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TRANSCRIPT
WORKSHOP
PROGRESS IN NON EQUILIBRIUMSTATISTICAL MECHANICS
Nice, June 8-12, 2015
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Participants
• Christophe Bahadoran [email protected]
• Giada Basile [email protected]
• Julien Barre [email protected]
• Michel Bauer [email protected]
• Jeremie Bec [email protected]
• Cedric Bernardin [email protected]
• Lorenzo Bertini [email protected]
• Thierry Bodineau [email protected]
• Freddy Bouchet [email protected]
• Raphal Chetrite [email protected]
• Abhishek Dhar [email protected]
• Roland Diel [email protected]
• Tadahisa Funaki [email protected]
• Thierry Goudon [email protected]
• Francois Huveneers [email protected]
• Milton Jara [email protected]
• Konstantin Khanin [email protected]
• Claudio Landim [email protected]
• Vivien Lecomte [email protected]
• Paul Lemire [email protected]
• Jani Lukkarinen [email protected]
• Kirone Mallick [email protected]
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• Matteo Marcozzi [email protected]
• Antoine Mellet [email protected]
• Cesare Nardini [email protected]
• Byron Jimenez Oviedo [email protected]
• Stefano Olla [email protected]
• Nicolas Perkowski [email protected]
• Christophe Pocquet [email protected]
• Sylvain Prolhac [email protected]
• Herbert Spohn [email protected]
• Gabriel Stoltz [email protected]
• Hugo Touchette [email protected]
• Dario Vincenzi [email protected]
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Schedule
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Titles and Abstracts
• Christophe Bahadoran (University of Clermont Ferrand)
Phase transitions in site-disordered asymmetric exclusion and zero-range processes
We review recent and ongoing work with T. Bodineau and T. Mountford/K. Ravis-hankar/E. Saada We consider TASEP and general asymmetric ZRP with site disorder.(”general” means that we do not restrict to constant jump rate or totally asymmetricjumps) We address the following questions: for ZRP, convergence to the critical in-variant measure, supercritical hydrodynamics, quenched strong local equilibrium. ForTASEP, existence of a flat segment for the current-density function via renormalization.
• Giada Basile (University of Roma La Sapienza)
Ficks law for the Lorentz gas in a low density regime
A simple microscopic model to validate the Ficks law is the Lorentz gas, namely a sys-tem of non interacting light particles in a distribution of scatterers, in contact with twomass reservoirs. We show that, in a low density regime, there exists a unique stationarysolution for the microscopic dynamics which converges to the stationary solution ofthe heat equation. In the same regime the macroscopic current in the stationary stateobeys the Ficks law.
• Michel Bauer (CEA Saclay)
Affinity and Fluctuations in a Mesoscopic Noria
Cycle affinities play an important in out-of-equilibrium phenomena. We shall explain,in the context of random processes, their remarkable invariance under a number of nat-ural probabilistic constructions and their intimate relationship with first passage non-equilibrium fluctuation relations.
• Lorenzo Bertini (University Roma La Sapienza)
Large deviations with respect to motion by curvature
Consider the Allen-Cahn equation in space dimensions d = 2 or d= 3. Under diffusiverescaling of space and time, for suitably prepared initial data the limiting dynamicsis described by the motion by mean curvature of the interface between the two stablephases. We consider a stochastic perturbation of the Allen-Cahn equation and analyzeits large deviations asymptotic in such sharp interface limit. The corresponding ratefunction is finite only when there exists a d-1 interface between the two stable phasesand can be written as the sum of two terms: the first takes into account ”wrong” mo-tions of the interface while the second is due to the possible occurrence of nucleation
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events. Our result relies on a previous analysis on the variational convergence of theaction functional associated to the Allen-Cahn equation and uses tools of the geometricmeasure theory.
• Thierry Bodineau (Ecole Polytechnique and CNRS)
Tagged particle in a deterministic dynamics of hard spheres
We consider a tagged particle in a diluted gas of hard spheres. Starting from the hamil-tonian dynamics of particles in the Boltzmann-Grad limit, we will show that the taggedparticle follows a Brownian motion after an appropriate rescaling. We use the linearBoltzmann equation as an intermediate level of description for one tagged particle in agas close to global equilibrium.
