Thermostatistics of small systems: Exact results in the microcanonical formalism

TL;DR

This paper investigates the validity of common approximations in the microcanonical formalism for small systems by comparing exact solutions of two models with and without these approximations.

Contribution

It provides exact results for small systems and assesses the accuracy of typical thermodynamic assumptions used in the microcanonical approach.

Findings

Standard approximations may not hold for systems with 10-100 particles.

Exact solutions reveal deviations from approximate methods.

Results highlight limitations of thermodynamic assumptions in small systems.

Abstract

Several approximations are made to study the microcanonical formalism that are valid in the thermodynamics limit. Usually it is assumed that: 1)Stirling approximation can be used to evaluate the number of microstates; 2) the surface entropy can be replace by the volumen entropy; and 3)derivatives can be used even if the energy is not a continuous variable. It is also assumed that the results obtained from the microcanonical formalism agree with those from the canonical one. However, it is not clear if these assumptions are right for very small systems (10-100 particles). To answer this questions, two systems with exact solutions (the Einstein model of a solid and the two-level system)have been solve with and without these approximations.