Thermodynamics of a subensemble of a canonical ensemble

TL;DR

This paper compares two methods for describing the thermodynamics of a subsystem in contact with a thermal bath, analyzing their differences and advantages through classical and quantum examples, and concludes the first method is superior.

Contribution

The paper provides a detailed comparison of two approaches to subsystem thermodynamics, highlighting their differences and establishing the superiority of the first approach.

Findings

The two approaches yield significantly different results for non-bilinear interactions.

Classical and quantum analyses show the first approach is more accurate.

The first approach is recommended for general use in subsystem thermodynamics.

Abstract

Two approaches to describe the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy $E_{S}$ is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function $Z_{S}$ is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation $E_{S}=-d(\ln Z_{S})/d\beta$ is used to obtain the mean system energy. % ($\beta\equiv 1/(k_{B}T)$, $k_{B}$ is the Boltzmann constant, %and $T$ is the temperature). Employing both classical and quantum mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.