Federico Bonetto

Aspects of ergodic, qualitative, and statistical theory of motion

1 General Qualitative Properties.- 2 Ergodicity and Ergodic Points.- 3 Entropy and Complexity.- 4 Markovian Pavements.- 5 Gibbs Distributions.- 6 General Properties of Gibbs and SRB Distributions.- 7 Analyticity, Singularity and Phase Transitions.- 8 Special Ergodic Theory Problems in Nonchaotic Dynamics.- 9 Some Special Topics in KAM Theory.- 10 Special Problems in Chaotic Dynamics.- A Nonequilibrium Thermodynamics? Twenty-Seven Comments.- Name Index.- Citations Index.

Fourier's law for a harmonic crystal with self-consistent stochastic reservoirs

We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the “exterior” left and right heat baths are at specified values TLand TR, respectively, while the temperatures of the “interior” baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fouriers law with a finite positive thermal conductivity which can also be computed using the Green–Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d>>1, the stationary infinite volume heat conductivity behaves as (ldd)−1where ldis the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

A fluctuation theorem for non-equilibrium relaxational systems driven by external forces

We discuss an extension of the fluctuation theorem to stochastic models that, in the limit of zero external drive, are not able to equilibrate with their environment, extending results presented by Sellitto (cond-mat/9809186). We show that if the entropy production rate is suitably defined, its probability distribution function verifies the Fluctuation Relation with the ambient temperature replaced by a (frequency-dependent) effective temperature. We derive modified Green-Kubo relations. We illustrate these results with the simple example of an oscillator coupled to a nonequilibrium bath driven by an external force. We discuss the relevance of our results for driven glasses and the diffusion of Brownian particles in out of equilibrium media and propose a concrete experimental strategy to measure the low frequency value of the effective temperature using the fluctuations of the work done by an ac conservative field. We compare our results to related ones that appeared in the literature recently.We discuss an extension of the fluctuation theorem to stochastic models that, in the limit of zero external drive, are not able to equilibrate with their environment, extending earlier results of Sellitto. We show that if the entropy production rate is suitably defined, its probability distribution function verifies the fluctuation relation with the ambient temperature replaced by a (frequency dependent) effective temperature. We derive modified Green–Kubo relations. We illustrate these results with the simple example of an oscillator coupled to a non-equilibrium bath driven by an external force. We discuss the relevance of our results for driven glasses and the diffusion of Brownian particles in out-of-equilibrium media and propose a concrete experimental strategy for measuring the low frequency value of the effective temperature using the fluctuations of the work done by an ac conservative field. We compare our results to related ones that appeared in the literature recently.

Global and local) fluctuations of phase space contraction in deterministic stationary nonequilibrium.

We studied numerically the validity of the fluctuation relation introduced in Evans et al. [Phys. Rev. Lett. 71, 2401-2404 (1993)] and proved under suitable conditions by Gallavotti and Cohen [J. Stat. Phys. 80, 931-970 (1995)] for a two-dimensional system of particles maintained in a steady shear flow by Maxwell demon boundary conditions [Chernov and Lebowitz, J. Stat. Phys. 86, 953-990 (1997)]. The theorem was found to hold if one considers the total phase space contraction sigma occurring at collisions with both walls: sigma=sigma( upward arrow )+sigma( downward arrow ). An attempt to extend it to more local quantities sigma( upward arrow ) and sigma( downward arrow ), corresponding to the collisions with the top or bottom wall only, gave negative results. The time decay of the correlations in sigma( upward arrow, downward arrow ) was very slow compared to that of sigma. (c) 1998 American Institute of Physics.

Note on the Kaplan-Yorke dimension and linear transport coefficients

A number of new relations between the Kaplan–Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small but non-zero external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the linear transport coefficients in terms of the Kaplan–Yorke dimension is derived. Alternatively, the Kaplan–Yorke dimension for a dissipative macroscopic system can be expressed in terms of the linear transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good.

Beta function and anomaly of the Fermi surface for a

We derive a perturbation theory, based on the renormalization group, for the Fermi surface of a one dimensional system of fermions in a periodic potential interacting via a short range, spin independent potential. The infrared problem is studied by writing the Schwinger functions in terms of running couplings. Their flow is described by a Beta function, whose existence and analyticity as a function of the running couplings is proved. If the fermions are spinless we prove that the Beta function is vanishing and the renormalization flow is bounded for any small interaction. If the fermions are spinning the Beta function is not vanishing but, if the conduction band is not filled or half filled and the interaction is repulsive, it is possible again to control the flow proving the partial asymptotic freedom of the theory. This is done showing that the Beta function is partially vanishing using the exact solution of the Mattis model, which is the spin analogue of the Luttinger model. In both these cases Schwinger functions are anomalous so that the system is a “Luttinger liquid”. Our results extend the work in [B.G.P.S.], where neither spin nor periodic potential were considered; an explicit proof of some technical results used but not explicitly proved there is also provided.

d=1

We study a d-dimensional coupled map lattice consisting of hyperbolic toral automorphisms (Arnold cat maps) that are weakly coupled by an analytic map. We construct the Sinai-Ruelle-Bowen measure for this system and study its marginals on the tori. We prove that they are absolutely continuous with respect to Lebesgue measure if and only if the coupling satisfies a non-degeneracycondition.

system of interacting fermions in a periodic potential

In this note we discuss a paradigmatic example of interacting particles subject to non conservative external forces and to the action of thermostats consisting of external (finite) reservoirs of particles. We then consider a model of granular materials of interest for experimental tests that had recently attracted lot of attentions. This model can be reduced to the previously discussed example under a number of assumptions, in particular that inelasticity due to internal collisions can be neglected for the purpose of measuring the large deviation functional for entropy production rate. We show that if the restitution coefficient in the granular material model is close to one, then the required assuptions are verified on a specific time scale and we predict a fluctuation relation for the entropy production rate measured on the same time scale.In this note we discuss a paradigmatic example of interacting particles subject to non conservative external forces and to the action of thermostats consisting of external (finite) reservoirs of particles. We then consider a model of granular materials of interest for experimental tests that had recently attracted lot of attentions. This model can be reduced to the previously discussed example under a number of assumptions, in particular that inelasticity due to internal collisions can be neglected for the purpose of measuring the large deviation functional for entropy production rate. We show that if the restitution coefficient in the granular material model is close to one, then the required assuptions are verified on a specific time scale and we predict a fluctuation relation for the entropy production rate measured on the same time scale.

Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature