abstracts
Abstracts - 1st Focus Week
The first focus week on NONEQUILIBRIUM SYSTEMS will take place in week two of the semester, from Monday June 2, 2008 to Friday June 6, 2008.
Frederico Bonetto, bonetto@math.gatech.edu
In recent years we have developed some perturbative tools to analyze Dynamical Systems. The goal is to obtain constructive and explicit expressions for the quantity of relevance having in mind mainly possible application to Nonequilibrium Statistical Mechanics. Among the results
are the construction of the SRB measure for Coupled Anosov Maps and for Anosov Flows. I'll review some of these results together with possible application to open problems in Dynamical Systems Theory. Recently, we have applied similar ideas to the case of Stochastic Dynamical Systems.
Giancarlo Benettin, benettin@math.unipd.it
The two-dimensional vs. the one-dimensional Fermi-Pasta Ulam problem
We shall introduce several two-dimensional FPU models, differing for the boundary conditions, and look in particular at the behavior in the thermodynamic limit. Accurate numerical computations indicate that:
(i) There are relevant qualitative differences between dimensions one and two.
(ii) In dimension two the equilibrium times do depend on the boundary conditions, but in any case they are much shorter than in dimension one. An extrapolation to the thermodynamic limit gives either 
or 
denoting the energy per degree of freedom.
(iii) The differences are related to differences in the structure of the Hamiltonian, if written in the normal modes coordinates.
Jean-Pierre Eckmann, Jean-Pierre.Eckmann@physics.unige.ch
A Model of Heat Transport
I will report on work with Pierre Collet. In it, we first define a deterministic scattering model for heat conduction which is continuous in space, and which has a Boltzmann type flavor, obtained by a closure based on memory loss between collisions. This model has, for stochastic driving forces at the boundary, close to Maxwellians, a unique non-equilibrium steady state. The techniques of proof, so far, only work for ‘equal‘ temperatures in a sense I will describe.
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Pierre Gaspard,
Heat conduction without mass transport is considered in a Hamiltonian multidisk hyperbolic billiard where energy transport takes place through binary collisions between neighboring particles. In a special limit, we show that heat conduction is controled by a master equation,
which allows us to compute estimates of heat conducitivity. The spectrum of Lyapunov exponents can also be obtained in this limit. This is a joint work with Thomas Gilbert.
Philippe Jacquet, Philippe.Jacquet@physics.unige.ch
Transport properties of a chain of dynamical quantum dots
In this talk I will present a model for electric and heat currents based on the scattering approach. The system is made of a chain of N quantum dots, each of which being coupled to a particle reservoir. The left and right ends of the system are moreover coupled to two particle reservoirs. These reservoirs can be described by any of the standardphysical distribution functions: Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein. I will argue that this chain of dynamical quantum dots can be seen, in particular, as an effective quantum version of the Eckmann-Young model with rotating discs. Considering the linear response regime I will first present the general transport properties of the system, discussing in particular the Onsager relations and the entropy production. Then I will analyse the chemical potential and temperature profiles as well as the electrical and heat currents, IL and JL, under the null currents condition:
Ii = Ji= 0 for i = 1, . . . ,N. The main results are related to some kind of universality. Finally, using the predictions of random matrix theory, I will argue that the currents are governed by Ohm's and Fourier's laws.
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Gianni Jona-Lasinio, gianni.jona@roma1.infn.it
Nonequilibrium thermodynamics: a self-contained macroscopic description of diffusive systems
From a fluctuation approach to irreversible phenomena developed in recent years one can extract a self-contained macroscopic description of stationary states where diffusion is the dominant process. Examples and simple theorems will be discussed, in particular the notion of equilibrium state from a nonequilibrium point of view.
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Antti Kupiainen, ajkupiai@mappi.helsinki.fi
Diffusion of energy in a coupled map lattice
We consider a lattice of coupled maps, each having a conserved ”energy”. We prove diffusion for the energy density of the coupled system by reducing it to a study of random walks in random environments.
