TY - JOUR
T1 - Boltzmann Entropy of a Freely Expanding Quantum Ideal Gas
AU - Pandey, Saurav
AU - Bhat, Junaid Majeed
AU - Dhar, Abhishek
AU - Goldstein, Sheldon
AU - Huse, David A.
AU - Kulkarni, Manas
AU - Kundu, Anupam
AU - Lebowitz, Joel L.
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2023/8
Y1 - 2023/8
N2 - 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.
AB - 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.
KW - Boltzmann entropy
KW - Macrovariables
KW - Non-equilibrium
KW - Quantum gas
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U2 - 10.1007/s10955-023-03154-y
DO - 10.1007/s10955-023-03154-y
M3 - Article
AN - SCOPUS:85168525325
SN - 0022-4715
VL - 190
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 8
M1 - 142
ER -