Kinetic Model of Mass Exchange with Dynamic Arrhenius Transition Rates

TL;DR

This paper develops a nonlinear kinetic model for mass exchange between grains with Arrhenius transition rates, analyzing its dynamics, regimes, and effects of noise, with applications in sintering and econophysics.

Contribution

It introduces a novel mass exchange model with dynamic Arrhenius rates, providing analytical and numerical insights into its regimes and noise effects.

Findings

System exhibits diffusive and growth-decay regimes.

Equilibrium states are determined by mass equipartition.

Noise can reverse the equilibrium mass difference.

Abstract

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.