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Oscar Lanford, lanford@math.ethz.ch
Discretization of expanding maps and percolation on a tree
We consider a smooth expanding map f of the circle to itself and discretize it by putting down a grid of N evenly spaced points on the circle, applying the map to each grid point, and rounding the result to the nearest grid point. Call the resulting map of the N–point grid to itself fN. We want to study the iterates of fN, for large N. One salient characteristic of these iterates is that their images shrink down as the order of iteration increases, in a way which becomes independent of N as N goes to infinity. We introduce a limiting regime in which this statement can be made precise and show that behavior in this limiting regime is governed by a particular percolation process on a binary tree. We derive a few preliminary facts about this process and call attention to some open questions.
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David Mukamel, fnmukaml@wisemail.weizmann.ac.il
Ordering and Criticality in one dimensional driven systems
Phase transitions, condensation and criticality in one-dimensional driven systems are discussed. Results derived for models of driven particles hopping between sites on a lattice, such as the Zero Range Process (ZRP), the Totally Asymmetric Simple Exclusion Process (TASEP) and generalizations of it are presented. Mechanisms leading to phenomena such as phase transitions which result in multiple condensates are demonstrated.
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Claude-Alain Pillet, (2 hours), pillet@univ-tln.fr
C* -dynamical systems and nonequilibrium quantum statistical mechanics
In the past, equilibrium statistical mechanics brought a lot to the theory of dynamical systems. More recently the dynamical systems way of thinking paid back by inspiring new ideas in nonequilibrium statistical mechanics. Some of these ideas could even make their ways to quantum statistical mechanics. The first part of this minicourse will be a gentle introduction to open quantum systems: a class of physical systems for which the above mentioned ideas have turned out to be very fruitful. In the second part I will present some recent applications: linear response and central limit theorem for locally interacting fermions and quantum fluctuations symetries (joint works with V. Jaksic, Y. Ogata and Y. Pautrat).
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Antonio Politi, Florence
Heat conductivity as a testing ground for the characterization of out-of-equilibrium steady states
In the last decade, numerical and theoretical studies of heat conduction in classical low-dimensional systems have been mainly focused on the characterization of the anomalous behaviour of the transport coefficient. Here, with reference to a stochastic model, we perform an exact analysis of the two-point correlations of the out-of-equilibrium steady state (both in the case of normal and anomalous conductivity). We show that the invariant measure can be effectively described as a product of independent Gaussian distributions aligned along directions that are localized in real space (with a power-law tail). Such predictions are compared with the numerical analysis of the
Fermi-Pasta-Ulam beta-model, finding a qualitative agreement.
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Errico Presutti, (2 hours), presutti@axp.mat.uniroma2.it
Persistence of randomness in macroscopic limits
I will discuss some physical phenomena where macroscopic systems exhibit stochastic behavior, in particular in the first lecture I will discuss the “spinodal decomposition”and in the second “interface fluctuations” and their relevance in tunnelling phenomena.
In Lecture 1 I will define and study the Glauber dynamics in Ising systems with mean field interactions. This can be reduced to a simple random walk with state space a finite number of points on the line. In the macroscopic limit the evolution becomes deterministic and ruled by a ODE. The case of interest for spinodal decomposition is when the ODE has one unstable and two stable stationary solutions and the initial state is the unstable one. I will show that if we observe the Markov process for sufficiently long times we will see deviations from its deterministic limit behavior and convergence toward one of the two stable solutions.
In Lecture 2 I will discuss some stochastic PDEs in one spatial dimension. In the absence of noise they have a one parameter familyz (the parameter is denoted by
) of stationary solutions which are fronts (monotonically increasing functions) and one is obtained from the other by translations 
. I will consider as initial state the front ¯m0 (which would be invariant in the absence of noise) and study the limit case where the noise is small but the time is long. I will prove that on a suitable limit the shape of the front is unchanged but its location becomes random and it is described by a brownian motion. I will then conclude with some recent works on large deviations, in particular on the probability to observe a drift in the limit motion of the front and applications to tunnelling.
