisions [2–5], which give rise to novel and interesting phenomena in strongly interacting matter, like the chiral magnetic effect [for a review, see Ref. [6] and refs. therein]. In order to describe the evolution of the system, second-order relativistic dissipative fluid dynamics [7,8] needs to be extended to a self-consistent magnetohydrodynamic framework [9,10]. In Ref. [11] the equations of motion of relativistic, nonresistive, second-order dissipative magnetohydrodynamics were derived from the Boltzmann-Vlasov equation. In a nonresistive, i.e., ideally conducting, fluid the electric field is not an independent degree of freedom but is related to the magnetic field by E¼−v×Band therefore can be eliminated from the equations of motion. While this is a common approximation in magnetohydrodynamics, it cannot be realized in a fully consistent manner in a system whose microscopic dynamics is described by the Boltzmann equation. The reason is that the electric conductivity σEis a fluid-dynamical transport coefficient and thus, like all other transport coefficients, proportional to the mean free path of the particles. Taking the limit σE→∞while keeping the values of all other transport coefficients finite is inconsistent. In this paper we will dispense with the assumption of infinite conductivity, and derive the equations of motion of resistive, second-order dissipative magnetohydrodynamics. As in our previous work [11] we assume a singlecomponent system of spin-zero particles with electric charge qundergoing binary elastic collisions. The fluiddynamical equations of motion are derived by using the 14-moment approximation in the framework developed in Refs. [8,12,13]. The electric field is now included explicitly, and the resistive magnetohydrodynamic equations of motion contain new terms with new transport coefficients due to the coupling of charged particles to the electric field. The electric conductivity σEis defined through Ohm’s law of magnetohydrodynamics, Jμ ind ¼σEEμ, where Jμ ind is the charge current induced by the electric field Eμ. We will show that the electric conductivity is related to the thermal conductivity κ, giving rise to a type of Wiedemann-Franz law, σE≡q2κ=T, where Tis the temperature of matter. The paper is organized as follows. In Sec. II we recall the equations of motion of magnetohydrodynamics. In Sec. III we derive the infinite set of equations of motion for the irreducible moments up to tensor-rank two of the deviation of the single-particle distribution function from local equilibrium. Section IV is devoted to truncating this infinite system applying the 14-moment approximation, to obtain the equations for resistive, second-order dissipative *Extreme Light Infrastructure–Nuclear Physics (ELI-NP), Horia Hulubei Institute for Nuclear Physics (IFIN-HH), Reactorului Strasse 30, Bucharest-Magurele RO-077125, Romania. PHYSICAL REVIEW D 99, 056017 (2019) 2470-0010=2019=99(5)=056017(11) 056017-1 © 2019 American Physical Society magnetohydrodynamics. The Navier-Stokes limit of these equations is discussed in Sec. V. The last section contains a summary of this work. We adopt natural Heaviside-Lorentz units ℏ¼c¼ kB¼ϵ0¼μ0¼1, and the Minkowski space-time metric gμν ¼diagð1;−1;−1;−1Þ. The fluid four-velocity is uμ¼γð1;vÞT, with γ¼ð1−v2Þ−1=2and normalization uμuμ≡1, while in the local rest (LR) frame