Overdamped dynamics of long-range systems on a one-dimensional lattice:   Dominance of the mean-field mode and phase transition
  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. 
Shamik Gupta, Alessandro Campa, Stefano Ruffo
cond-mat.stat-mech