Field Driven Thermostated System : A Non-Linear Multi-Baker Map

In this paper, we discuss a simple model for a field driven, thermostated random walk that is constructed by a suitable generalization of a multi-baker map. The map is a usual multi-baker, but perturbed by a thermostated external field that has many of the properties of the fields used in systems with Gaussian thermostats. For small values of the driving field, the map is hyperbolic and has a unique SRB measure that we solve analytically to first order in the field parameter. We then compute the positive and negative Lyapunov exponents to second order and discuss their relation to the transport properties. For higher values of the parameter, this system becomes non-hyperbolic and posseses an attractive fixed point.Comment: 6 pages + 5 figures, to appear in Phys. Rev.

Chaotic properties of systems with Markov dynamics

We present a general approach for computing the dynamic partition function of a continuous-time Markov process. The Ruelle topological pressure is identified with the large deviation function of a physical observable. We construct for the first time a corresponding finite Kolmogorov-Sinai entropy for these processes. Then, as an example, the latter is computed for a symmetric exclusion process. We further present the first exact calculation of the topological pressure for an N-body stochastic interacting system, namely an infinite-range Ising model endowed with spin-flip dynamics. Expressions for the Kolmogorov-Sinai and the topological entropies follow.Comment: 4 pages, to appear in the Physical Review Letter

Distribution of label spacings for genome mapping in nanochannels

In genome mapping experiments, long DNA molecules are stretched by confining them to very narrow channels, so that the locations of sequence-specific fluorescent labels along the channel axis provide large-scale genomic information. It is difficult, however, to make the channels narrow enough so that the DNA molecule is fully stretched. In practice its conformations may form hairpins that change the spacings between internal segments of the DNA molecule, and thus the label locations along the channel axis. Here we describe a theory for the distribution of label spacings that explains the heavy tails observed in distributions of label spacings in genome mapping experiments.Comment: 18 pages, 4 figures, 1 tabl

Field dependent collision frequency of the two-dimensional driven random Lorentz gas

In the field-driven, thermostatted Lorentz gas the collision frequency increases with the magnitude of the applied field due to long-time correlations. We study this effect with computer simulations and confirm the presence of non-analytic terms in the field dependence of the collision frequency as predicted by kinetic theory.Comment: 6 pages, 2 figures. Submitted to Phys. Rev.

Wave packet autocorrelation functions for quantum hard-disk and hard-sphere billiards in the high-energy, diffraction regime

We consider the time evolution of a wave packet representing a quantum particle moving in a geometrically open billiard that consists of a number of fixed hard-disk or hard-sphere scatterers. Using the technique of multiple collision expansions we provide a first-principle analytical calculation of the time-dependent autocorrelation function for the wave packet in the high-energy diffraction regime, in which the particle's de Broglie wave length, while being small compared to the size of the scatterers, is large enough to prevent the formation of geometric shadow over distances of the order of the particle's free flight path. The hard-disk or hard-sphere scattering system must be sufficiently dilute in order for this high-energy diffraction regime to be achievable. Apart from the overall exponential decay, the autocorrelation function exhibits a generally complicated sequence of relatively strong peaks corresponding to partial revivals of the wave packet. Both the exponential decay (or escape) rate and the revival peak structure are predominantly determined by the underlying classical dynamics. A relation between the escape rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the counterpart classical system, previously known for hard-disk billiards, is strengthened by generalization to three spatial dimensions. The results of the quantum mechanical calculation of the time-dependent autocorrelation function agree with predictions of the semiclassical periodic orbit theory.Comment: 24 pages, 13 figure

Non-equilibrium Lorentz gas on a curved space

The periodic Lorentz gas with external field and iso-kinetic thermostat is equivalent, by conformal transformation, to a billiard with expanding phase-space and slightly distorted scatterers, for which the trajectories are straight lines. A further time rescaling allows to keep the speed constant in that new geometry. In the hyperbolic regime, the stationary state of this billiard is characterized by a phase-space contraction rate, equal to that of the iso-kinetic Lorentz gas. In contrast to the iso-kinetic Lorentz gas where phase-space contraction occurs in the bulk, the phase-space contraction rate here takes place at the periodic boundaries

Lattice gas with ``interaction potential''

We present an extension of a simple automaton model to incorporate non-local interactions extending over a spatial range in lattice gases. {}From the viewpoint of Statistical Mechanics, the lattice gas with interaction range may serve as a prototype for non-ideal gas behavior. {}From the density fluctuations correlation function, we obtain a quantity which is identified as a potential of mean force. Equilibrium and transport properties are computed theoretically and by numerical simulations to establish the validity of the model at macroscopic scale.Comment: 12 pages LaTeX, figures available on demand ([email protected]

The Kolmogorov-Sinai Entropy for Dilute Gases in Equilibrium

We use the kinetic theory of gases to compute the Kolmogorov-Sinai entropy per particle for a dilute gas in equilibrium. For an equilibrium system, the KS entropy, h_KS is the sum of all of the positive Lyapunov exponents characterizing the chaotic behavior of the gas. We compute h_KS/N, where N is the number of particles in the gas. This quantity has a density expansion of the form h_KS/N = a\nu[-\ln{\tilde{n}} + b + O(\tilde{n})], where \nu is the single-particle collision frequency and \tilde{n} is the reduced number density of the gas. The theoretical values for the coefficients a and b are compared with the results of computer simulations, with excellent agreement for a, and less than satisfactory agreement for b. Possible reasons for this difference in b are discussed.Comment: 15 pages, 2 figures, submitted to Phys. Rev.

Chaos properties and localization in Lorentz lattice gases

The thermodynamic formalism of Ruelle, Sinai, and Bowen, in which chaotic properties of dynamical systems are expressed in terms of a free energy-type function - called the topological pressure - is applied to a Lorentz Lattice Gas, as typical for diffusive systems with static disorder. In the limit of large system sizes, the mechanism and effects of localization on large clusters of scatterers in the calculation of the topological pressure are elucidated and supported by strong numerical evidence. Moreover it clarifies and illustrates a previous theoretical analysis [Appert et al. J. Stat. Phys. 87, chao-dyn/9607019] of this localization phenomenon.Comment: 32 pages, 19 Postscript figures, submitted to PR