of the fluid, uμ LR ¼ð1;0ÞT. The rank-two projection operator onto the three-space orthogonal to uμis defined as Δμν ≡gμν −uμuν. For any four-vector, Aμ, we define its projection onto the three-dimensional subspace orthogonal to uμas Ahμi≡ Δμ νAν. A straightforward generalization is the symmetric and traceless projection tensor of rank-2l, denoted by Δμ1μl ν1νl, such that the irreducible projections are Ahμ1μli≡ Δμ1μl ν1νlAν1νl[14]. As an example, the rank-four symmetric and traceless projection operator is defined as Δμν αβ ≡ 1 2ðΔμ αΔν βþΔμ βΔν αÞ−1 3ΔμνΔαβ. The four-momentum kμof particles is normalized to their rest mass squared, kμkμ¼m2 0. The energy of a particle in the LR frame of the fluid is defined as Ek≡kμuμand coincides with the on-shell energy k0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2þm2 0 p. The three-momentum of particles, k, is defined through the orthogonal projection of the four-momenta, khμi≡Δμ νkν,in the LR frame. The comoving derivative of a quantity Ais denoted by an overdot, i.e., _ A≡uμ∂μA, while the threespace gradient is ∇νA≡Δα ν∂αA, hence in the LR frame they reduce to the usual time and spatial derivatives ∂tA and ∇A. Furthermore, we use the decomposition ∂μuν¼ uμ_ uνþ1 3θΔμν þσμν þωμν, where we define the expansion scalar, θ≡∇ μuμ, the shear tensor σμν ≡∇ hμuνi¼ 1 2ð∇μuνþ∇νuμÞ−1 3θΔμν, and the vorticity ωμν ≡ 1 2ð∇μuν−∇νuμÞ. II. EQUATIONS OF MOTION OF MAGNETOHYDRODYNAMICS The equations of motion of magnetohydrodynamics are [see Eqs. (24) and (25) of Ref. [11] ] ∂μJμ f¼0;ð1Þ ∂νTμν ¼−FμλJext;λ:ð2Þ Here, Jμ f¼nfuμþVμ fð3Þ is the electric-charge four-current of the fluid, where nf¼ uνJν fis the electric-charge density and Vμ f¼Δμ νJν fis the electric-charge diffusion current. The electric-charge four-current is related to the particle four-current Nμ fby Jμ f≡qNμ f. Similarly, nf¼qnf, where nfis the particle density in the fluid, and Vμ f¼qVμ f, where Vμ fis the particle diffusion current. For the sake of generality, we have also added a source term from an external charge current Jμ ext in the energy-momentum equation (2). The total energy-momentum tensor of the system is given by Tμν ≡Tμν em þTμν f:ð4Þ It consists of an electromagnetic contribution which, for nonpolarizable, nonmagnetizable fluids, reads [15–17] Tμν em ≡−FμλFν λþ1 4gμνFαβFαβ:ð5Þ Here, Fμν ≡Eμuν−EνuμþϵμναβuαBβ;ð6Þ is the Faraday tensor, which we have decomposed in terms of the fluid four-velocity uμ, as well as the electric and magnetic field four-vectors Eμ≡Fμνuνand Bμ≡ 1 2ϵμναβFαβuν, respectively, with ϵμναβ being the LeviCivita tensor. The second part of the energy-momentum tensor (4) is the contribution from the fluid. For a nonpolarizable, nonmagnetizable fluid it reads Tμν f≡εuμuν−PΔμν þ2WðμuνÞþπμν;ð7Þ where we defined the energy density ε≡Tμν fuμuν, the isotropic pressure P≡−1 3Tμν fΔμν, the energy-momentum diffusion current Wμ≡Δμ αTαβ fuβ, and the shear-stress tensor πμν ≡Δμν αβTαβ f. Maxwell’s equations read [9] ∂μFμν ¼Jν;ϵμναβ∂μFαβ ¼0;ð8Þ where Jμ≡Jμ fþJμ ext is the total electric charge fourcurrent. These equations imply that ∂νTμν em ¼−FμλJλ:ð9Þ From this and Eq. (2) follows that the energy-momentum tensor of the fluid satisfies [18] ∂νTμν f¼FμλJf;λ:ð10Þ III. EQUATIONS OF MOTION FOR THE IRREDUCIBLE MOMENTS The relativistic Boltzmann equation coupled to an electromagnetic field [14,15], the