Boltzmann Entropy of a Freely Expanding Quantum Ideal Gas

Saurav Pandey, Junaid Majeed Bhat, Abhishek Dhar, Sheldon Goldstein, David A. Huse, Manas Kulkarni, Anupam Kundu, Joel L. Lebowitz

Research output: Contribution to journalArticlepeer-review


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, SB , 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.

Original languageEnglish (US)
Article number142
JournalJournal of Statistical Physics
Issue number8
StatePublished - Aug 2023

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Boltzmann entropy
  • Macrovariables
  • Non-equilibrium
  • Quantum gas


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