Self-Diffusion in Simple Models: Systems with Long-Range Jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self diffusion coefficient, $D_N(\rho)$, in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set of $N$ neighboring sites. We obtain positive upper and lower bounds on $F_N(\rho)=N((1-\r)-[D_N(\rho)/D_N(0)])/(\rho(1-\rho))$ for $\rho\in [0,1]$. Computer simulations for the square, triangular and one dimensional lattice suggest that $F_N$ becomes effectively independent of $N$ for $N\ge 20$.Comment: 24 pages, in TeX, 1 figure, e-mail addresses: [email protected], [email protected], [email protected]

Dynamics of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastic particles

The time dependence of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastically colliding spheres is investigated by kinetic theory. We determine the full time dependence of the coefficients of an expansion around the Gaussian state in Generalized Laguerre polynomials. Approximating this system of equations to sixth order, we find that the asymptotic state, where the mean energy T follows Haff's law with time independent cooling rate, is reached within a few collisions per particle. Two-dimensional molecular dynamics simulations confirm our results and show exponential behavior in the high-energy tails.Comment: 11 pages, 13 eps figures, to be published in Granular Matte