ICMP 2003

Nonequilibrium statistical mechanics

Session organized by G. Gallavotti (Rome), S. Olla (Cergy-Pontoise).


L. BarreiraLisbon
L. BertiniRoma 1
C. ForsterVienna
P. GaspardBruxelles
F. GolseParis
J. LebowitzRutgers
L. Rey-BelletAmherst
D. UeltschiUniversity of Arizona
H. T. YauNew York
Luís Barreira
Departamento de Matemática
Instituto Superior Técnico
1049-001 Lisboa

Poincaré recurrence: old and new

The classical Poincaré recurrence only gives information of qualitative nature. On the other hand it is clearly a matter of intrinsic difficulty and not of lack of interest that less is known concerning the quantitative behavior of recurrence. I will discuss recent developments obtained jointly with Benoît Saussol. These include the almost everywhere coincidence between the recurrence rate and the pointwise dimension in the case of hyperbolic dynamics. I will also discuss the almost product structure of recurrence, which closely imitates the product structure provided by the families of local stable and unstable manifolds, and the almost product structure of hyperbolic measures.

Lorenzo Bertini
Dipartimento di Matematica
Università di Roma La Sapienza
P.le A. Moro 2
00185 Roma

Large deviations for boundary driven lattice gases

We describe a dynamical large deviations for the empirical measure in stochastic lattice gases which describes the asymptotic probability of fluctuations from the hydrodynamic scaling limit. We then introduce the quasi potential by a variational problem on the rate functional. If the system is in contact with particles reservoirs at the boundary with different chemical potential, the dynamic is not reversible and this variational problem is non trivial. In the case of one dimensional symmetric simple exclusion process, we characterize the solution obtaining a dynamical proof of a formula found Derrida, Lebowitz, and Speer via combinatorial techniques

[Collaboration with A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim]

Christina Forster
Institute for Experimental Physics
University of Vienna
1090 Vienna, Boltzmanngasse 5

Analysis of Lyapunov modes in real and reciprocal space

The phase-space trajectory of many-body systems is Lyapunov unstable. For hard-particle systems the perturbations associated with specific Lyapunov exponents exhibit spacial structures to which we refer as Lyapunov modes. The mode belonging to the maximum exponent is strongly localized in physical space: only a small fraction of the particles contributes to the norm of the associated tangent vector at any instant of time. This ratio vanishes in the thermodynamic limit. In contrast, the perturbations generating the smallest-positive Lyapunov exponents are delocalized and have a wave-like appearance reminiscent of the modes of fluctuating hydrodynamics. The degeneracy of the associated exponents is determined by the orthonormality, the spatial symmetry, and the polarization (transverse or longitudinal) of the perturbations. The transverse modes do not propagate, the longitudinal modes do with a velocity of about 1/3 of the sound velocity. As expected, the amplitudes of pure modes scale with the inverse square root of the particle number. Mixed degenerate modes are analysed by Fourier transform techniques and are found to behave in accordance with theoretical expectations. The dispersion relations allow the construction of the relevant part of the Lyapunov spectrum for very large systems close to the thermodynamic limit.

Pierre Gaspard

The fractality of the hydrodynamic modes of diffusion

Transport by normal diffusion can be decomposed into hydrodynamic modes which relax exponentially toward the equilibrium state. In chaotic systems with two degrees of freedom, the fine scale structures of these modes are singular and fractal, characterized by a Hausdorff dimension given in terms of Ruelle's topological pressure. For long-wavelength modes, the Hausdorff dimension is related to the diffusion coefficient and the Lyapunov exponent. In the infinite-wavelength limit, the hydrodynamic modes lead to the nonequilibrium steady states, which also present a singular character. This singular character is a consequence of the mixing property of the dynamics. These results are illustrated with several systems such as the hard-disk and Yukawa-potential Lorentz gases.

