Chaotic scattering theory, thermodynamic formalism, and transport coefficients

The foundations of the chaotic scattering theory for transport and reaction-rate coefficients for classical many-body systems are considered here in some detail. The thermodynamic formalism of Sinai, Ruelle, and Bowen [D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading, MA, 1978)] is employed to obtain an expression for the escape rate for a phase-space trajectory of a system to leave a finite region of phase space for the first time. This expression relates the escape rate to the difference between the sum of the positive Lyapunov exponents and the Kolmogorov-Sinai entropy for the fractal set of phase-space trajectories that are trapped forever in the finite region. This relation is well known for systems of a few degrees of freedom and is extended here to systems with many degrees of freedom. The formalism is applied to smooth hyperbolic systems, to cellular-automata lattice gases, and to hard-sphere systems. In the last case, the geometric constructions of Sinai and co-workers [Russ. Math. Surv. 25, 137 (1970); 42, 181 (1987)] for billiard systems are used to describe the relevant chaotic scattering phenomena. Some applications of this formalism to nonhyperbolic systems are also discussed.