Entropy growth during free expansion of an ideal gas

Abstract To illustrate Boltzmann’s construction of an entropy function that is defined for a microstate macroscopic system, we present here the simple example free expansion one dimensional gas non-interacting point particles. The requires to define macrostates, corresponding variables. We macrostate M by specifying fraction particles in rectangular boxes Δ x v single particle position-velocity space { , }. verify when number large Boltzmann entropy, S B ( t ), typical nonequilibrium ensemble coincides with Gibbs coarse-grained time-evolved one-particle distribution associated this ensemble. ) approaches its maximum possible value dynamical evolution given initial state. rate approach depends on size definition macrostate, going zero at any fixed time → 0. Surprisingly different curves collapse scaled as: ∼ τ /Δ . find explicit expression limit also consider different, more hydrodynamical, macrostates which monotone increasing, unlike previous has small decaying oscillations near value. Our system non-ergodic, non-chaotic and non-interacting; our results thus these concepts are not as relevant sometimes claimed, observing irreversibility increase. Rather, notions conditions, typicality, numbers coarse-graining important factors. demonstrate ideas through extensive simulations well analytic results.