Maxwell-Boltzmann Distribution and Liouville Equation

Maxwell-Boltzmann Distribution and Liouville Equation Francesco R. Ruggeri Hanwell, N.B. Dec. 9, 2025 The Liouville equation, d density (pi’s, xi’s)/dt partial = { H, density }, where H is the Hamiltonian and {}, Poisson brackets, is sometimes used to derive the Maxwell-Boltzmann distribution (MB) according to (1). (1) argues that the Hamiltonian governs smooth motion in phase space (e.g. free particle motion or motion in a potential) and not stochastic motion due to collisions. (1) thus concludes that one cannot use the Liouville approach for a gas and so cannot use the BBKGY approach to derive the MB distribution. Alternatives to the Liouville approach for finding the MB distribution are the maximization of Shannon’s entropy subject to the single constraint Sum over i ei p(ei) = Etotal or reaction time reversal balance. We suggest that stochasticity exists at the free particle collision point because it is the interaction which determines the outcomes. We argue that this holds in both the classical MB and quantum mechanical free particle exp(-iEt+ipx) cases for elastic collisions. It seems that one needs to only focus on the collisions themselves which conserve momentum and energy and whatever additional bias may appear. We suggest that the MB case is one for which all other bias is removed. Thus, one does not need the Liouville, maximum entropy or reaction balance cases to find the MB distribution. It follows from bias and conservation features of collisions, we argue. An interesting feature which arises from the quantum case is that exp(-iEt+ipx), derived solely from considerations of bias removal and energy and momentum conservation for a free particle collision, together with Lorentz invariance, actually describes the motion of the free particle, i.e. -Et+px is the free particle relativistic and nonrelativistic action=Lagrangian * t. In other words, linking the collision with Lorentz invariance associates bias and conservation with the trajectory motion which is also found in the Liouville approach. Liouville Approach The Liouville approach seems to focus on Hamiltonian mechanics, but such a procedure is based on Newtonian motion through a potential V(x), i.e. smooth motion. This motion does not occur in a gas which involves the collision of particles as pointed out in (1). (1) states clearly that the Liouville approach cannot be used to derive the MB distribution. In the quantum mechanical case, even smooth motion V(x) cannot be assumed for a particle and one must consider again stochastic collisions, we argue, between the particle and V(x) such that only rms averages are smooth. We suggest that the physics of both gases and quantum interactions is based on stochastic collision type behaviour. Maximization of Entropy or Time Reversal Reaction Balance Another approach used to calculate the MB distribution is maximization of Shannon’s entropy: - Sum over i p(ei) ln(p(ei)) ((1)) subject to the single constraint Sum over i ei p(ei) =Etotal ((2)) Shannon’s entropy form follows from: Probability(N runs for N very large) = 1/ number of states= Product over i p(ei) power (N p(ei)) → ln(Number of states) = - Sum over p(ei) ln(p(ei)) ((3)) Again, there is no focus on the collision itself and bias and conservation laws it contains. A third approach is time reversal reaction balance. This at least focuses on the collision, but does not directly deal with the notion of bias and conservation. It states that for an elastic collision: e1+e2 = e3 + e4 ((4a)) and p(e1)p(e2) = p(e3)p(4) ((4b)) The removal of bias and conservation in a collision should be based on only the incident particles. One does not necessarily need the product outcomes in an equation (i.e. ((4b))) to obtain the MB distribution. Focus on Bias and Conservation in a Collision We argue that stochasticity arises from a collision in a gas and in an interaction for a quantum particle in general. Like (1), we would not use Hamilton or Lagrangian equations of motion which are smooth in phase space and focus only on the bias and conservation features of an interaction. We first consider the MB case. Let us imagine that particles with different energies collide. At the interaction point, it is as if all of the energy is incorporated into one entity regardless of how it is originally distributed over the incident particles. As a result, Removal of all bias means that one retains no information/bias of how a certain total amount of energy was originally distributed among incident particles ((5)) ((5)) must be quantified. Consider