2012/09/30 by Shamik Gupta, Alessandro Campa, Stefano Ruffo · 1 citation
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Condensed matter physics #Differential equation #Fokker–Planck equation #Lattice (music) #Mean field theory #Mode coupling #Phase space #Phase transition #Physics #Quantum mechanics #Statistical Mechanics and Entropy #Statistical physics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.86.061130
published as Phys. Rev. E 86, 061130 (2012) · 13 pages, 6 figures; v2: revised version, close to the published version
openalex publication_date 2012/12/26 · arxiv created 2012/12/27 · arxiv updated 2013/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the overdamped dynamics of a paradigmatic long-range system of particles residing on the sites of a one-dimensional lattice and in the presence of thermal noise. The internal degree of freedom of each particle is a periodic variable that 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/rα; 0<α<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 nonuniform 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 α=0. Our analysis also allows us to compute the growth and decay rates of spatial Fourier modes of density fluctuations. The growth rates compare very well with numerical simulations.