so-called BoltzmannVlasov equation reads, kμ∂μfkþqFμνkν ∂ ∂kμfk¼C½f;ð11Þ DENICOL, MOLNÁR, NIEMI, and RISCHKE PHYS. REV. D 99, 056017 (2019) 056017-2 where fkis the single-particle distribution function, C½fis the usual collision term in the Boltzmann equation, see e.g., Eq. (54) of Ref. [11]. A state of local thermal equilibrium is specified by a single-particle distribution function of the form [19] f0k¼½exp ðβ0Ek−α0Þþa−1;ð12Þ with α0¼μβ0, where μis the (in general space-time dependent) chemical potential and β0¼1=T the (spacetime dependent) inverse temperature, while a¼1for fermions/bosons and a→0for Boltzmann particles. Unless α0,β0, and uμare constants (i.e., independent of space-time coordinates, such that equilibrium is global instead of local), the distribution function f0kis not a solution of the Boltzmann equation (11). However, it is a convenient starting point to derive the equations of motion for dissipative fluid dynamics using the method of moments [8,13]. To this end, one decomposes fk¼f0kþδfk;ð13Þ where δfkis the deviation of the solution fkof the Boltzmann equation from the local-equilibrium distribution function f0k. In the following, we will use the notation h  i ≡ZdK fk;h  i0≡ZdK f0k; h  iδ≡ZdK δfk;ð14Þ where dK ≡gd3k=½ð2πÞ3k0is the Lorentz-invariant measure in momentum space and gis the degeneracy factor of the state with momentum k. From Eq. (13) follows immediately that h  i ¼ h  i0þhi δ. The particle four-current and the energy-momentum tensor of the fluid are given as the following moments of fk, Nμ f≡hkμi;T μν f≡hkμkνi;ð15Þ and, consequently, we identify the fluid-dynamical variables introduced in the previous section as, nf¼hEki,Vμ f¼hkhμii,ε¼hE2 ki,P¼−1 3hΔμνkμkνi, Wμ¼hEkkhμii,πμν ¼hkhμkνii. For reasons of symmetry, hEr kkhμ1kμnii0≡0for n≥1, thus Vμ f¼hkhμiiδ, Wμ¼hEkkhμiiδ,πμν ¼hkhμkνiiδ. Now, following Refs. [8,13] we define the symmetric and traceless irreducible moments of δfk, ρμ1μn r≡hEr kkhμ1kμniiδ:ð16Þ Note that the tensors khμ1kμliare irreducible with respect to Lorentz transformations that leave the fluid 4-velocity invariant and form a complete and orthogonal set [14]. In terms of the irreducible moments (16) the corrections to the equilibrium values of particle density, nf0, energy density, ε0, and isotropic pressure, P0, are δnf≡nf−nf0¼ρ1,δε ≡ε−ε0¼ρ2, and Π≡P− P0¼ðρ2−m2 0ρ0Þ=3. The particle and energy-momentum diffusion currents orthogonal to the fluid velocity are Vμ f¼ ρμ 0and Wμ¼ρμ 1, while the shear-stress tensor is πμν ¼ρμν 0. So far, the local equilibrium state introduced in Eq. (12) has not been defined: the equilibrium variables α0,β0, and uμmust be properly specified in the context of the Boltzmann equation. The first step is to define temperature and chemical potential by introducing matching conditions, nf¼nf0ðα0;β0Þand ε¼ε0ðα0;β0Þ. These conditions define α0and β0such that the particle density and energy density of the system are identical to those of a local equilibrium state characterized by f0k. This implies ρ1¼ρ2¼0. For the sake of completeness, we shall continue with the derivation of the equations of motion for the irreducible moments without specifying the fluid four-velocity. In this way, the equations of motion derived in this paper can be made compatible with any definition of uμ. Equations (1) and (10) lead to equations of motion for α0, β0, and