Critical dynamical fluctuations in reaction-diffusion processes
Benoit Dagallier, Claudio Landim
TL;DR
The paper rigorously analyzes a one-dimensional reaction-diffusion system formed by a Glauber+Kawasaki dynamics at a dynamical critical point. It proves that critical slowdown is governed by a single slow observable—the global density or magnetisation—whose fluctuations converge to a nonlinear stochastic differential equation, while all other observables remain fast and Gaussian with explicitly computable space-time covariance. The authors develop a refined relative-entropy framework, including tilted reference measures, to decouple slow and fast modes and derive close macroscopic equations for the slow magnetisation, even in the presence of nonlinearity that defeats local equilibrium. A detailed analysis of the adjoint generators and higher-order correlations, augmented by renormalisation and large-deviation controls, yields a comprehensive description of both slow non-Gaussian fluctuations and fast Gaussian fluctuations, providing a rigorous handle on dynamical phase transitions in short-range interacting particle systems. The work lays groundwork for extensions to higher dimensions and more complex Kawasaki dynamics, and offers a precise mechanism for the observed dynamical universality via slow-fast mode projection.
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.