Find the arrow of time in the closure assumption

The exact one-particle equation depends on a two-particle distribution, which in turn depends on higher correlations. Molecular chaos closes this hierarchy by approximating the distribution of incoming collision partners by a product of one-particle distributions at the encounter. The gas should be dilute enough that partners have generally lost relevant correlation with each other before meeting. The assumption concerns the incoming collision surface, not arbitrary independent positions and velocities at every instant.

A collision itself creates correlations between outgoing velocities through its shared conservation laws. Imposing independence before and after every encounter would discard that structure indiscriminately. Reversing all velocities in an evolved state turns outgoing correlations into carefully prepared incoming correlations; that reversed ensemble need not obey the original closure assumption. Thus the reduced equation’s time direction is consistent with reversible microscopic dynamics once its restricted statistical assumptions are made explicit.

Differentiate H in time and insert the kinetic equation. The extra derivative of f integrates to zero when particle number is conserved. For periodic space, an isolated specularly reflecting vessel, or sufficiently decaying whole-space data with no entropy flux at infinity, the streaming contribution produces no net change in the integrated H. The collision contribution can then be symmetrized under exchanging the two particles and reversing the elastic collision map.

Write A for the incoming product of distributions and B for the outgoing product. The symmetrized integrand is minus a positive kernel times (A−B) times the logarithm of A/B. Because logarithm increases monotonically, these two factors have the same sign. H therefore cannot increase and kinetic entropy cannot decrease. In an open vessel, entropy can enter or leave through the boundary, so the total time derivative includes an entropy flux as well as nonnegative production.

Collision production vanishes when the paired products balance on all allowed scattering channels. Under ordinary nondegenerate elastic collision rules, this gives a local Maxwellian. Its density, flow velocity, and temperature may still vary in space. Streaming can disturb that local equilibrium, so vanishing collision production at one instant is not by itself proof of a uniform final equilibrium. A quantitative convergence rate also requires further hypotheses beyond the sign inequality.

The Markov lab offers an elementary entropy analogy. Reset and set p=q=0.25, then Step twice to inspect two transitions from state zero. Predict state-one probability 0.375. Reset and set p=q=0.5, then Step once; predict probability 0.5. The symmetric chain moves toward a uniform law, but it is a finite exact probability model, not a gas collision operator or a proof of the Boltzmann theorem. General asymmetric chains need not increase ordinary Shannon entropy.