Polydispersity and optimal relaxation in the hard sphere fluid

TL;DR

This paper investigates how mass heterogeneity in polydisperse hard sphere gases affects the kinetic relaxation rate, revealing that an optimal mixture with a finite number of species minimizes relaxation time, depending on spatial dimension.

Contribution

It introduces a dynamical equipartition principle showing that the optimal relaxation is achieved with a finite number of species, not a continuous distribution.

Findings

Optimal mixture depends on spatial dimension, with five species in 1D and one in 4D or higher.

No continuous mass distribution minimizes relaxation time.

The framework applies to dilute granular and colloidal systems.

Abstract

We consider the mass heterogeneity in a gas of polydisperse hard particles as a key to optimizing a dynamical property: the kinetic relaxation rate. Using the framework of the Boltzmann equation, we study the long time approach of a perturbed velocity distribution toward the equilibrium Maxwellian solution. We work out the cases of discrete as well as continuous distributions of masses, as found in dilute fluids of mesoscopic particles such as granular matter and colloids. On the basis of analytical and numerical evidence, we formulate a dynamical equipartition principle that leads to the result that no such continuous dispersion in fact minimizes the relaxation time, as the global optimum is characterized by a finite number of species. This optimal mixture is found to depend on the dimension d of space, ranging from five species for d=1 to a single one for d>=4. The role of the collisional kernel is also discussed, and extensions to dissipative systems are shown to be possible.