On the parametric dependences of a class of non-linear singular maps

We discuss a two-parameter family of maps that generalize piecewise linear, expanding maps of the circle. One parameter measures the effect of a non-linearity which bends the branches of the linear map. The second parameter rotates points by a fixed angle. For small values of the nonlinearity parameter, we compute the invariant measure and show that it has a singular density to first order in the nonlinearity parameter. Its Fourier modes have forms similar to the Weierstrass function. We discuss the consequences of this singularity on the Lyapunov exponents and on the transport properties of the corresponding multibaker map. For larger non-linearities, the map becomes non-hyperbolic and exhibits a series of period-adding bifurcations.Comment: 17 pages, 13 figures, to appear in Discrete and Continuous Dynamical Systems, series B Higher resolution versions of Figures 5 downloadable at http://www.glue.umd.edu/~jrd

Chaotic Scattering Theory of Transport and Reaction-Rate Coefficients

The chaotic scattering theory is here extended to obtain escape-rate expressions for the transport coefficients appropriate for a simple classical fluid, or for a chemically reacting system. This theory allows various transport coefficients such as the coefficients of viscosity, thermal conductivity, etc., to be expressed in terms of the positive Lyapunov exponents and Kolmogorov-Sinai entropy of a set of phase space trajectories that take place on an appropriate fractal repeller. This work generalizes the previous results of Gaspard and Nicolis for the coefficient of diffusion of a particle moving in a fixed array of scatterers.Comment: 27 pages LaTeX, no figure

A Note on the Ruelle Pressure for a Dilute Disordered Sinai Billiard

The topological pressure is evaluated for a dilute random Lorentz gas, in the approximation that takes into account only uncorrelated collisions between the moving particle and fixed, hard sphere scatterers. The pressure is obtained analytically as a function of a temperature-like parameter, beta, and of the density of scatterers. The effects of correlated collisions on the topological pressure can be described qualitatively, at least, and they significantly modify the results obtained by considering only uncorrelated collision sequences. As a consequence, for large systems, the range of beta-values over which our expressions for the topological pressure are valid becomes very small, approaching zero, in most cases, as the inverse of the logarithm of system size.Comment: 15 pages RevTeX with 2 figures. Final version with some typo's correcte

On thermostats and entropy production

The connection between the rate of entropy production and the rate of phase space contraction for thermostatted systems in nonequilibrium steady states is discussed for a simple model of heat flow in a Lorentz gas, previously described by Spohn and Lebowitz. It is easy to show that for the model discussed here the two rates are not connected, since the rate of entropy production is non-zero and positive, while the overall rate of phase space contraction is zero. This is consistent with conclusions reached by other workers. Fractal structures appear in the phase space for this model and their properties are discussed. We conclude with a discussion of the implications of this and related work for understanding the role of chaotic dynamics and special initial conditions for an explanation of the Second Law of Thermodynamics.Comment: 14 pages, 1 figur