Entropy growth during free expansion of an ideal gas
Abstract
To illustrate Boltzmann's construction of an entropy function that is defined for a single microstate of a system, we present here the simple example of the free expansion of a one dimensional gas of hard point particles. The construction requires one to define macrostates, corresponding to macroscopic observables. We discuss two different choices, both of which yield the thermodynamic entropy when the gas is in equilibrium. We show that during the free expansion process, both the entropies converge to the equilibrium value at long times. The rate of growth of entropy, for the two choice of macrostates, depends on the coarse graining used to define them, with different limiting behaviour as the coarse graining gets finer. We also find that for only one of the two choices is the entropy a monotonically increasing function of time. Our system is nonergodic, nonchaotic and essentially noninteracting; our results thus illustrate that these concepts are not very relevant for the question of irreversibility and entropy increase. Rather, the notions of typicality, large numbers and coarsegraining are the important factors. We demonstrate these ideas through extensive simulations as well as analytic results.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.07742
 Bibcode:
 2021arXiv210907742C
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 15 pages, 12 figures