Radius of curvature approach to the Kolmogorov-Sinai entropy of dilute hard particles in equilibrium

TL;DR

This paper develops a new method using the radius of curvature to estimate the Kolmogorov-Sinai entropy in dilute hard particle gases, achieving excellent agreement with simulations.

Contribution

It introduces a novel approach to estimate the constant in the entropy expansion by solving a differential equation for eigenvalue distribution.

Findings

The method accurately estimates the constant B in the entropy expansion.

Results show very good agreement with existing simulation data.

Eigenvalue distribution predictions match new simulation results.

Abstract

We consider the Kolmogorov-Sinai entropy for dilute gases of $N$ hard disks or spheres. This can be expanded in density as $h_{\mathrm{KS}} \propto n N [\ln n a^d+ B + O(n a^d)+O(1/N)]$, with $a$ the diameter of the sphere or disk, $n$ the density, and $d$ the dimensionality of the system. We estimate the constant $B$ by solving a linear differential equation for the approximate distribution of eigenvalues of the inverse radius of curvature tensor. We compare the resulting values of $B$ both to previous estimates and to existing simulation results, finding very good agreement with the latter. Also, we compare the distribution of eigenvalues of the inverse radius of curvature tensor resulting from our calculations to new simulation results. For most of the spectrum the agreement between our calculations and the simulations again is very good.