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Boltzmann entropy of a freely expanding quantum ideal gas
Statistical Mechanics · Submitted 22 Mar 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 $\log{2}\leq\Delta s_B^f\leq 2\log{2}$ for fermions, and $0\leq\Delta s_B^f\leq\log{2}$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.99 | 0.92 | 0.04 | asserted 0.84 |
| s1 | This quantum Boltzmann entropy, $S_B$, essentially counts the "number" of independent wavefunctions (microstates) giving rise to a specified macrostate. | background 0.73 | 0.15 | 0.05 | asserted 0.98 |
| s2 | 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. | background 0.99 | 0.12 | 0.08 | asserted 0.95 |
| s3 | 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. | method 0.85 | 0.59 | 0.04 | asserted 1.00 |
| s4 | For the quantum gas, we use a non-classical choice of the $f$-macrovariables. | method 0.79 | 0.85 | 0.04 | asserted 1.00 |
| s5 | For both choices, the corresponding entropies $s_B^f$ and $s_B^U$ grow and eventually saturate. | result 1.00 | 0.90 | 0.17 | asserted 1.00 |
| s6 | As in the classical case, the growth rate of $s_B^f$ depends on the momentum coarse-graining scale. | result 0.97 | 0.65 | 0.05 | asserted 1.00 |
| s7 | 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 $\log{2}\leq\Delta s_B^f\leq 2\log{2}$ for fermions, and $0\leq\Delta s_B^f\leq\log{2}$ for bosons. | result 1.00 | 0.87 | 0.29 | asserted 1.00 |
| s8 | 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. | result 1.00 | 0.89 | 0.30 | asserted 1.00 |
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Original 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 $\log{2}\leq\Delta s_B^f\leq 2\log{2}$ for fermions, and $0\leq\Delta s_B^f\leq\log{2}$ 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.
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We also note that thermalization, or, more generally, equilibration, in free fermion systems has been established theoretically [ref] and numerically [ref] in various settings.
Heat flows from hot to cold: A simple rigorous example of thermalization in an isolated macroscopic quantum system · Why do we have thermalization in an integrable model? · 0.93
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