727 research outputs found

    Entropy, Thermostats and Chaotic Hypothesis

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    The chaotic hypothesis is proposed as a basis for a general theory of nonequilibrium stationary states. Version 2: new comments added after presenting this talk at the Meeting mentioned in the Acknowledgement. One typo corrected.Comment: 6 page

    Separating Solution of a Quadratic Recurrent Equation

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    In this paper we consider the recurrent equation Λp+1=1pq=1pf(qp+1)ΛqΛp+1q\Lambda_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda_{q}\Lambda_{p+1-q} for p1p\ge 1 with fC[0,1]f\in C[0,1] and Λ1=y>0\Lambda_1=y>0 given. We give conditions on ff that guarantee the existence of y(0)y^{(0)} such that the sequence Λp\Lambda_p with Λ1=y(0)\Lambda_1=y^{(0)} tends to a finite positive limit as pp\to \infty.Comment: 13 pages, 6 figures, submitted to J. Stat. Phy

    Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems

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    We consider a dynamical system with state space MM, a smooth, compact subset of some Rn{\Bbb R}^n, and evolution given by TtT_t, xt=Ttxx_t = T_t x, xMx \in M; TtT_t is invertible and the time tt may be discrete, tZt \in {\Bbb Z}, Tt=TtT_t = T^t, or continuous, tRt \in {\Bbb R}. Here we show that starting with a continuous positive initial probability density ρ(x,0)>0\rho(x,0) > 0, with respect to dxdx, the smooth volume measure induced on MM by Lebesgue measure on Rn{\Bbb R}^n, the expectation value of logρ(x,t)\log \rho(x,t), with respect to any stationary (i.e. time invariant) measure ν(dx)\nu(dx), is linear in tt, ν(logρ(x,t))=ν(logρ(x,0))+Kt\nu(\log \rho(x,t)) = \nu(\log \rho(x,0)) + Kt. KK depends only on ν\nu and vanishes when ν\nu is absolutely continuous wrt dxdx.Comment: 7 pages, plain TeX; [email protected], [email protected], [email protected], to appear in Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 8, Issue

    Fluctuation theorem for stochastic dynamics

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    The fluctuation theorem of Gallavotti and Cohen holds for finite systems undergoing Langevin dynamics. In such a context all non-trivial ergodic theory issues are by-passed, and the theorem takes a particularly simple form. As a particular case, we obtain a nonlinear fluctuation-dissipation theorem valid for equilibrium systems perturbed by arbitrarily strong fields.Comment: 15 pages, a section rewritte

    Topics in chaotic dynamics

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    Various kinematical quantities associated with the statistical properties of dynamical systems are examined: statistics of the motion, dynamical bases and Lyapunov exponents. Markov partitons for chaotic systems, without any attempt at describing ``optimal results''. The Ruelle principle is illustrated via its relation with the theory of gases. An example of an application predicts the results of an experiment along the lines of Evans, Cohen, Morriss' work on viscosity fluctuations. A sequence of mathematically oriented problems discusses the details of the main abstract ergodic theorems guiding to a proof of Oseledec's theorem for the Lyapunov exponents and products of random matricesComment: Plain TeX; compile twice; 30 pages; 140K Keywords: chaos, nonequilibrium ensembles, Markov partitions, Ruelle principle, Lyapunov exponents, random matrices, gaussian thermostats, ergodic theory, billiards, conductivity, gas.

    The largest eigenvalue of rank one deformation of large Wigner matrices

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    The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration

    In-flight dissipation as a mechanism to suppress Fermi acceleration

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    Some dynamical properties of time-dependent driven elliptical-shaped billiard are studied. It was shown that for the conservative time-dependent dynamics the model exhibits the Fermi acceleration [Phys. Rev. Lett. 100, 014103 (2008)]. On the other hand, it was observed that damping coefficients upon collisions suppress such phenomenon [Phys. Rev. Lett. 104, 224101 (2010)]. Here, we consider a dissipative model under the presence of in-flight dissipation due to a drag force which is assumed to be proportional to the square of the particle's velocity. Our results reinforce that dissipation leads to a phase transition from unlimited to limited energy growth. The behaviour of the average velocity is described using scaling arguments.Comment: 4 pages, 5 figure

    Motion of a random walker in a quenched power law correlated velocity field

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    We study the motion of a random walker in one longitudinal and d transverse dimensions with a quenched power law correlated velocity field in the longitudinal x-direction. The model is a modification of the Matheron-de Marsily (MdM) model, with long-range velocity correlation. For a velocity correlation function, dependent on transverse co-ordinates y as 1/(a+|{y_1 - y_2}|)^alpha, we analytically calculate the two-time correlation function of the x-coordinate. We find that the motion of the x-coordinate is a fractional Brownian motion (fBm), with a Hurst exponent H = max [1/2, (1- alpha/4), (1-d/4)]. From this and known properties of fBM, we calculate the disorder averaged persistence probability of x(t) up to time t. We also find the lines in the parameter space of d and alpha along which there is marginal behaviour. We present results of simulations which support our analytical calculation.Comment: 8 pages, 4 figures. To appear in Physical Review

    Thermodynamic formalism for field driven Lorentz gases

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    We analytically determine the dynamical properties of two dimensional field driven Lorentz gases within the thermodynamic formalism. For dilute gases subjected to an iso-kinetic thermostat, we calculate the topological pressure as a function of a temperature-like parameter \ba up to second order in the strength of the applied field. The Kolmogorov-Sinai entropy and the topological entropy can be extracted from a dynamical entropy defined as a Legendre transform of the topological pressure. Our calculations of the Kolmogorov-Sinai entropy exactly agree with previous calculations based on a Lorentz-Boltzmann equation approach. We give analytic results for the topological entropy and calculate the dimension spectrum from the dynamical entropy function.Comment: 9 pages, 5 figure

    Damage spreading and coupling in Markov chains

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    In this paper, we relate the coupling of Markov chains, at the basis of perfect sampling methods, with damage spreading, which captures the chaotic nature of stochastic dynamics. For two-dimensional spin glasses and hard spheres we point out that the obstacle to the application of perfect-sampling schemes is posed by damage spreading rather than by the survey problem of the entire configuration space. We find dynamical damage-spreading transitions deeply inside the paramagnetic and liquid phases, and show that critical values of the transition temperatures and densities depend on the coupling scheme. We discuss our findings in the light of a classic proof that for arbitrary Monte Carlo algorithms damage spreading can be avoided through non-Markovian coupling schemes.Comment: 6 pages, 8 figure
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