Classical dynamics on graphs

We consider the classical evolution of a particle on a graph by using a time-continuous Frobenius-Perron operator which generalizes previous propositions. In this way, the relaxation rates as well as the chaotic properties can be defined for the time-continuous classical dynamics on graphs. These properties are given as the zeros of some periodic-orbit zeta functions. We consider in detail the case of infinite periodic graphs where the particle undergoes a diffusion process. The infinite spatial extension is taken into account by Fourier transforms which decompose the observables and probability densities into sectors corresponding to different values of the wave number. The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a Frobenius-Perron operator corresponding to a given sector. The diffusion coefficient is obtained from the hydrodynamic modes of diffusion and has the Green-Kubo form. Moreover, we study finite but large open graphs which converge to the infinite periodic graph when their size goes to infinity. The lifetime of the particle on the open graph is shown to correspond to the lifetime of a system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure

Viscosity in the escape-rate formalism

We apply the escape-rate formalism to compute the shear viscosity in terms of the chaotic properties of the underlying microscopic dynamics. A first passage problem is set up for the escape of the Helfand moment associated with viscosity out of an interval delimited by absorbing boundaries. At the microscopic level of description, the absorbing boundaries generate a fractal repeller. The fractal dimensions of this repeller are directly related to the shear viscosity and the Lyapunov exponent, which allows us to compute its values. We apply this method to the Bunimovich-Spohn minimal model of viscosity which is composed of two hard disks in elastic collision on a torus. These values are in excellent agreement with the values obtained by other methods such as the Green-Kubo and Einstein-Helfand formulas.Comment: 16 pages, 16 figures (accepted in Phys. Rev. E; October 2003

On the derivation of Fourier's law in stochastic energy exchange systems

We present a detailed derivation of Fourier's law in a class of stochastic energy exchange systems that naturally characterize two-dimensional mechanical systems of locally confined particles in interaction. The stochastic systems consist of an array of energy variables which can be partially exchanged among nearest neighbours at variable rates. We provide two independent derivations of the thermal conductivity and prove this quantity is identical to the frequency of energy exchanges. The first derivation relies on the diffusion of the Helfand moment, which is determined solely by static averages. The second approach relies on a gradient expansion of the probability measure around a non-equilibrium stationary state. The linear part of the heat current is determined by local thermal equilibrium distributions which solve a Boltzmann-like equation. A numerical scheme is presented with computations of the conductivity along our two methods. The results are in excellent agreement with our theory.Comment: 19 pages, 5 figures, to appear in Journal of Statistical Mechanics (JSTAT

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

Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases II: Open Systems

We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e. the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is finite in at least some directions, and with absorbing boundary conditions, the moving particle escapes the system with probability one. However, there is a set of zero Lebesgue measure of initial phase points for the moving particle, such that escape never occurs. Typically, this set of points forms a fractal repeller, and the Lyapunov spectrum is calculated here for trajectories on this repeller. For this calculation, we need the solution of the recently introduced extended Boltzmann equation for the nonequilibrium distribution of the radius of curvature matrix and the solution of the standard Boltzmann equation. The escape-rate formalism then gives an explicit result for the Kolmogorov Sinai entropy on the repeller.Comment: submitted to Phys Rev

Quantum fingerprints of classical Ruelle-Pollicot resonances

N-disk microwave billiards, which are representative of open quantum systems, are studied experimentally. The transmission spectrum yields the quantum resonances which are consistent with semiclassical calculations. The spectral autocorrelation of the quantum spectrum is shown to be determined by the classical Ruelle-Pollicot resonances, arising from the complex eigenvalues of the Perron-Frobenius operator. This work establishes a fundamental connection between quantum and classical correlations in open systems.Comment: 6 pages, 2 eps figures included, submitted to PR

Comparison of averages of flows and maps

It is shown that in transient chaos there is no direct relation between averages in a continuos time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincare map. In contrast to permanent chaos, results obtained from the Poincare map can even be qualitatively incorrect. The reason is that the return time between intersections on the Poincare surface becomes relevant. However, after introducing a true-time Poincare map, quantities known from the usual Poincare map, such as conditionally invariant measure and natural measure, can be generalized to this case. Escape rates and averages, e.g. Liapunov exponents and drifts can be determined correctly using these novel measures. Significant differences become evident when we compare with results obtained from the usual Poincare map.Comment: 4 pages in Revtex with 2 included postscript figures, submitted to Phys. Rev.

Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

Two-dimensional random Lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. In systems of finite spatial extension, the asymptotic decay of the survival probability is exponential and characterized by an escape rate, which can be related to the average positive Lyapunov exponent and to the dimension of the fractal repeller of the system. For infinite systems, the survival probability obeys a stretched exponential law of the form P(c,t)~exp(-Ct^{1/2}). The transition between the two regimes is studied and we show that, for a given trap density, the non-integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space. Nevertheless, the repeller remains fractal. We determine the special scaling properties of this fractal.Comment: 40 pages, 10 figures, preprint for Physica

Methods of calculation of a friction coefficient: Application to the nanotubes

In this work we develop theoretical and numerical methods of calculation of a dynamic friction coefficient. The theoretical method is based on an adiabatic approximation which allows us to express the dynamic friction coefficient in terms of the time integral of the autocorrelation function of the force between both sliding objects. The motion of the objects and the autocorrelation function can be numerically calculated by molecular-dynamics simulations. We have successfully applied these methods to the evaluation of the dynamic friction coefficient of the relative motion of two concentric carbon nanotubes. The dynamic friction coefficient is shown to increase with the temperature.Comment: 4 pages, 6 figure

Transport and dynamics on open quantum graphs

We study the classical limit of quantum mechanics on graphs by introducing a Wigner function for graphs. The classical dynamics is compared to the quantum dynamics obtained from the propagator. In particular we consider extended open graphs whose classical dynamics generate a diffusion process. The transport properties of the classical system are revealed in the scattering resonances and in the time evolution of the quantum system.Comment: 42 pages, 13 figures, submitted to PR