Multicanonical distribution: Statistical equilibrium of multiscale systems

TL;DR

This paper introduces a multicanonical formalism for describing the statistical equilibrium of complex multiscale systems, modeling hierarchical structures as nested heat reservoirs with fluctuating temperatures, and applies it to turbulence data.

Contribution

It develops a novel multicanonical framework for multiscale systems and derives explicit distributions using hypergeometric functions, with practical application to turbulence statistics.

Findings

Multicanonical distribution expressed via hypergeometric functions.

Accurately models acceleration statistics in Lagrangian turbulence.

Provides a new approach to statistical equilibrium in hierarchical systems.

Abstract

A multicanonical formalism is introduced to describe statistical equilibrium of complex systems exhibiting a hierarchy of time and length scales, where the hierarchical structure is described as a set of nested "internal heat reservoirs" with fluctuating "temperatures." The probability distribution of states at small scales is written as an appropriate averaging of the large-scale distribution (the Boltzmann-Gibbs distribution) over these effective internal degrees of freedom. For a large class of systems the multicanonical distribution is given explicitly in terms of generalized hypergeometric functions. As a concrete example, it is shown that generalized hypergeometric distributions describe remarkably well the statistics of acceleration measurements in Lagrangian turbulence.