Fluctuation suppression and enhancement in interacting particle systems

TL;DR

This paper studies how interactions in particle systems influence fluctuations, revealing that positive interactions suppress fluctuations while negative interactions enhance them, with implications for physical systems and sampling methods.

Contribution

It provides a theoretical analysis of fluctuation behavior in interacting particle systems with positive and negative definite potentials, connecting to physical and computational applications.

Findings

Positive definite interactions reduce fluctuations compared to Monte Carlo sampling.

Negative definite interactions increase fluctuations, potentially diverging over time.

Fluctuations vanish for positive definite kernels at zero temperature, diverge for negative ones.

Abstract

We investigate in this work the effects of interaction on the fluctuation of empirical measures. The systems with positive definite interaction potentials tend to exhibit smaller fluctuation compared to the fluctuation in standard Monte Carlo sampling while systems with negative definite potentials tend to exhibit larger fluctuation. Moreover, if the temperature goes to zero, the fluctuation for positive definite kernels in the long time tends to vanish to zero, while the fluctuation for negative definite kernels in the long time tends to blow up to infinity. This phenomenon may gain deeper understanding to some physical systems like the Poisson-Boltzmann system, and may help to understand the properties of some particle based variational inference sampling methods.