uμ: _ α0¼1 D20 ½−J30ðnf0θþ∂μVμ fÞþJ20ðε0þP0þΠÞθþJ20ð∂μWμ−Wμ_ uμ−πμνσμνÞþJ20qEμVf;μ;ð17Þ _ β0¼1 D20 ½−J20ðnf0θþ∂μVμ fÞþJ10ðε0þP0þΠÞθþJ10ð∂μWμ−Wμ_ uμ−πμνσμνÞþJ10qEμVf;μ;ð18Þ and _ uμ¼1 ε0þP0nf0 β0 ð∇μα0−h0∇μβ0Þ−Π_ uμþ∇μΠ−4 3Wμθ−Wνðσμν −ωμνÞ−_ Wμ−Δμ ν∂κπκν þ1 ε0þP0 ½qnf0Eμ−qBbμνVf;ν;ð19Þ RESISTIVE DISSIPATIVE MAGNETOHYDRODYNAMICS FROM …PHYS. REV. D 99, 056017 (2019) 056017-3 where h0≡ðε0þP0Þ=nf0is the enthalpy per particle in equilibrium and the thermodynamic integrals Jnq and Dnq are defined in Appendix A. Note that these equations extend Eqs. (70)–(72) of Ref. [11] by terms proportional to the electric field Eμ.1 For a given fluid four-velocity uμ, the equations of motion (1) and (10) only specify five of the 14 independent variables α0,β0,Π,Vμ f,Wμ, and πμν. In order to close the system of equations of motion, one needs to specify additional equations of motion that can be provided by a suitable truncation of the infinite set of equations of motion for the irreducible moments ρμ1μl r. The latter equations are obtained by calculating the comoving derivative _ ρhμ1μli r≡ Δμ1μl ν1νluα∂αρν1νl r, using the Boltzmann equation (11), for details see Refs. [8,11,13]. For the irreducible moments of tensor-rank zero one obtains _ ρr−Cr−1¼αð0Þ rθþG3r D20 ∂μVμ f−G2r D20 ∂μWμþθ 3m2 0ðr−1Þρr−2−ðrþ2Þρr−3G2r D20 Π þrρμ r−1þG2r D20 Wμ_ uμ−∇μρμ r−1þðr−1Þρμν r−2þG2r D20 πμνσμν −G2r D20 qEνVν f−ðr−1ÞqEνρν r−2:ð20Þ This equation is very similar to Eq. (75) of Ref. [11] except for the terms proportional to Wμand the last two terms which constitute the contributions from the electric field. The equation of motion for irreducible moments of tensor-rank one reads _ ρhμi r−Chμi r−1¼αð1Þ r∇μα0−αh r_ Wμþrρμν r−1_ uν−1 3∇μðm2 0ρr−1−ρrþ1Þ−Δμ αð∇νραν r−1þαh r∂κπκαÞ þ1 3½m2 0ðr−1Þρμ r−2−ðrþ3Þρμ r−4αh rWμθþðr−1Þρμνλ r−2σνλ þ1 5σμν½2m2 0ðr−1Þρr−2;ν−ð2rþ3Þρr;ν−5αh rWνþðρr;νþαh rWνÞωμν þ1 3½m2 0rρr−1−ðrþ3Þρrþ1−3αh rΠ_ uμþαh r∇μΠ−αh rqBbμνVf;ν−qBbμνρr−1;ν þðαh rnf0þβ0Jrþ1;1ÞqEμþ1 3½ðrþ2Þρr−m2 0ðr−1Þρr−2qEμ−ðr−1Þρμν r−2qEν:ð21Þ Here we introduced a new dimensionless antisymmetric tensor bμν ≡−ϵμναβuαbβ, where the unit four-vector in the direction of the magnetic field and orthogonal to uμis bμ≡Bμ B, with bμbμ¼−1and B≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffi −BμBμ p[21]. Equation (21) differs from Eq. (76) of Ref. [11] by the last three terms taking into account the electric field, as well as by the additional terms proportional to Wμ. Finally, the equation of motion for the irreducible moments of tensor-rank two is _ ρhμνi r−Chμνi r−1¼2αð2Þ rσμν þ2 15 ½m4 0ðr−1Þρr−2−m2 0ð2rþ3Þρrþðrþ4Þρrþ2σμν þ2 5_ uhμ½m2 0rρνi r−1−ðrþ5Þρνi rþ1 −2 5½∇hμðm2 0ρνi r−1−ρνi rþ1Þ þ rρμνγ r−1_ uγ−Δμν αβ∇λραβλ r−1þðr−1Þρμνλκ r−2σλκ þ2ρλhμ rωνi λ þ1 3½m2 0ðr−1Þρμν r−2−ðrþ4Þρμν rθþ2 7½2m2 0ðr−1Þρκhμ r−2−ð2rþ5Þρκhμ rσνi κ−2qBbαβΔμν ακgλβρκλ r−1 þ2qEhμρνi r−ðr−1ÞΔμν αβqEλραβλ r−2þ2 5qEðαðm2 0ρβÞ r−2−ρβÞ rÞ:ð22Þ This equation differs from Eq. (77) of Ref. [11] by the last two terms, which constitute the contributions from a nonvanishing electric field. 