Berlin 2005 – scientific programme

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DY: Dynamik und Statistische Physik

DY 10: Fluid Dynamics I

DY 10.1: Invited Talk

Friday, March 4, 2005, 10:00–10:30, TU H3010

On the connections between chaos theory and statistical mechanics — •Henk van Beijeren — Institute for Theoretical Physics, Utrecht University

The past years have seen a surge of activity on the connections between chaos theory and statistical mechanics. In the Gaussian thermostat formalism, developed by Hoover, Evans et al, the irreversible entropy production in a stationary non-equilibrium system is related to the sum of all of its Lyapunov exponents. In the escape-rate formalism of Gaspard and and Nicolis, transport coefficients determining the rate of escape of systems from phase space through an open boundary are related tot the Kolmogorov-Sinai entropy and the sum of all positive Lyapunov exponents on a small subset of phase space. In Ruelle’s thermodynamic formalism 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 wil discuss some outstanding open problems.