  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.
