On the connections between chaos theory and statistical mechanics
by Henk van Beijeren
Abstract: The past years have seen a surge of activity on the connections between chaos theory and statistical mechanics. Among the connections known I want to mention:
1) the Gaussian thermostat formalism, developed by Hoover, Evans et al. Here the irreversible entropy production in a stationary non-equilibrium system is related to the sum of all of its Lyapunov exponents.
2) the escape-rate formalism of Gaspard and and Nicolis, in which transport coefficients determining the rate of escape of systems from phase space through an open boundary are related to the Kolmogorov-Sinai entropy and the sum of all positive Lyapunov exponents on a small subset of phase space.
3) Ruelle's thermodynamic formalism, in which chaotic as well as transport properties can be obtained from a single dynamical partition function. This is even more ambitious, but for the majority of many-particle systems calculation of the dynamical partition function is a very hard task.
Here I will briefly introduce dynamical systems and discuss their characteristic properties. I will show how quantities like Lyapunov exponents, Kolmogorov-Sinai entropies and topological pressures may be calculated for a dilute Lorentz gas (disordered billiard), which is a system with fixed scatterers on
random positions, with which a point particle makes elastic collisions. Comparisons of the results with computer simulation results show a very good agreement.
For a dilute hard sphere gas in equilibrium both the KS entropy (equal to the sum of all positive Lyapunov exponents) and the largest Lyapunov exponent can be calculated analytically to leading orders in the density. Again, comparisons to computer simulations show good agreement. The smallest positive Lyapunov exponents for these systems show very interesting collective behavior, which can also be
explained through kinetic theory calculations.
Finally I will discuss some outstanding open problems.
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