Shamik Gupta, Alessandro Campa, Stefano Ruffo
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 $1/r^\alpha$; $0 \le \alpha < 1$. 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 model has 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 states at $T=1/2$. Thus, the stationary state of the model is the same as that of its mean-field counterpart which has $\alpha=0$. We justify this mean-field dominance by performing linear stability analysis of both the uniform and non-uniform stationary solutions of the Fokker-Planck equation. Our analysis also allows us to compute the growth and decay rates of spatial Fourier modes of density fluctuations. These rates compare very well with numerical simulations.
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http://arxiv.org/abs/1209.6380
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