François Golse
Institut Universitaire de France
& Laboratoire Jacques-Louis Lions
Boite Courrier 187
75252 Paris Cedex 05

On the statistics of free-path lengths for the periodic Lorentz gas

Consider the motion of a gas of point particles in a periodic array of spherical obstacles. Collisions involving two or more particles are neglected; only the collisions between the particles and the obstacles are taken into account. This talk reviews some results bearing on the distribution of free-path lengths for these particles, more precisely (1) upper and lower bounds for that distribution in any space dimension, and (2) the asymptotic evaluation of the tail of that distribution in the small obstacle limit, in space dimension two. Applications to kinetic theory are discussed.

Joel L. Lebowitz
Center for Mathematical Sciences Research
Rutgers, The State University
110 Frelinghuysen Road
Piscataway, NJ 08854-8019

Large deviations and fluctuations in some model stationary nonequilibrium systems

I will present exact expressions for the probability of macroscopic density profiles corresponding to large deviations and/or small fluctuations from typical profiles in some stationary nonequilibrium model systems. These are one dimensional lattice systems with exclusion dynamics (symmetric or asymmetric) in contact with infinite reservoirs having different densities. In contrast to those equilibrium systems, the large deviation functionals for these systems are highly non-local. The fluctuation field may or may not be obtainable from the large deviation function and may or may not be Gaussian. In the maximal current phase of the asymmetric simple exclusion process the fluctuation field is the sum of the derivative of a Brownian motion and of a Brownian excursion. (This decomposition may hold also for a larger class of processes.)

[Work done jointly with B. Derrida, E. Speer and C. Enaud.]

Luc Rey Bellet
Lederle Graduate Research Tower
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA 01003

Non equilibrium statistical mechanics of open classical systems

We consider a chain of anharmonic oscillators coupled to two reservoirs modeled by free phonons fields at positive and different temperatures. We discuss existence and uniqueness of stationary states, rate of convergence to the stationary state, and heat flow, entropy production and their fluctuations (Gallavotti-Cohen fluctuation theorem).

Daniel Ueltschi
University of Arizona

On non-equilibrium stationary states between reservoirs of quantum particles

Two reservoirs in contact give rise to a non-equilibrium stationary state. In the case where reservoirs contain non-interacting fermions, and the contact involves local interactions, the stationary state is characterized by a convergent perturbative series (Dyson series). We compute energy and particle flows at lowest orders, and establish properties such as strictly positive entropy production.

[Collaboration with J. Froehlich and M. Merkli; math-ph/0212062]

Horng-Tzer Yau
Courant Institute of Mathematical Sciences
251 Mercer Street
New York University
New York, NY, 10012

Diffusion of random Schrödinger equation in scaling limits

We prove that the random Schrödinger equation converges to a heat equation in scaling limits in dimension d=3.


E. Celeghini
Firenze and INFN

Nonequilibrium quantum statistics

E. Carlen
Georgia Tech., USA

Y. Elskens, M. Kiessling, V. Ricci
CNRS Marseille; Rutgers; Roma I

The Vlasov limit for a relativistic particle system interacting with a scalar wave field

Y. Elskens
CNRS Marseille

The analogue of Van Kampen-Case modes for a finite number of degrees of freedom

Y. Elskens, D. Escande
CNRS Marseille

Derivation of the Fokker-Planck equation from strongly chaotic hamiltonian dynamics

I. Lebed
CAHI, Moscow

The pair function theory in hydrdynamics

F. Dunlop

Stationary states of one-dimensional interfaces

S. Grobinsky, G. Schutz, H. Spohn

Condensation in the zero range process: stationary and dynamical properties

P. Kocian, K. Lendi

PCI, Impact of Von Neumann conditions for density operators of kernel and inhomogeneity in non-Markovian quantum master equations

J.P. Eckmann, E. Zabey

Strange heat flux in (an)harmonic networks

J.P. Eckmann, E. Moses, D. Sergi
Univ. of Geneva and Weizmann Institute

Specialists and generalists in mathematical physics