Fluctuations of composite observables and stability of statistical systems

TL;DR

This paper investigates how fluctuations of composite observables relate to the stability of statistical systems, establishing a theorem linking partial and total dispersions and discussing ensemble stability.

Contribution

It introduces a theorem connecting the anomalous or normal nature of partial dispersions to the overall dispersion in statistical systems.

Findings

Global dispersion is anomalous if any partial dispersion is anomalous.

Global dispersion is normal only if all partial dispersions are normal.

The theorem is illustrated with examples of statistical systems.

Abstract

Thermodynamic stability of statistical systems requires that susceptibilities be semipositive and finite. Susceptibilities are known to be related to the fluctuations of extensive observable quantities. This relation becomes nontrivial, when the operator of an observable quantity is represented as a sum of operators corresponding to the extensive system parts. The association of the dispersions of the partial operator terms with the total dispersion is analyzed. A special attention is paid to the dependence of dispersions on the total number of particles N in the thermodynamic limit. An operator dispersion is called thermodynamically normal, if it is proportional to N at large values of the latter. While, if the dispersion is proportional to a higher power of N, it is termed thermodynamically anomalous. The following theorem is proved: The global dispersion of a composite operator, which is a sum of linearly independent self-adjoint terms, is thermodynamically anomalous if and only if at least one of the partial dispersions is anomalous, the power of N in the global dispersion being defined by the largest partial dispersion. Conversely, the global dispersion is thermodynamically normal if and only if all partial dispersions are normal. The application of the theorem is illustrated by several examples of statistical systems. The notion of representative ensembles is formulated. The relation between the stability and equivalence of statistical ensembles is discussed.