Role of initial conditions in $1D$ diffusive systems: compressibility, hyperuniformity and long-term memory

TL;DR

This paper investigates how initial conditions influence long-term fluctuations in one-dimensional diffusive systems, revealing a universal role of a static compressibility parameter across different initial states.

Contribution

It analytically links initial condition effects to a static compressibility, unifying various initial state classes within a universal framework for diffusive systems.

Findings

Long-term memory of initial conditions is governed by a static compressibility.

Hyperuniform initial states form a universality class with quenched-like fluctuations.

Monte Carlo simulations confirm the analytical predictions.

Abstract

We analyse the long-lasting effects of initial conditions on fluctuations in one-dimensional diffusive systems. We consider both the fluctuations of current for non-interacting diffusive particles starting from a step-like initial density profile, and the mean-square displacement of tracers in homogeneous systems with single-file diffusion. For these two cases, we show analytically (via the propagator and Macroscopic Fluctuation Theory, respectively) that the long-term memory of initial conditions is mediated by a single static quantity: a generalized compressibility that quantifies the density fluctuations of the initial state. We thereby identify a universality class of hyperuniform initial states whose dynamical variances coincide with the `quenched' cases studied previously; we also describe a continuous family of other classes among which equilibrated (or `annealed') initial conditions are but one family member. We verify our predictions through extensive Monte Carlo simulations.