Nonanalyticities of entropy functions of finite and infinite systems

TL;DR

This paper investigates the nonanalytic behavior of microcanonical entropy in finite and infinite systems, revealing how finite system nonanalyticities relate to phase transitions in the thermodynamic limit.

Contribution

It provides an exact analysis of the location and nature of nonanalytic points in microcanonical entropy for a solvable classical spin model, highlighting differences between finite and infinite systems.

Findings

Finite systems have a fixed nonanalytic point at the same energy value.

In the thermodynamic limit, the nonanalytic point shifts to a different energy value.

Care is needed when inferring infinite system properties from finite system nonanalyticities.

Abstract

In contrast to the canonical ensemble where thermodynamic functions are smooth for all finite system sizes, the microcanonical entropy can show nonanalytic points also for finite systems, even if the Hamiltonian is smooth. The relation between finite and infinite system nonanalyticities is illustrated by means of a simple classical spin-like model which is exactly solvable for both, finite and infinite system sizes, showing a phase transition in the latter case. The microcanonical entropy is found to have exactly one nonanalytic point in the interior of its domain. For all finite system sizes, this point is located at the same fixed energy value $\epsilon_{c}^{finite}$, jumping discontinuously to a different value $\epsilon_{c}^{infinite}$ in the thermodynamic limit. Remarkably, $\epsilon_{c}^{finite}$ equals the average potential energy of the infinite system at the phase transition point. The result, supplemented with results on nonanalyticities of the microcanonical entropy for other models, indicates that care is required when trying to infer infinite system properties from finite system nonanalyticities.