1Note that terms proportional to Wμalso did not appear in Ref. [11], since the equations derived in that reference employed the Landau frame [20], where uμis defined as an eigenvector of the energy-momentum tensor, uμTμν ¼εuν, leading to Wμ¼0. DENICOL, MOLNÁR, NIEMI, and RISCHKE PHYS. REV. D 99, 056017 (2019) 056017-4 In Eqs. (20)–(22),αh r,αðlÞ r, and Gij are thermodynamic coefficients, which are explicitly given in Appendix A, while the linearized collision integral is defined as Chμ1μli r−1≡Δμ1μl ν1νlZdK Er−1 kkν1kνlC½f ¼−X Nl n¼0 AðlÞ rn ρμ1μl n;ð23Þ where the coefficient AðlÞ rn ∼λmfp contains time scales proportional to the mean free path of the particles. Note that the last equality of the above equation is obtained using the moment expansion of the single-particle distribution function first introduced in Ref. [8], which, for the sake of completeness, is listed in Appendix A. IV. EQUATIONS OF MOTION IN THE 14-MOMENT APPROXIMATION In order to obtain a closed system of fluid-dynamical equations of motion, we now truncate the infinite set (20)– (22) of equations of motion for the irreducible moments. The simplest and most widely used truncation is the socalled 14-moment approximation [7]. First, all irreducible tensor moments ρμ1μl rfor l>2are explicitly set to zero in Eqs. (21)–(22). Second, the remaining scalar ρr, vector ρμ r, and rank-2 tensor moments ρμν rare expressed as linear combinations of the lowest-order moments ρ0≡−3Π=m2 0, ρμ 0≡Vμ f,ρμ 1≡Wμ, and ρμν 0≡πμν, i.e., in terms of quantities appearing in Jμ fand Tμν f, cf. Eqs. (3),(7). The relations affecting this truncation are Eqs. (A5)–(A7). Equation (20) then leads to an equation of motion for the bulk viscous pressure τΠ_ ΠþΠ¼−ζθ −δΠΠΠθþλΠππμνσμν −lΠV∇μVμ f−τΠVVμ f_ uμ−λΠVVμ f∇μα0 −lΠW∇μWμ−τΠWWμ_ uμ−λΠWWμ∇μα0−δΠVEqVν fEν−δΠWEqWνEν:ð24Þ Similarly, from Eq. (21) we obtain an equation for the particleand energy-diffusion currents, τV_ Vhμi f−τVh−1 0_ WhμiþVμ f−h−1 0Wμ¼κ∇μα0−τVVf;νωνμ −δVVVμ fθ−λVVVf;νσμν þτVh−1 0Wνωνμ −δWWWμθ−λWWWνσμν −lVΠ∇μΠþlVπΔμν∇λπλ νþτVΠΠ_ uμ −τVππμν _ uνþλVΠΠ∇μα0−λVππμν∇να0−δVBqBbμνVf;ν−δWBqBbμνWν þδVEqEμþδVΠEqΠEμþδVπEqπμνEν:ð25Þ The equation of motion for the shear-stress tensor follows from Eq. (22), τπ_πhμνiþπμν ¼2ησμν þ2τππhμ λωνiλ−δπππμνθ−τπππλhμσνi λþλπΠΠσμν −τπVVhμ f_ uνiþlπV∇hμVνi fþλπVVhμ f∇νiα0−τπWWhμ_ uνiþlπW∇hμWνiþλπWWhμ∇νiα0 −δπBqBbαβΔμν ακgλβπκλ þδπVEqEhμVνi fþδπWEqEhμWνi:ð26Þ The coefficients appearing in these equations are listed in Appendix B. Note that Eq. (25) represents the relaxation equation for the heat flow defined by qμ≡Wμ−h0Vμ f:ð27Þ In case we choose the local rest frame following Landau’s picture (which imposes Wμ≡0), the heat flow is simply given in terms of the particle diffusion alone, qμ¼−h0Vμ f. On the other hand, choosing the rest frame according to Eckart’s picture (which requires Vμ f≡0), leads to a heatflow that is solely given by the flow of energy and momentum, qμ¼Wμ. Since the relaxation equations (24)–(26) contain both diffusive quantities, the equations of motion derived in this paper are consistent with either choice of local rest frame. The coefficients proportional to the electric field in the equation for the bulk viscous pressure are δΠVE ¼m2 0 3Að0Þ 00 Fð1Þ 20 −G20 D20 −β0 h0 ∂Fð1Þ 10 ∂β0; δΠWE ¼m2 0 3Að0Þ 00 Fð1Þ 21 −β0 h0 ∂Fð1Þ 11 ∂β0:ð28Þ The coefficients proportional to the electric field in the equation for the particle-diffusion current are RESISTIVE DISSIPATIVE MAGNETOHYDRODYNAMICS FROM …PHYS. REV. D 99, 056017 (2019) 056017-5 δVE ¼1 Að1Þ 00 ð−nf0h−1 0þβ0J11Þ;ð29Þ δVΠE¼−1 m2 0Að1Þ 00 2þm2 0Fð1Þ 20 −m2 0 β0 h0 ∂Fð0Þ 10 ∂β0; δVπE¼1 Að1Þ 00 Fð2Þ 20 −β0 h0 ∂Fð2Þ 10 ∂β0;ð30Þ and the