The Kolmogorov-Sinai entropy for dilute systems of hard particles in equilibrium

Abstract

In an equilibrium system, the Kolmogorov-Sinai entropy, $h_{\mathrm{KS}}$, equals the sum of the positive Lyapunov exponents, the exponential rates of divergence of infinitesimal perturbations. Kinetic theory may be used to calculate the Kolmogorov-Sinai entropy for dilute gases of many hard disks or spheres in equilibrium at low number density $n$. The density expansion of $h_{\mathrm{KS}}$ is $N\bar{\nu }A[\ln n+B+O(n)]$, where $\bar{\nu }$ is the single-particle collision frequency. Previous calculations of $A$ were succesful. Calculations of $B$, however, were unsatisfactory. In this paper, I show how the probability distribution of the stretching factor can be determined from a nonlinear differential equation by an iterative method. From this the Kolmogorov-Sinai entropy follows as the average of the logarithm of the stretching factor per unit time. I calculate approximate values of $B$ and compare these to results from existing simulations. The agreement is good.

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