On the merit of a Central Limit Theorem-based approximation in statistical physics

TL;DR

This paper rigorously examines a Central Limit Theorem-based approximation in statistical physics, identifying its limitations at low/intermediate temperatures and its failure to detect quantum criticalities, while confirming its high-temperature accuracy.

Contribution

It provides a rigorous analysis of the conditions under which the CLT-based approximation is valid or fails in statistical physics, clarifying its applicability and limitations.

Findings

Fails at low and intermediate temperatures

Inadequate for quantum criticalities

Accurately predicts high-temperature behavior

Abstract

The applicability conditions of a recently reported Central Limit Theorem-based approximation method in statistical physics are investigated and rigorously determined. The failure of this method at low and intermediate temperature is proved as well as its inadequacy to disclose quantum criticalities at fixed temperatures. Its high temperature predictions are in addition shown to coincide with those stemming from straightforward appropriate expansions up to (k_B T)^(-2). Our results are clearly illustrated by comparing the exact and approximate temperature dependence of the free energy of some exemplary physical systems.