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

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.

13 pages, 6 figures; v2: revised version, close to the published version

References in corpus (4)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Collisionless relaxation in non-neutral plasmas
• Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems
• Synchronization of oscillators with long range power law interactions

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• Spontaneous synchronization and nonequilibrium statistical mechanics of coupled phase oscillators
• Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions
• Dynamics of coupled oscillator systems in presence of a local potential