• Freddy Bouchet (ENS Lyon and CNRS)
Large deviation theory and the EyringKramers formula for non gradient dynamics.Applications to abrupt transitions for turbulent atmosphere jets
(joint with Julien Reygner, Eric Simonnet, and Tomas Tangarife)
We will present the proof of a new formula for the transition rates between two basinsof attraction of dynamical systems with weak noises, for non gradient dynamics, inarbitrary dimensions. This formula extend both the large deviation results of Freidlin-Wentzell theory (by computing the prefactor of the large deviation estimate), and theEyring-Kramers formula valid only for gradient dynamics. We discuss applications toturbulent flows. For the computation of transitions rates for the metastable turbulentdynamics of atmosphere jets, described by the stochastic barotropic quasi-geostrophicmodel, we will discuss analytical results (based on averaging) and numerical results(based on the adaptive multilevel splitting algorithm).
• Abhishek Dhar (ICTS Bangalore)
Understanding anomalous transport in one-dimensional systems through fluctuatinghydrodynamics
The theory of fluctuating hydrodynamics makes detailed predictions on the form ofequilibrium correlations of conserved quantities in one-dimensional anharmonic chains.Using the connection between transport coefficients and equilibrium correlations sug-gested by the Green-Kubo formula, one then hopes to be able to make predictions ontransport properties. In this talk, the theoretical predictions of fluctuating hydrodynam-ics are compared with direct equilibrium simulation results for the Fermi-Pasta-Ulamchain. Some issues on connecting this with results of non-equilibrium simulations isdiscussed. It is also explained, within the scope of hydrodynamic theory, why the Rotormodel shows normal transport.
• Tadahisa Funaki (University of Tokyo)
Scaling limits for weakly pinned Gaussian random fields under the presence of twopossible candidates
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We study the scaling limit and prove the law of large numbers for weakly pinned Gaus-sian random fields under the critical situation that two possible candidates of the limitsexist at the level of large deviation principle, under the condition that the strength ofpinning is sufficiently large and the spatial dimension d satisfies d≥ 3. The talk is basedon joint works with E. Bolthausen and T. Chiyonobu, see http://arxiv.org/abs/1406.7766
• Francois Huveneers (University Paris Dauphine)
Lack of thermalization in quantum chains: Many-Body Localization
Some isolated quantum systems may fail to be their own heat bath (= fail to thermalize).This phenomenon, dubbed as Many-Body Localization, has the same physical originas the (one-body) Anderson localization. It is a topic of rather intense investigationssince about ten years. In the talk, I’ ll describe the basic phenomenology as well as partof my own work on the topic (from collaborations with W. De Roeck, M. Mueller, M,Schiulaz, D. Abanin).
• Milton Jara (IMPA Rio de Janeiro)
From anomalous to normal diffusion on one-dimensional stochastic chains
Conservation of momentum plays a fundamental role on the non-equilibrium propertiesof one-dimensional chains of anharmonic oscillators. This claim can be made rigorousfor harmonic chains subjected to stochastic perturbations. If the stochastic perturba-tion conserves both energy and momentum, energy superdiffuses. If the stochasticperturbation conserves only energy, the energy diffuses. Introducing a parameter thatregulates the strength of the violation of momentum conservation, we obtain a familyof operators connecting the fractional Laplacian that governs the anomalous diffusionand the usual Laplacian.
• Konstantin Khanin (University of Toronto)
Directed Polymers and the random Hamilton-Jacobi equation
• Claudio Landim (IMPA Rio de Janeiro)
Condensing zero-range processes
Zero-range processes with decreasing jump rates exhibit a condensation transition,where a positive fraction of all particles condenses on a single lattice site when thetotal density exceeds a critical value.
Consider a condensing zero-range process with N particles evolving on a fixed and fi-nite set S. In the diffusive time scale N2 the fraction of particles at each site evolves asa diffusion whose drift is unbounded. The process remains absorbed at the boundaryonce it attains it, and it performs, after this hitting time, a diffusion on a lower dimen-sional simplex, similar to the original one, until all particles concentrate on a singlesite.
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In a longer time scale N1+α , α > 1, the site which concentrates all particles evolvesas a random walk on S whose transition rates are proportional to the capacities of theunderlying random walk.
