No Anomalous Fluctuations Exist in Stable Equilibrium Systems

TL;DR

This paper proves a theorem showing that in stable equilibrium systems, all global observable fluctuations are normal unless at least one component exhibits anomalous fluctuations, ruling out anomalies in Bose-Einstein condensates.

Contribution

A rigorous theorem linking the normality of global fluctuations to that of their components in equilibrium systems, clarifying misconceptions about fluctuation anomalies.

Findings

Global fluctuations are normal if all partial fluctuations are normal.

Anomalous fluctuations occur only if at least one partial fluctuation is anomalous.

Fictitious fluctuation anomalies in calculations are explained.

Abstract

An equilibrium statistical system is known to be stable if the fluctuations of global observables are normal, when their dispersions are proportional to the number of particles, or to the system volume. A general theorem is rigorously proved for the case, when an observable is a sum of linearly independent terms: The dispersion of a global observable is normal if and only if all partial dispersions of its terms are normal, and it is anomalous if and only if at least one of the partial dispersions is anomalous. This theorem, in particular, rules out the possibility that in a stable system with Bose-Einstein condensate some fluctuations of either condensed or noncondensed particles could be anomalous. The conclusion is valid for arbitrary systems, whether uniform or nonuniform, interacting weakly or strongly. The origin of fictitious fluctuation anomalies, arising in some calculations, is elucidated.