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Fundamental temperature exclusively determines the validity of superstatistics

Constanza Farías, Sergio Davis

TL;DR

The paper addresses the range of validity of superstatistics by deriving necessary and sufficient conditions expressed solely in terms of the fundamental inverse temperature $β_F$ and its derivatives. It shows that a steady-state model is superstatistical if and only if $(-1)^n β_F^{(n)}(E;S)≥0$ for all $n≥0$, enabling explicit formulas for all moments and cumulants of the conditional distribution $P(β|E,S)$. The authors provide recurrence relations and a differential-equation framework to compute cumulants from $β_F$, and illustrate the theory with the $q$-canonical, Gaussian, and a simple correction to the canonical ensemble. This work clarifies the precise mathematical criteria for superstatistics and implies that noncanonical $β_F$ must be infinitely differentiable and non-polynomial, with direct implications for modeling nonequilibrium steady states in physics and beyond.

Abstract

The theory of superstatistics is a generalization of Boltzmann-Gibbs statistical mechanics which admits temperature fluctuations, and generates non-canonical ensembles from the distribution function of these fluctuations. Recently, some results have been presented showing that superstatistics is not universally applicable, but several conditions on the so-called fundamental inverse temperature function $β_F$ must be met by any superstatistical model. In this work we provide a set of neccessary and sufficient conditions for a non-equilibrium steady state model to be expressible by superstatistics, showing that $β_F$ by itself determines the existence of a superstatistical distribution of temperature.

Fundamental temperature exclusively determines the validity of superstatistics

TL;DR

The paper addresses the range of validity of superstatistics by deriving necessary and sufficient conditions expressed solely in terms of the fundamental inverse temperature and its derivatives. It shows that a steady-state model is superstatistical if and only if for all , enabling explicit formulas for all moments and cumulants of the conditional distribution . The authors provide recurrence relations and a differential-equation framework to compute cumulants from , and illustrate the theory with the -canonical, Gaussian, and a simple correction to the canonical ensemble. This work clarifies the precise mathematical criteria for superstatistics and implies that noncanonical must be infinitely differentiable and non-polynomial, with direct implications for modeling nonequilibrium steady states in physics and beyond.

Abstract

The theory of superstatistics is a generalization of Boltzmann-Gibbs statistical mechanics which admits temperature fluctuations, and generates non-canonical ensembles from the distribution function of these fluctuations. Recently, some results have been presented showing that superstatistics is not universally applicable, but several conditions on the so-called fundamental inverse temperature function must be met by any superstatistical model. In this work we provide a set of neccessary and sufficient conditions for a non-equilibrium steady state model to be expressible by superstatistics, showing that by itself determines the existence of a superstatistical distribution of temperature.
Paper Structure (10 sections, 2 theorems, 65 equations)

This paper contains 10 sections, 2 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. The superstatistical framework
  3. Fundamental temperature determines the superstatistical class of models
  4. Cumulants using recurrence relations and differential equations
  5. Examples
  6. The $q$-canonical ensemble
  7. The Gaussian ensemble
  8. A simple correction to the canonical ensemble
  9. Concluding remarks
  10. Proof of theorem 1

Key Result

Theorem 1

A steady-state model $S$ having fundamental inverse temperature $\beta_F$ is a superstatistical model (including the canonical case) if and only if holds for all integer $n \geq 0$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • proof : Proof of theorem 1