• Vivien Lecomte (CNRS and University of Paris 7)
Symmetries of large deviations of additive observables: what one gains from forgettingprobabilities and turning to the quantum world
The study of rare or anomalous fluctuations in classical, stochastic systems has leadin recent years to the understanding of dynamical phase transitions occurring in phe-nomena where different classes of trajectories enter in competition. Example systemsinclude ’exclusion processes’ (lattice gases in which particles interact only throughan exclusion rule: particles cannot occupy the same site). In such systems, jammedand non-jammed histories can compete. The dynamical phase transition is reflectedat the mathematical level in a singularity of a large deviation function. Although be-ing classical, the dynamics of such systems can be mapped to the thermodynamics ofan isolated quantum spin chain. Classical ’rare events’ are mapped to quantum typ-ical configurations. This mapping is known at the formal level for many years, butit has not been fully exploited. We discuss (i) how the understanding of finite-sizeproperties of the classical dynamical phase transition brings a new light on an examplequantum phase transition and (ii) how standard quantum rotational symmetries allowto map non-equilibrium to equilibrium current fluctuations in the SSEP driven out-of-equilibrium by its boundaries, giving an answer to the existence or not of singularities.
• Jani Lukkarinen (University of Helsinki)
Wick polynomials and time-evolution of cumulants
In this joint work with Matteo Marcozzi, we show how Wick polynomials of randomvariables can be defined combinatorially as the unique choice which removes all ”inter-nal contractions” from the related cumulant expansions, also in a non-Gaussian case.We d iscuss how an expansion in terms of the Wick polynomials can be used for deriva-tion of a hierarchy of equations for the time-evolution of cumulants. These methods arethen applied to simplify the formal derivation of the Boltzmann-Peierls equation in thekinetic scaling limit of the discrete nonlinear Schrodinger equation (DNLS) with suit-able random initial data. We also present a reformulation of the standard perturbationexpansion using cumulants which could simplify the problem of a rigorous derivationof the Boltzmann-Peierls equation by separating the analysis of the solutions to theBoltzmann-Peierls equation from the analysis of the corrections. This latter scheme isgeneral and not tied to the DNLS evolution equations.
• Kirone Mallick (CEA Saclay)
Tagged particle in single-file diffusion
Single-file diffusion is a one-dimensional interacting infinite-particle system in whichthe order of particles never changes. The mean-square displacement of a tagged particle
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(or tracer) in the single-file exhibits an anomalously slow sub-diffusive growth. Westudy the statistics of the tracer’s position using Macroscopic Fluctuation Theory; wecompute the large deviation function for impenetrable Brownian particles and for anarbitrary single-file system, we apply perturbation techniques and derive an explicitformula for the variance in terms of the transport coefficients.
• Antoine Mellet (University of Maryland)
Anomalous diffusion phenomena: A kinetic approach
We will review some results concerning the derivation of fractional diffusion equa-tions from kinetic equations and in particular some applications to the description ofanomalous energy transport in FPU chains.
• Cesare Nardini (University of Edinburgh)
Kinetic Theory for driven out of equilibrium long-range interacting systems and 2Dturbulence
Long-range interacting systems include gravitational systems, plasma in the low den-sity limit, two-dimensional and simple models for geophysical fluids. In many physicalcontext, long-range interacting systems are found to be out of equilibrium because ofexternal drving. Examples come from climate dynamics, plasma physics and, recently,experimental setups with cold-atoms driven by laser light. Geophysical flows, for ex-ample, are characterised by their self-organisation into large scale coherent structuressuch as jet-streams, cyclones and anti-cyclones. The description of these structures,of their evolution and the fluctuations they undergo are outstanding problems at theinterface between climate science and statistical mechanics. In order to address thedescription of driven long-range interacting systems in a theoretical way, we concen-trate in this talk on models as simple as possible that still retain the following twomain characteristics: non-local (i.e. long-range) nature of the interactions and brokendetailed balance (i.e. non-equilibrium dynamics). We present results both for particlesystems and two-dimensional flows and we show that their dynamics can be describedvery accurately in the limit where there is a separation of time-scales between the evo-lution of the mean state and the evolution of the fluctuations around it. The maintheoretical tool developed is kinetic theory. We will in particular give a generalisationof the Lenard-Balescu equation to situations where detailed balance is broken and theaccuracy of the results obtained will be discussed through comparisons with direct nu-merical simulations. Ongoing work on the refinement of kinetic theory that permits todescribe fluctuations of the mean state around its average will also be discussed.
