Landau theory of restart transitions

TL;DR

This paper introduces a Landau-like theoretical framework to analyze phase transitions in systems with stochastic restart, revealing how optimal restart rates change behaviorally, with applications to chemical reactions and diffusion.

Contribution

It presents a unified Landau theory for restart transitions, linking phase change behavior to first passage time moments in stochastic processes.

Findings

Optimal restart rate exhibits first or second order transitions.

The framework applies to chemical reactions and diffusion processes.

Transitions depend on system-specific parameters.

Abstract

We develop a Landau like theory to characterize the phase transitions in resetting systems. Restart can either accelerate or hinder the completion of a first passage process. The transition between these two phases is characterized by the behavioral change in the order parameter of the system namely the optimal restart rate. Like in the original theory of Landau, the optimal restart rate can undergo a first or second order transition depending on the details of the system. Nonetheless, there exists no unified framework which can capture the onset of such novel phenomena. We unravel this in a comprehensive manner and show how the transition can be understood by analyzing the first passage time moments. Power of our approach is demonstrated in two canonical paradigm setup namely the Michaelis Menten chemical reaction and diffusion under restart.