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.

arXiv commentsubmitted to pre

References in corpus (9)

• Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients
• Chaotic Scattering Theory of Transport and Reaction-Rate Coefficients
• Lyapunov Exponents and KS Entropy for the Lorentz Gas at Low Densities
• Largest Lyapunov Exponent for Many Particle Systems at Low Densities
• The Kolmogorov-Sinai Entropy for Dilute Gases in Equilibrium
• Goldstone modes in Lyapunov spectra of hard sphere systems
• B. B. G. K. Y. Hierarchy Methods for Sums of Lyapunov Exponents for Dilute Gases
• The Lyapunov spectrum of the many-dimensional dilute random Lorentz gas
• Front propagation techniques to calculate the largest Lyapunov exponent of dilute hard disk gases

Cited by in corpus (7)

• What does dynamical systems theory teach us about fluids?
• Lyapunov spectra of billiards with cylindrical scatterers: comparison with many-particle systems
• On the Kolmogorov-Sinai entropy of many-body Hamiltonian systems
• Radius of curvature approach to the Kolmogorov-Sinai entropy of dilute hard particles in equilibrium
• State-to-State Cosmology: a new view on the cosmological arrow of time and the past hypothesis
• Chaoticity of the Wet Granular Gas
• Equivalence of kinetic-theory and random-matrix approaches to Lyapunov spectra of hard-sphere systems