Diffusion, effusion, and chaotic scattering: An exactly solvable Liouvillian dynamics

Abstract

We study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area-preserving mapping of an infinitely extended strip onto itself. This exactly solvable mapping sustains chaotic behaviors and diffusion processes. The relationship between the diffusion coefficient, the Lyapunov exponent, and the entropy per unit time is derived. The long-lived classical resonances of the Liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation. In this sense, there is a quasi-isomorphism between the resonance spectrum of the Liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation. Furthermore, we show that a fractal repeller is associated to each non-equilibrium state in the isolated and finite multibaker chain. The nonequilibrium states are all unstable with respect to the equilibrium, validating a weak form of the second principle of thermodynamics for the present dynamical system. Consequences of nonequilibrium fractals on classical measurements are discussed. We then describe the open multibaker chain as a scattering system. Fractal properties of chaotic scattering are here shown to be related to diffusion in the chain.