Boltzmann entropy of a freely expanding quantum ideal gas

Abstract

We study the time evolution of the Boltzmann entropy of a microstate during the non-equilibrium free expansion of a one-dimensional quantum ideal gas. This quantum Boltzmann entropy, $S_B$, essentially counts the "number" of independent wavefunctions (microstates) giving rise to a specified macrostate. It generally depends on the choice of macrovariables, such as the type and amount of coarse-graining, specifying a non-equilibrium macrostate of the system, but its extensive part agrees with the thermodynamic entropy in thermal equilibrium macrostates. We examine two choices of macrovariables: the $U$-macrovariables are local observables in position space, while the $f$-macrovariables also include structure in momentum space. For the quantum gas, we use a non-classical choice of the $f$-macrovariables. For both choices, the corresponding entropies $s_B^f$ and $s_B^U$ grow and eventually saturate. As in the classical case, the growth rate of $s_B^f$ depends on the momentum coarse-graining scale. If the gas is initially at equilibrium and is then released to expand to occupy twice the initial volume, the per-particle increase in the entropy for the $f$-macrostate, $\Delta s_B^f$, satisfies $\log 2\le \Delta s_B^f\le 2\log 2$ for fermions, and $0\le \Delta s_B^f\le \log 2$ for bosons. For the same initial conditions, the change in the entropy $\Delta s_B^U$ for the $U$-macrostate is greater than $\Delta s_B^f$ when the gas is in the quantum regime where the final stationary state is not at thermal equilibrium.

31 pages, 14 figures

References in corpus (2)

• Full counting statistics in a propagating quantum front and random matrix spectra
• Boltzmann's entropy during free expansion of an interacting ideal gas

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