Overdamped dynamics of long-range systems on a one-dimensional lattice: Dominance of the mean-field mode and phase transition

TL;DR

This paper analyzes the overdamped dynamics of a long-range one-dimensional lattice system with thermal noise, revealing a phase transition at a critical temperature and dominance of the mean-field mode.

Contribution

It demonstrates that the spatial density dynamics are governed by a stable uniform state that transitions at T=1/2, aligning with the mean-field model, and provides analytical growth rate calculations.

Findings

Phase transition at T=1/2 between uniform and non-uniform states.

Mean-field mode dominates the dynamics.

Analytical growth rates match numerical simulations.

Abstract

We consider the overdamped dynamics of a paradigmatic long-range system of particles residing on the sites of a one-dimensional lattice, in the presence of thermal noise. The internal degree of freedom of each particle is a periodic variable which is coupled to those of other particles with an attractive XY-like interaction. The coupling strength decays with the interparticle separation $r$ in space as $1/r^\alpha$; ~$0 < \alpha < 1$. We study the dynamics of the model in the continuum limit by considering the Fokker-Planck equation for the evolution of the spatial density of particles. We show that the equation allows a linearly stable stationary state which is always uniform in space, being non-uniform in the internal degrees below a critical temperature $T=1/2$ and uniform above, with a phase transition between the two at $T=1/2$. The state is the same as the equilibrium state of the mean-field version of the model, obtained by considering $\alpha=0$. Our analysis also lets us to compute the growth and decay rates of spatial Fourier modes of density fluctuations. The growth rates compare very well with numerical simulations.