Critical dynamical fluctuations in reaction-diffusion processes

TL;DR

This paper investigates the critical fluctuations in a one-dimensional reaction-diffusion process, revealing how the global density causes slowdown at criticality and characterizing the non-Gaussian nature of magnetization fluctuations.

Contribution

It introduces a detailed analysis of dynamical phase transition effects on fluctuations, highlighting the role of global density and non-Gaussian behavior at critical points.

Findings

Magnetization fluctuations are non-Gaussian and described by a non-linear SDE.

Fast modes converge to a Gaussian field with explicit space-time covariance.

Slow dynamics are driven by the global density, causing critical slowdown.

Abstract

We consider a one-dimensional microscopic reaction-diffusion process obtained as a superposition of a Glauber and a Kawasaki dynamics. The reaction term is tuned so that a dynamical phase transition occurs in the model as a suitable parameter is varied. We study dynamical fluctuations of the density field at the critical point. We characterise the slowdown of the dynamics at criticality, and prove that this slowdown is induced by a single observable, the global density (or magnetisation). We show that magnetisation fluctuations are non-Gaussian and characterise their limit as the solution of a non-linear SDE. We prove, furthermore, that other observables remain fast: the density field acting on the fast modes (i.e. on mean-0 test functions) and with Gaussian scaling converges, in the sense of finite dimensional distributions, to a Gaussian field with space-time covariance that we compute explicitly. The proof relies on a decoupling of slow and fast modes relying in particular on a relative entropy argument. Major technical difficulties include the fact that local equilibrium does not hold due to the non-linearity, and proving replacement estimates on diverging time intervals due to critical slowdown.