Boltzmann entropy of a freely expanding quantum ideal gas
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arXiv
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2023
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| author | Pandey, Saurav Bhat, Junaid Majeed Dhar, Abhishek Goldstein, Sheldon Huse, David A. Kulkarni, Manas Kundu, Anupam Lebowitz, Joel L. |
| author_facet | Pandey, Saurav Bhat, Junaid Majeed Dhar, Abhishek Goldstein, Sheldon Huse, David A. Kulkarni, Manas Kundu, Anupam Lebowitz, Joel L. |
| contents | 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, $Δs_B^f$, satisfies $\log{2}\leqΔs_B^f\leq 2\log{2}$ for fermions, and $0\leqΔs_B^f\leq\log{2}$ for bosons. For the same initial conditions, the change in the entropy $Δs_B^U$ for the $U$-macrostate is greater than $Δs_B^f$ when the gas is in the quantum regime where the final stationary state is not at thermal equilibrium. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_12330 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boltzmann entropy of a freely expanding quantum ideal gas Pandey, Saurav Bhat, Junaid Majeed Dhar, Abhishek Goldstein, Sheldon Huse, David A. Kulkarni, Manas Kundu, Anupam Lebowitz, Joel L. Statistical Mechanics Quantum Gases Quantum Physics 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, $Δs_B^f$, satisfies $\log{2}\leqΔs_B^f\leq 2\log{2}$ for fermions, and $0\leqΔs_B^f\leq\log{2}$ for bosons. For the same initial conditions, the change in the entropy $Δs_B^U$ for the $U$-macrostate is greater than $Δs_B^f$ when the gas is in the quantum regime where the final stationary state is not at thermal equilibrium. |
| title | Boltzmann entropy of a freely expanding quantum ideal gas |
| topic | Statistical Mechanics Quantum Gases Quantum Physics |
| url | https://arxiv.org/abs/2303.12330 |