727 research outputs found

### Entropy, Thermostats and Chaotic Hypothesis

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

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

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

We consider a dynamical system with state space $M$, a smooth, compact subset
of some ${\Bbb R}^n$, and evolution given by $T_t$, $x_t = T_t x$, $x \in M$;
$T_t$ is invertible and the time $t$ may be discrete, $t \in {\Bbb Z}$, $T_t =
T^t$, or continuous, $t \in {\Bbb R}$. Here we show that starting with a
continuous positive initial probability density $\rho(x,0) > 0$, with respect
to $dx$, the smooth volume measure induced on $M$ by Lebesgue measure on ${\Bbb
R}^n$, the expectation value of $\log \rho(x,t)$, with respect to any
stationary (i.e. time invariant) measure $\nu(dx)$, is linear in $t$, $\nu(\log
\rho(x,t)) = \nu(\log \rho(x,0)) + Kt$. $K$ depends only on $\nu$ and vanishes
when $\nu$ is absolutely continuous wrt $dx$.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

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

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

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

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

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

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

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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