Response Theory and Phase Transitions for the Thermodynamic Limit of Interacting Identical Systems

TL;DR

This paper investigates how large networks of interacting stochastic systems respond to perturbations, revealing conditions for phase transitions driven by mutual interactions, and clarifying their distinction from critical transitions.

Contribution

It derives new response relations and conditions for phase transitions in the thermodynamic limit of coupled stochastic systems, including non-equilibrium cases.

Findings

Phase transitions occur due to system interactions and are marked by diverging linear response.

Such phase transitions do not necessarily involve divergence of autocorrelation times.

Results clarify differences between endogenous phase transitions and critical phenomena.

Abstract

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers-Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker-Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai-Zwanzig model and of the Bonilla-Casado-Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.