The existence of Burnett coefficients in the periodic Lorentz gas

The linear super-Burnett coefficient gives corrections to the diffusion equation in the form of higher derivatives of the density. Like the diffusion coefficient, it can be expressed in terms of integrals of correlation functions, but involving four different times. The power law decay of correlations in real gases (with many moving particles) and the random Lorentz gas (with one moving particle and fixed scatterers) are expected to cause the super-Burnett coefficient to diverge in most cases. Here we show that the expression for the super-Burnett coefficient of the periodic Lorentz gas converges as a result of exponential decay of correlations and a nontrivial cancellation of divergent contributions.Comment: 8 pages, no figure

Deterministic diffusion in flower shape billiards

We propose a flower shape billiard in order to study the irregular parameter dependence of chaotic normal diffusion. Our model is an open system consisting of periodically distributed obstacles of flower shape, and it is strongly chaotic for almost all parameter values. We compute the parameter dependent diffusion coefficient of this model from computer simulations and analyze its functional form by different schemes all generalizing the simple random walk approximation of Machta and Zwanzig. The improved methods we use are based either on heuristic higher-order corrections to the simple random walk model, on lattice gas simulation methods, or they start from a suitable Green-Kubo formula for diffusion. We show that dynamical correlations, or memory effects, are of crucial importance to reproduce the precise parameter dependence of the diffusion coefficent.Comment: 8 pages (revtex) with 9 figures (encapsulated postscript

A strong pair correlation bound implies the CLT for Sinai Billiards

For Dynamical Systems, a strong bound on multiple correlations implies the Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is derived for dynamically Holder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension.Comment: 13 page

Normal transport properties for a classical particle coupled to a non-Ohmic bath

We study the Hamiltonian motion of an ensemble of unconfined classical particles driven by an external field F through a translationally-invariant, thermal array of monochromatic Einstein oscillators. The system does not sustain a stationary state, because the oscillators cannot effectively absorb the energy of high speed particles. We nonetheless show that the system has at all positive temperatures a well-defined low-field mobility over macroscopic time scales of order exp(-c/F). The mobility is independent of F at low fields, and related to the zero-field diffusion constant D through the Einstein relation. The system therefore exhibits normal transport even though the bath obviously has a discrete frequency spectrum (it is simply monochromatic) and is therefore highly non-Ohmic. Such features are usually associated with anomalous transport properties

Decay of the Sinai Well in D dimensions

We study the decay law of the Sinai Well in $D$ dimensions and relate the behavior of the decay law to internal distributions that characterize the dynamics of the system. We show that the long time tail of the decay is algebraic ($1/t$), irrespective of the dimension $D$.Comment: 14 pages, Figures available under request. Revtex. Submitted to Phys. Rev. E.,e-mail: [email protected]

The characteristic exponents of the falling ball model

We study the characteristic exponents of the Hamiltonian system of $n$ ($\ge 2$) point masses $m_1,\dots,m_n$ freely falling in the vertical half line $\{q|\, q\ge 0\}$ under constant gravitation and colliding with each other and the solid floor $q=0$ elastically. This model was introduced and first studied by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic (Lyapunov) exponents of the above dynamical system are nonzero, provided that $m_1\ge\dots\ge m_n$ (i. e. the masses do not increase as we go up) and $m_1\ne m_2$

Nonequilibrium stochastic processes: Time dependence of entropy flux and entropy production

Based on the Fokker-Planck and the entropy balance equations we have studied the relaxation of a dissipative dynamical system driven by external Ornstein-Uhlenbeck noise processes in absence and presence of nonequilibrium constraint in terms of the thermodynamically inspired quantities like entropy flux and entropy production. The interplay of nonequilibrium constraint, dissipation and noise reveals some interesting extremal nature in the time dependence of entropy flux and entropy production.Comment: RevTex, 17 pages, 9 figures. To appear in Phys. Rev.

A multibaker map for shear flow and viscous heating

A consistent description of shear flow and the accompanied viscous heating as well the associated entropy balance is given in the framework of a deterministic dynamical system. A laminar shear flow is modeled by a Hamiltonian multibaker map which drives velocity and temperature fields. In an appropriate macroscopic limit one recovers the Navier-Stokes and heat conduction equations along with the associated entropy balance. This indicates that results of nonequilibrium thermodynamics can be described by means of an abstract, sufficiently chaotic and mixing dynamics. A thermostating algorithm can also be incorporated into this framework.Comment: 11 pages; RevTex with multicol+graphicx packages; eps-figure

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.

Upper bound for the time derivative of entropy for nonequilibrium stochastic processes

We have shown how the intrinsic properties of a noise process can set an upper bound for the time derivative of entropy in a nonequilibrium system. The interplay of dissipation and the properties of noise processes driving the dynamical systems in presence and absence of external forcing, reveals some interesting extremal nature of the upper bound.Comment: RevTex, 13 pages, 6 figure