Exact Expressions of Entropy for Classical Non-interacting Many-body Systems

Abstract

In the thermodynamic limit, the equilibrium state of a many-body system can be characterized by three pairs of conjugate thermodynamic variables: $E/T,V/P,N/\mu$. In this limit, the thermodynamic properties in all ensembles are equivalent up to the leading order of $E,V,N$. However, for systems of finite size, this ensemble equivalence is no longer exact, and the thermodynamic properties may differ substantially among ensembles. To quantify these finite-size effects rigorously, it is desirable to develop a universal ensemble theory applicable to systems of arbitrary size, providing exact expressions for entropy and, thereby, giving rise to the precise value of all equilibrium thermodynamic quantities. In this work, we propose a theory that determines the exact entropy expressions for classical non-interacting many-body systems of arbitrary size across all statistical ensembles, based on only two postulates: \textbf{stationarity}, requiring that the physical laws be invariant under time translation, and \textbf{unbiasedness}, requiring that the equilibrium mixed state maximize the entropy subject to the prescribed constraints. Moreover, we show that the entropy expressions obtained in different ensembles converge to the common asymptotic form $S \asymp \ln\!\left( \left( \frac{4\pi m e E}{3N} \right)^{3N/2} \cdot \left(\frac{V}{N}\right)^N \right)+N$, consistent with the predictions of the large deviation theory.