  We study a nonlinear kinetic model of mass exchange between interacting
grains. The transition rates follow the Arrhenius equation with an activation
energy that depends on the grain mass. We show that the activation parameter
can be absorbed in the initial conditions for the grain masses, and that the
total mass is conserved. We obtain numerical solutions of the coupled,
nonlinear, ordinary differential equations of mass exchange for the two-grain
system, and we compare them with approximate theoretical solutions in specific
neighborhoods of the phase space. Using phase plane methods, we determine that
the system exhibits regimes of diffusive and growth-decay (reverse diffusion)
kinetics. The equilibrium states are determined by the mass equipartition and
separation nullcline curves. If the transfer rates are perturbed by white
noise, numerical simulations show that the system still exhibits diffusive and
growth-decay regimes, although the noise can reverse the sign of equilibrium
mass difference. Finally, we present theoretical analysis and numerical
simulations of a system with many interacting grains. Diffusive and
growth-decay regimes are established as well, but the approach to equilibrium
is considerably slower. Potential applications of the mass exchange model
involve coarse-graining during sintering and wealth exchange in econophysics.