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Luc Rey-Bellet, lr7q@math.umass.edu
Large devaitions for billiards and nonuniformly hyperbolic dynamical systems
This is a joint work with Lai-Sang Young. The famous Fluctuation Theorem refers to a symmetry of the large deviations of the entropy production. For deterministic (thermostatted) nonequilibrium systems it is therefore important to establish large deviation principles. For uniformly hyperbolic dynamical systems this is a well-known consequence of the thermodynamic formalism. In this talk we will discuss large deviation results for dynamical systems which are nonuniformly hyperbolic, such as Sinai billiards with or without external forces, Henon maps, quadratic maps, etc. . . .
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Lamberto Rondoni, rondoni@calvino.polito.it
Onset of diffusive behaviour in confined transport systems
We investigate the onset of diffusive behaviour in polygonal channels for discs of finite size,modelling simple microporous membranes. It is well established that the point-particle case displays anomalous transport, because of slow correlation decay in the absence of defocussing collisions. We investigate which features of point-particle transport survive in the case of finite-sized particles. We quantify the time scales over which the transport of discs shows features typical of the point particles, or is driven toward diffusive behaviour. We illustrate how, and at what stage a normal behaviour, consistent with kinetic theory sets in, as particle numbers grow and mean free paths diminish.
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David Ruelle, (2 hours), ruelle@ihes.fr
Nonequilibrium Statistical Mechanics and Smooth Dynamical Systems
There are many different approaches to nonequilibrium statistical mechanics, involving different mathematical idealizations. The relation between these approaches remains often unclear, reflecting the fact that nonequilibrium statistical mechanics (not close to equilibrium) is currently still in a formative stage. The approach that we wish to discuss here relates classical nonequilibrium statistical mechanics with the ergodic theory of a smooth dynamical system on a compact manifold. This approach excludes quantum systems, but is otherwise physically quite reasonable. Its chief interest is that it can use the powerful techniques and results of the theory of differentiable dynamical systems (in particular hyperbolic dynamical systems). This allows one to formulate, and in part solve, problems that remain otherwise quite inaccessible.
David P. Sanders, dps@hp.fciencias.unam.mx
Rare events and long-range correlations in systems with many random walkers
In the first half of this talk, I will describe analytical and numerical results on mean first-passage times to rare events in systems consisting of many independent random walkers [1]. The rare events in question consist of large density fluctuations; in particular, we study how long we must wait to see such fluctuations occur. In the second half, the model is modified so that the walkers also carry an ”energy”, i.e. a conserved quantity which is locally exchanged between the walkers at each time step. I will present analytical results on the long-range correlations which develop in the coupled mass–energy transport system, when the system is subjected to a gradient of ”temperature”.
[1] How rare are diffusive rare events?, David P. Sanders and Hern´an Larralde, Europhysics Letters, accepted for publication.
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Benjamin Schlein, schlein@math.lmu.de
Dynamics of Bose-Einstein condensates
In this talk I am going to report on recent results, obtained in collaboration with L. Erdos and H.-T. Yau, concerning the time evolution of Bose-Einstein condensates. In particular, I am going to present a rigorous derivation, from first principle many body quantum dynamics, of a one-particle nonlinear Schroedinger equation, known as the Gross-Pitaevskii equation, describing the dynamics of the condensate wave function.
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Angelo Vulpiani, Angelo.Vulpiani@roma1.infn.it
Some aspects of the Fluctuation-Dissipation Relation
We illustrate the relation between the relaxation of spontaneous fluctuations, and the response to an external perturbation. The Fluctuation-Dissipation Relation (FDR) has been originally developed in the framework of statistical mechanics of Hamiltonian systems, nevertheless a generalized FDR holds under rather general hypotheses, regardless of the Hamiltonian, or equilibrium nature of the system. Some examples, beyond the standard applications of statistical mechanics, are discussed: fluids and granular media.
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Lai-Sang Young, lsy@courant.nyu.edu
Reliability of Neural Oscillator Networks
A fluctuating stimulus I(t) is presented to a network of oscillators, driving it out of equilibrium and eliciting a response R(t). Depending on the network’s initial state, R(t) may vary. Reliability refers to the degree of trial-to-trial variability when I(t) is presented multiple times. We discuss this problem in the context of neuroscience, where it arises naturally and where intrinsically active neurons are sometimes modeled as phase oscillators. Results of a part numerical, part theoretical study will be reported.