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 sBf and sBU grow and eventually saturate. As in the classical case, the growth rate of sBf 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, ΔsBf , satisfies log2≤ΔsBf≤2log2 for fermions, and 0≤ΔsBf≤log2 for bosons. For the same initial conditions, the change in the entropy ΔsBU for the U-macrostate is greater than ΔsBf when the gas is in the quantum regime where the final stationary state is not at thermal equilibrium.