Dynamical fluctuations in the Riesz gas

TL;DR

This paper studies the fluctuations in a Riesz gas of particles performing Brownian motion with long-range interactions, revealing different growth behaviors depending on the interaction exponent s.

Contribution

It provides a detailed analysis of fluctuation growth and correlation structures in Riesz gases, highlighting phase transitions at s=1.

Findings

For 0 < s < 1, fluctuations grow as t^{s/(2(1+s))}.

For s > 1, fluctuations grow as t^{1/4} with amplitude depending on s.

Tagged particle correlations resemble fractional Brownian motion.

Abstract

We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the $|x-y|^{-s}$ Riesz potential, causing the over-damped motion of particles. We investigate fluctuations of the integrated current and the position of a tagged particle. We show that for $0 < s < 1$, the standard deviations of both quantities grow as $t^{\frac{s}{2(1+s)}}$. When $s>1$, the interactions are effectively short-ranged, and the universal sub-diffusive $t^\frac{1}{4}$ growth emerges with only amplitude depending on the exponent. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.