Boltzmann's entropy during free expansion of an interacting ideal gas

TL;DR

This paper investigates how Boltzmann's entropy evolves during the free expansion of an interacting one-dimensional gas, revealing convergence to a universal growth curve and discussing effects of finite size and shocks.

Contribution

It demonstrates the typicality of entropy growth for interacting gases and shows the equivalence of different macro-variable choices under local thermal equilibrium.

Findings

Entropy growth converges to a universal curve for large systems.

Different macro-variables yield identical limiting entropy growth.

Finite size and coarse-graining cause oscillations and shocks in entropy evolution.

Abstract

In this work we study the evolution of Boltzmann's entropy in the context of free expansion of a one dimensional interacting gas inside a box. Boltzmann's entropy is defined for single microstates and is given by the phase-space volume occupied by microstates with the same value of macrovariables which are coarse-grained physical observables. We demonstrate the idea of typicality in the growth of the Boltzmann's entropy for two choices of macro-variables -- the single particle phase space distribution and the hydrodynamic fields. Due to the presence of interaction, the growth curves for both these entropies are observed to converge to a monotonically increasing limiting curve, on taking the appropriate order of limits, of large system size and small coarse graining scale. Moreover, we observe that the limiting growth curves for the two choices of entropies are identical as implied by local thermal equilibrium. We also discuss issues related to finite size and finite coarse gaining scale which lead interesting features such as oscillations in the entropy growth curve. We also discuss shocks observed in the hydrodynamic fields.