Non-equilibrium fluctuations for the stirring process with births and deaths

TL;DR

This paper analyzes non-equilibrium fluctuations in a one-dimensional stirring process with boundary-driven particle injection and removal, revealing that fluctuations follow an Ornstein-Uhlenbeck process with specific boundary conditions.

Contribution

It derives sharp bounds on space-time $v$-functions for the process, establishing their order and boundary behavior, advancing understanding of non-equilibrium fluctuation phenomena.

Findings

Fluctuations are described by an Ornstein-Uhlenbeck process.

$v$-functions of degree 2 and 3 are of order $N^{-1}$.

$v$-functions of degree ≥ 4 are of order $N^{-1- ext{positive}}$.

Abstract

We consider the one-dimensional stirring process on the segment $\{-N,\ldots,N\}$, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of size $K$. We investigate the non-equilibrium fluctuations of the system, starting from a product measure associated with a smooth initial profile. Given our initial state, the fluctuations are given by an Ornstein-Uhlenbeck process whose characteristic operators are the Laplacian and gradient operators. The domains of these operators include functions with boundary conditions that depend on the hydrodynamic profile. A central ingredient in our analysis is the derivation of sharp bounds on the space and space-time $v$-functions of arbitrary degree for the centered occupation variables. In particular, we prove that the $v$-functions of degree $2$ and $3$ are of order $N^{-1}$, while those of degree at least $4$ are of order $N^{-1-\zeta}$ for some $\zeta > 0$.