Singular behavior of fluctuations in a relaxation process

TL;DR

This paper demonstrates the existence of a singular point in the probability distribution of an extensive variable in a non-interacting Gaussian model, linking it to phase transitions in a dual constrained system, applicable in both equilibrium and non-equilibrium states.

Contribution

It introduces a novel connection between singularities in probability distributions and phase transitions via a dual system framework, extending to non-equilibrium conditions.

Findings

Identifies a singular point in the distribution $P(M)$ of the Gaussian model.

Shows the singularity corresponds to a phase transition in a dual constrained system.

Demonstrates the mechanism's generality in equilibrium and non-equilibrium states.

Abstract

Carrying out explicitly the computation in a paradigmatic model of non-interacting systems, the Gaussian Model, we show the existence of a singular point in the probability distribution $P(M)$ of an extensive variable $M$. Interpreting $P(M)$ as a thermodynamic potential of a dual system obtained from the original one by applying a constraint, we discuss how the non-analytical point of $P(M)$ is the counterpart of a phase-transition in the companion system. We show the generality of such mechanism by considering both the system in equilibrium or in the non-equilibrium state following a temperature quench.