Boltzmann's Entropy During Free Expansion of an Interacting Gas

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. Our interacting particle model is a gas of hard point particles with alternating masses, a system known to have good ergodicity properties. 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 interactions, 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 to interesting features such as oscillations in the entropy growth curve. We also discuss shocks observed in the hydrodynamic fields.