coefficient coupling Wμto the magnetic field is δWB ¼Fð1Þ 11 Að1Þ 00 :ð31Þ Finally, the coefficients proportional to the electric field in the equation for the shear-stress tensor are δπVE ¼2 5Að2Þ 00 4þm2 0Fð1Þ 20 −m2 0 β0 h0 ∂Fð1Þ 10 ∂β0; δπWE ¼2m2 0 5Að2Þ 00 Fð1Þ 21 −β0 h0 ∂Fð1Þ 11 ∂β0:ð32Þ The thermodynamic integral FðlÞ rn is defined in Eq. (A4). In the limit of a massless Boltzmann gas with constant cross section σ,Jnq ≡Inq ¼ðnþ1Þ! 2ð2qþ1Þ!! β2−n 0P0, and hence Að1Þ 00 ¼4=ð9λmfpÞ,Að2Þ 00 ¼3=ð5λmfpÞ, where λmfp ¼1=ðn0σÞ is the mean free path of the particles. In the massless limit, m0¼0, the coefficients δΠVE ¼δΠWE ¼0, while δVΠE formally diverges ∼1=m2 0. However, the bulk viscous pressure is Π¼−m2 0ρ0=3, which cancels this divergence, and the remaining term is ∼ρ0Eμ.Now,Eμis of order one in gradients (see below and Ref. [22]), while ρ0is actually of second order, since the coefficient αð0Þ rin the Navier-Stokes term in Eq. (20) vanishes in the massless limit for all r. Thus, the respective term is of third order in gradients and, for this reason, we neglect it in the massless limit. In Table Iwe list the m0¼0values of those coefficients in Eq. (25), which are not proportional to Π. Similarly, in Table II we list the m0¼0values of those coefficients in Eq. (26), which are not proportional to Π. V. NAVIER-STOKES LIMIT, OHMIC CURRENT, AND WIEDEMANN–FRANZ LAW In the Navier-Stokes limit, all second-order terms are discarded from the relaxation equations (24)–(26).We employ the power-counting advertised in Ref. [22], i.e., Eμis of order one, i.e., of the same order as gradients of α0,β0, and uμ, or of the same order as the dissipative quantities Π,Vμ f,Wμ, and πμν. On the other hand, the magnetic field is of order zero, like other thermodynamic quantities. For the bulk viscous pressure and shear-stress tensor, this ultimately leads to Π¼−ζθ and πμν ¼2ησμν − δπBqBbαβΔμν ακgλβπκλ, see the discussion in Sec. IV. B of Ref. [11], where these equations have already been analyzed. However, for the Navier-Stokes limit of the diffusion currents, the electric field has a non-negligible impact. For the sake of simplicity and comparison to Ref. [11], we work in the Landau frame, where Wμ¼0. To first order, the particle-diffusion current becomes Vμ f¼κ∇μα0þδVEqEμ−δVBqBbμνVf;ν:ð33Þ The Ohmic induction current is given by the second term of Eq. (33) (after multiplying by q), Jμ ind ≡σEEμ;ð34Þ with the electric conductivity σE≡q2δVE:ð35Þ As originally noted by Einstein [23], the electric conductivity and the particle-diffusion coefficient must be related by TABLE I. The coefficients for the diffusion equation for a Boltzmann gas with constant cross section in the ultrarelativistic limit, in the 14-moment approximation, with τð1Þ 00 ¼τV. κτ V½λmfpδVV½τVδWW½τVλVV½τVλWW½τVλVπ½τVlVπ½τVτVπ½τVδVB½τVδWB½τVδVE½τVδVπE½τV 3=ð16σÞ9=41−β0=33=5−β0=4β0=20 β0=20 β0=20 5β0=12 −β2 0=12 P0β2 0=12 0 TABLE II. The coefficients for the shear-stress equation for a Boltzmann gas with constant cross section in the ultrarelativistic limit, in the 14-moment approximation, with τð2Þ 00 ¼τπ. ητ π½λmfpδππ½τπτππ½τπλπV½τπlπV½τπτπV½τπλπW½τπlπW½τπτπW½τπδπB½τπδπVE½τπδπWE½τπ 4=ð3σβ0Þ5=34=310=700002=522β0=58=50 DENICOL, MOLNÁR, NIEMI, and RISCHKE PHYS. REV. D 99, 056017 (2019) 056017-6 σE¼q2β0κ;ð36Þ which is the kinetic-theory version of the famous Wiedemann–Franz