Mesoscopic Fluctuations in Statistical Systems

TL;DR

This work develops a unified theory of mesoscopic fluctuations, viewing them as heterophase fluctuations of one phase embedded in another host. It introduces a general, quasi-equilibrium statistical framework built on fibered/integrated Hilbert spaces, manifold indicator functions, and averaging over phase configurations, yielding an effective Hamiltonian with phase probabilities $w_f$ that minimize a thermodynamic potential. The approach spans quantum and classical systems and is illustrated through diverse models (magnetic, ferroelectric, density fluctuations, structural fluctuations, frustrated materials, superfluid and superconducting fluctuations), linking microscopic phase structure to macroscopic thermodynamics and observable signatures such as Debye-Waller and Mössbauer factors. The framework enables quantitative predictions of phase coexistence, nucleation temperatures, and the impact of mesoscopic fluctuations on transition behavior, with potential applicability to condensed matter, trapped atomic systems, and complex social or biological systems. Overall, it provides a cohesive, adaptable methodology for describing nanoscale phase separation and its thermodynamic consequences across a wide range of disciplines.

Abstract

The fluctuations are termed mesoscopic, when their typical size is essentially larger then the average distance between the nearest neighbors, while being much smaller than the overall system size. Since the features of mesoscopic fluctuations are essentially different from those of the surrounding matter, they can be interpreted as fluctuations of one phase occurring inside another host phase. In condensed matter, these fluctuations are of nanosize. They can occur in many-body systems of different nature, for instance, they are typical for condensed matter, can appear in systems of trapped atoms, and also arise in biological and social systems. A survey of the experimental evidence for the occurrence of mesoscopic fluctuations in different materials and systems is given. The main attention is paid to a general theoretical approach for describing them. Applications of the approach are also discussed.

Figures (2)

• Figure 1: Free energies as functions of the frustration parameter $g$ under the fixed cohesive-energy parameter $u = -1$: (a) Free energy of the frustrated system $f$ (solid line), compared with the free energies of a pure crystalline solid $f_{sol}$ (dashed line) and a pure random matter $f_{ran}$ (dashed-dotted line); (b) Detalized region of the frustration parameter $g$, where the free energy of the frustrated matter crosses the free energy of the crystalline solid.
• Figure 2: The probability of the crystalline $w_1$ (solid line) and random $w_2$ (dashed line) fractions in a frustrated material as functions of the frustration parameter $g$, under fixed $u = -1$. The point of the phase transition between the frustrated matter and crystalline solid is $g_0$ = 0.128; the upper bound for the existence of the frustrated matter is $g_c=0.231632$.