Dynamical fluctuations in the Riesz gas

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.

13 pages; v2: minor updates, added references

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• Tracer and current fluctuations in driven diffusive systems
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Inferring intermediate states by leveraging the many-body Arrhenius law
• Dynamical Spreading and Memory Retention Under Power Law Potential
• Expansion into the vacuum of stochastic gases with long-range interactions
• Exact fluctuation and long-range correlations in a single-file model under resetting
• Exact large-scale correlations in diffusive systems with general interactions: explicit characterisation without the Dean--Kawasaki equation
• Macroscopic fluctuation theory of interacting Brownian particles
• Current fluctuations in finite-sized one-dimensional non-interacting passive and active systems