law. For the massless Boltzmann gas, the validity of this relation can be easily checked using the relation δVE ¼3 16 nf0β0λmfp and the fact that κ¼3 16 nf0λmfp [8]. As noted in Ref. [24], this relation must also hold for a different reason: in a state of constant Tand uμand in the absence of dissipation, an electric field induces a chargedensity gradient such that (in our conventions for metric and chemical potential), ∇μα0¼−qβ0Eμ:ð37Þ This relation can also be found from the second-order transport equation (25), setting all dissipative quantities to zero, which leads to the condition κ∇μα0¼−δVEqEμ. This relation together with Eq. (37) then confirms the Einstein relation (36). Note that in the presence of a magnetic field Eq. (34) no longer holds [25]. Using Eq. (21) in the Navier-Stokes approximation we obtain ρμ r¼κμν r∇να0þδμν rqEν;ð38Þ hence the conductivity tensor can be defined similarly to Eq. (36) σμν E;r ¼q2δμν r:ð39Þ The rank-two tensor coefficients may be decomposed in the direction parallel and orthogonal to the magnetic field in terms of the projection operators, bμbν,Ξμν ≡Δμν þbμbν, and the tensor bμν as κμν r¼κr⊥Ξμν −κrjjbμbν−κr×bμν;ð40Þ δμν r¼δr⊥Ξμν −δrjjbμbν−δr×bμν:ð41Þ In order to calculate the transport coefficients κμν ror δμν r,we will follow the inversion procedure of Ref. [11], hence in the 14-moment approximation (N1¼1), setting ∇μα0¼0 we get δ0jj ¼β0αð1Þ r Að1Þ r0 ;δ0⊥¼δ0jj1þqBFð1Þ 1−r;0þαh r Að1Þ r02−1 ; δ0×¼δ0⊥qBFð1Þ 1−r;0þαh r Að1Þ r0 :ð42Þ Comparing with Eqs. (101) of Ref. [11], we conclude that δ0jj ¼β0κ0jj;δ0⊥¼β0κ0;δ0×¼β0κ0×;ð43Þ confirming that Eq. (36) also holds in tensorial form, σμν E;r ¼q2β0κμν r, and irrespective of the limit of a massless Boltzmann gas. VI. CONCLUSIONS AND OUTLOOK BasedonthemomentexpansionoftheBoltzmannequation for a single-component gas of charged spin-zero particles coupled to an electromagnetic field, we have derived the equations of motion of resistive, second-order dissipative magnetohydrodynamics in the 14-moment approximation. New transport coefficients appear due to the coupling to the electric field. We computed these coefficients in the limit of a masslessBoltzmanngas.WeanalyzedtheNavier-Stokeslimit of the dissipative quantities and recovered Ohm’slaw.We foundthattheelectricalconductivityandtheparticlediffusion satisfy the well-known Einstein relation, which constitutes a type of Wiedemann-Franz law. In future studies, one should address the generalization to particles with nonzero spin. Then, particles have a microscopic dipole moment which generates nonvanishing macroscopic magnetization and polarization fields [17,24]. The spin of the particles also gives rise to spin-vorticity coupling terms, which leads to the so-called chiral vortical effect [26]. This may necessitate an extension of the standard fluid-dynamical conservation laws by an equation of motion for the macroscopic spin tensor [27,28]. ACKNOWLEDGMENTS The authors acknowledge enlightening discussion with G. Moore. E. M. acknowledges the warm hospitality of the Department of Physics of the University of Jyväskylä, where part of this work was done. