  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.
