Boltzmann Entropy of a Freely Expanding Quantum Ideal Gas

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, Δ s_B^f , satisfies log2≤Δ s_B^f≤ 2log2 for fermions, and 0≤Δ s_B^f≤log2 for bosons. For the same initial conditions, the change in the entropy Δ s_B^U for the U -macrostate is greater than Δ s_B^f when the gas is in the quantum regime where the final stationary state is not at thermal equilibrium.