three particles with energies e1,e2 and e3 that collide. Then, for independent particles, the probability is: P(e1)P(e2)P(e3) ((6a)) Next, consider a particle with e1+e2, another with e3 and one with e=0 which collide. P(e=0) is a constant and one may take this a 1 and consider unnormalized probabilities. In such a case, the total energy e1+e2+e3+0 (the same as ((6a))) is associated with the probability: P(e1+e2) P(e3) P(e=0) ((6b)) As a result, P(e1+e2) = P(e1)P(e2) ((7)) for unnormalized particles ((7)) if one cannot distinguish between ((6a)) and ((6b)) as we have pointed out in previous notes. We have also called this the AND condition of probabilities. ((7)) has the solution: P(ei) = C exp(-ei/T) the MB distribution ((8)) We argue, however, that stochasticity in a collision should also occur for free particles with no notion of an equilibrium in a gas. This is more akin to the Liouville approach which follows trajectories, but again it is not the trajectory which is of interest, but the collision which removes bias and has conservation laws. In the free particle case, there can be no real value weight for P1(ei) or P2(p=momentum) because there is no thermal equilibrium. Each must have the same real value weight, say 1, but still the variable ei and p (one dimension here) must appear. This suggests that if all bias is to be removed and conservation is to occur, ((8)) is modified to: exp(-iEt+ipx) ((9)) ((9)) contains E and p because one has both energy and momentum conservation explicitly shown and displays x,t because one wishes the result to be Lorentz invariant. If one only had exp(ip) then p for E1 and p for E2 (i.e. p=m1v1=m2v2 nonrelativistically) would Lorentz transform to: p from E1 → g(v) p + g(v) vE1 and p from E2 →g(v) p + g(v) vE2 where g(v) = 1/sqrt(1-vv/cc) ((10)) Bias and conservation at the collision level are the necessary ingredients to statistically deal with an MB gas and quantum free particle collisions as well. The quantum approach, however, seems to have an association with the Liouville in the sense that even though one considers collisions with bias removal and conservation, the result exp(-iEt+ipx) contains the Lorentz invariant: -Et+px ((11)) ((11)) is both the free particle relativistic (-mo sqrt(1-vv/cc)) t ) and nonrelativistic (.5movv t) (with v=x/t) action which is linked to a free particle trajectory. In the quantum free particle case, and not the MB one, the free particle smooth trajectory is linked to the collision properties itself. In other words, a free particle moves through space in a certain way linked to its action = Lagrangian * t, but this motion already “sets it up” for a non-biased collision which conserves energy and momentum. exp(-iEt+ipx) is derived through bias removal and energy/momentum considerations for a stochastic collision, but then Lorentz invariance is imposed and this is linked to the collision to trajectory motion. In other words, in the quantum free particle case, the behaviour at the collision point is linked with the trajectory motion which is akin to the Liouville approach in that the trajectory is considered as being linked to a stochastic collision. This is perhaps why one sees Liouville approaches used to derive the Schrodinger equation, but we argue that these start with real valued classical ideas and then insert and complex “i” is a somewhat arbitrary manner. Conclusion In conclusion, we argue that the MB distribution follows from bias removal and energy conservation considerations of the stochastic collisions which occur in a gas. Like (1) we argue that one should not use the Liouville equation to find the MB distribution and that even maximization of entropy and time reversal action-reaction are not needed. We suggest that all one needs is the observation that p(ei+ej) = p(ej)p(ei) (p not normalized). We then extend these ideas to free particle quantum mechanical collision and obtain exp(-iEt+ipx) which is Lorentz invariant. Thus, we have gone one step beyond bias removal and conservation and have introduced x and t into the probability. The probability is now formally associated with the trajectory (as in the Liouville case) as -Et+px is both the free particle relativistic and nonrelativistic action= Lagrangian * t. Thus, in the quantum case, stochasticity is part of the free particle motion even when it is not interacting, i.e. there is a dx= hbar/p and dt=hbar/E. References 1. Wang, H. Liouville Equation in Statistical Mechanics is not Applicable to Gases Composed of Colliding Molecules (2022 or later) https://arxiv.org/pdf/2302.10225