Bernoulli's formula and Poisson's equations for a confined quantum gas: Effects due to a moving piston

TL;DR

This paper investigates the nonequilibrium thermodynamics of a quantum gas confined in a cavity with a moving piston, highlighting quantum non-adiabatic effects on pressure and internal energy, relevant for nano-scale heat engines.

Contribution

It derives quantum non-adiabatic corrections to Bernoulli's formula and Poisson's equations for a confined quantum gas with a moving piston, considering both quantum and classical regimes.

Findings

QNA contribution proportional to piston's velocity squared

Coefficient depends on temperature, density, and dimensionality

Relevance for nano-scale heat engine design

Abstract

We study a nonequilibrium equation of states of an ideal quantum gas confined in the cavity under a moving piston with a small but finite velocity in the case that the cavity wall suddenly begins to move at time origin. Confining to the thermally-isolated process, quantum non-adiabatic (QNA) contribution to Poisson's adiabatic equations and to Bernoulli's formula which bridges the pressure and internal energy is elucidated. We carry out a statistical mean of the non-adiabatic (time-reversal-symmetric) force operator found in our preceding paper (K. Nakamura et al, Phys. Rev. E Vol.83, 041133, (2011)) in both the low-temperature quantum-mechanical and high temperature quasi-classical regimes. The QNA contribution, which is proportional to square of the piston's velocity and to inverse of the longitudinal size of the cavity, has a coefficient dependent on temperature, gas density and dimensionality of the cavity. The investigation is done for a unidirectionally-expanding 3-d rectangular parallelepiped cavity as well as its 1-d version. Its relevance in a realistic nano-scale heat engine is discussed.