Hydrodynamic Limit of Brownian Particles Interacting with Short and Long Range Forces

We investigate the time evolution of a model system of interacting particles, moving in a $d$-dimensional torus. The microscopic dynamics are first order in time with velocities set equal to the negative gradient of a potential energy term $\Psi$ plus independent Brownian motions: $\Psi$ is the sum of pair potentials, $V(r)+\gamma^d J(\gamma r)$, the second term has the form of a Kac potential with inverse range $\gamma$. Using diffusive hydrodynamical scaling (spatial scale $\gamma^{-1}$, temporal scale $\gamma^{-2}$) we obtain, in the limit $\gamma\downarrow 0$, a diffusive type integro-differential equation describing the time evolution of the macroscopic density profile.Comment: 37 pages, in TeX (compile twice), to appear on J. Stat. Phys., e-mail addresses: [email protected], [email protected]

Note on a diffraction-amplification problem

We investigate the solution of the equation \partial_t E(x,t)-iD\partial_x^2 E(x,t)= \lambda |S(x,t)|^2 E(x,t)$, for x in a circle and S(x,t) a Gaussian stochastic field with a covariance of a particular form. It is shown that the coupling \lambda_c at which diverges for t>=1 (in suitable units), is always less or equal for D>0 than D=0.Comment: REVTeX file, 8 pages, submitted to Journal of Physics