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Center CRC-TR 211 “Strong-interaction matter under extreme conditions”—Project No. 315477589—TRR 211. G. S. D. thanks for Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support. E. M. is supported by the Bundesministerium für Bildung und Forschung (BMBF) and by the Research Council of Norway, (NFR) Project No. 255253/F50. H. N. is supported by the Academy of Finland, Project No. 297058. D. H. R. is partially supported by the High-end Foreign Experts Project No. GDW20167100136 of the State Administration of Foreign Experts Affairs of China. APPENDIX A: SOME USEFUL FORMULAS Following Refs. [8,13], we recall that the single-particle distribution function fkcan be expanded around f0kas, fk¼f0kþf0kð1−af0kÞX ∞ l¼0X Nl n¼0 ρμ1μl nkhμ1kμliHðlÞ kn; ðA1Þ RESISTIVE DISSIPATIVE MAGNETOHYDRODYNAMICS FROM …PHYS. REV. D 99, 056017 (2019) 056017-7 where the coefficient HðlÞ knis a polynomial in energy and defined as, HðlÞ kn¼ð−1Þl l!J2l;lX Nl i¼nX i m¼0 aðlÞ in aðlÞ im Em k:ðA2Þ The coefficients aðlÞ ij are calculated via the Gram-Schmidt orthogonalization procedure and are expressed in terms of thermodynamic integrals Jnq, for more details see e.g., Ref. [8]. Any irreducible moment of arbitrary order rand tensor rank lcan always be expressed as a linear combination of irreducible moments of all orders nand the same tensor rank, ρμ1μl r¼X Nl n¼0 ρμ1μl nFðlÞ −r;n;ðA3Þ where FðlÞ rn ¼l! ð2lþ1Þ!!ZdKE−r kHðlÞ knðΔαβkαkβÞlf0kð1−af0kÞ: ðA4Þ In the 14-moment approximation the above expressions simplify considerably, hence using Eq. (A3) with the summation limits N0¼2,N1¼1,N2¼0for different tensor rank, we obtain the following relations, ρr≡X N0 n¼0;≠1;2 ρnFð0Þ −r;n ¼−3 m2 0 ΠFð0Þ −r;0 ≡−3 m2 0 ΠJr0D30 þJrþ1;0G23 þJrþ2;0D20 J20D20 þJ30G12 þJ40D10 ;ðA5Þ ρμ r≡X N1 n¼0 ρμ nFð1Þ −r;n ¼Vμ fFð1Þ −r;0þWμFð1Þ −r;1 ≡Vμ f Jrþ2;1J41 −Jrþ3;1J31 D31 þWμ−Jrþ2;1J31 þJrþ3;1J21 D31 ; ðA6Þ ρμν r≡X N2 n¼0 ρμν nFð2Þ −r;n ¼πμνFð2Þ −r;0≡πμν Jrþ4;2 J42 :ðA7Þ Note that Eqs. (A5)–(A7) are the same as Eqs. (115)–(117) of Ref. [11], except for Eq. (A6), which now contains a term proportional to Wμwhen compared to Eq. (116) of Ref. [11]. Furthermore, for r; n ≥0,FðlÞ −r;n ¼δrn, however Eqs. (A5)–(A7) are to be used for irreducible moments not only with positive but also with negative rgiven by ρμ1μl −r¼X Nl n¼0 ρμ1μl nFðlÞ rn :ðA8Þ Truncating the sum as in Eqs. (A5)–(A7), the coefficients of Eq. (A8) can be written similarly as in Eq. (67) of Ref. [8], ρ−r¼−3 m2 0 γð0Þ rΠþOðKnÞ;ðA9Þ ρμ −r¼γVð1Þ rVμ fþγWð1Þ rWμþOðKnÞ;ðA10Þ ρμν −r¼γð2Þ rπμν þOðKnÞ:ðA11Þ In Ref. [8] the coefficients γðlÞ rwere calculated explicitly in the Landau frame, hence γVð1Þ r≡γð1Þ r, while γWð1Þ ris a new coefficient in the Eckart frame. Note that, in the 14-moment approximation, γð0Þ r≡Fð0Þ r0,γVð1Þ r≡Fð1Þ r0,γWð1Þ r≡Fð1Þ r1, γð2Þ r≡Fð2Þ r0. The usual thermodynamic integrals are defined in local equilibrium such that, Inq ≡ð−1Þq ð2qþ1Þ!! ZdK En−2q kðΔαβkαkβÞqf0k;ðA12Þ Jnq ≡ð−1Þq ð2qþ1Þ!! ZdK En−2q kðΔαβkαkβÞqf0kð1−af0kÞ: ðA13Þ Here we also recall the following coefficients appearing in Eqs. (20)–(22), αð0Þ r≡ð1−rÞIr1−Ir0−nf0 D20 ðh0G2r−G3rÞ;ðA14Þ αð1Þ r≡Jrþ1;1−h−1 0Jrþ2;1;ðA15Þ αð2Þ r≡Irþ2;1þðr−1ÞIrþ2;2;ðA16Þ αh r≡−β0 ε0þP0 Jrþ2;1;ðA17Þ and Dnq ≡Jnþ1;qJn−1;q −J2 nq;ðA18Þ Gnm ≡Jn;0Jm;0−Jn−1;0Jmþ1;0:ðA19Þ In the limit of a massless Boltzmann gas with constant cross section, Jnq ≡Inq ¼ðnþ1Þ! 2ð2qþ1Þ!! β2−n 0P0, and thus αh 0¼ −h−1 0¼−β0=4as well αh 1¼1, hence the coefficients of interest are DENICOL, MOLNÁR, NIEMI, and RISCHKE PHYS. REV. D 99, 056017 (2019) 056017-8