In a strong uniform Newtonian gravitational field, the entropy of a monoatomic ideal gas depends on the container's cross-sectional area and is extensive when particle number per unit area is fixed.
Box of Ideal Gas in Free Fall
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We study the \textit{quantum} partition function of non-relativistic, ideal gas in a (non-cubical) box falling freely in arbitrary curved spacetime with centre 4-velocity u^a. When perturbed energy eigenvalues are properly taken into account, we find that corrections to various thermodynamic quantities include a very specific, sub-dominant term which is independent of \textit{kinematic} details such as box dimensions and mass of particles. This term is characterized by the dimensionless quantity, \Xi=R_00 \Lambda^2, where R_00=R_ab u^a u^b and \Lambda=\beta \hbar c, and, quite intriguingly, produces Euler relation of homogeneity two between entropy and energy -- a relation familiar from black hole thermodynamics.
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A note on entropy of matter in presence of gravity: status of extensivity of entropy
In a strong uniform Newtonian gravitational field, the entropy of a monoatomic ideal gas depends on the container's cross-sectional area and is extensive when particle number per unit area is fixed.