• Stefano Olla (university Paris Dauphine)
Non-equilibrium macroscopic evolution of chain of oscillators with conservative noise
In the non-equilibrium evolution of a one dimensional chain of anharmonic oscilla-tors we expect two main space-time scales: an hyperbolic scale where the evolution isballistic-mechanic, dominated by tension gradients and governed by Euler equations,
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and a super-diffusive scale where the evolution, at constant tension, depends on thegradients of the temperature and is governed by a fractional heat equation. This con-jecture can be proven for an harmonic chain with random exchange of momentumbetween nearest neighbor particles.
• Nicolas Perkowski (University Paris Dauphine)
Paracontrolled KPZ equation
I will present several results that can be obtained by analyzing the KPZ equation us-ing paracontrolled distributions. For example, we will see that the solution to KPZis given by the value function of an optimal control problem, where a Brownian mo-tion is steered through a white noise potential. We will also study discretizations laSasamoto-Spohn and prove their convergence to the KPZ equation or a modified ver-sion of it. This is joint work with Massimiliano Gubinelli.
• Christophe Poquet (University Roma Tor Vergata)
Subgeometric convergence to non-equilibrium stationary states for coupled rotors
We will consider a chain composed of three coupled rotors, attached to thermal baths(with possibly different temperatures) at each extremity. An important feature of thissystem is that when the middle rotor oscillates rapidly, the energy of this rotor decreasesvery slowly, due to averaging phenomena. We will construct an effective dynamicsfor the middle rotor, using averaging techniques, which will allow us to prove theergodicity of the process at subgeometric rate. This is a work made in collaborationwith N. Cuneo and J.-P. Eckmann.
• Sylvain Prolhac (University of Toulouse)
Current fluctuations for totally asymmetric exclusion on the relaxation scale
The asymmetric simple exclusion process (ASEP) is a Markov process describing parti-cles hopping on a one-dimensional lattice with a preferred direction. Many exact resultshave been obtained for the current fluctuations of this model using Bethe ansatz, bothin the infinite volume limit L→ ∞ and in the stationary state obtained in the long timelimit T → ∞. The crossover between these two regimes corresponds to the relaxationscale T ∼ L3/2 characteristic of KPZ universality, observed in interface growth mod-els, driven diffusive systems and directed polymers in random media. This crossoverhas been studied recently in the totally asymmetric model (TASEP) where all particleshop in the same direction. Explicit formulas have been derived in the thermodynamiclimit for the eigenvalues and the normalization of the infinitely many Bethe eigenstatesthat contributes to the crossover T ∼ L3/2. High precision numerics indicate that theseresults are also valid for ASEP with general asymmetry of hopping rates, as expectedfrom KPZ universality.
• Herbert Spohn (Technical University of Munich)
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Low temperature dynamics of the one-dimensional discrete nonlinear Schrodinger equa-tion
I discuss the dynamical phase diagram of the DNLS. For this gathering the most inter-esting part is the low temperature regime which is characterized by universal scalingproperties in the same class as a generic anharmonic chain (Levy heat peak and KPZsound peaks). This is joint work with C. Mendl.
• Gabriel Stoltz (CERMICS, Ecole des Ponts & Inria Rocquencourt)
Langevin dynamics with space-time periodic nonequilibrium forcing
I present results on the ballistic and diffusive behavior of Langevin dynamics undera space-time periodic driving force. In the hyperbolic scaling, a non-trivial averagevelocity can be observed even if the external forcing vanishes. More surprisingly, anaverage velocity in the direction opposite to the forcing may develop at the linear re-sponse level – a phenomenon called negative mobility. The diffusive limit of (possiblystrongly) forced systems is studied using appropriate solutions of Poisson equations,extending recent works on pointwise estimates of the resolvent for the generator asso-ciated with Langevin dynamics.
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Practical Informations
• Campus: The conference will take place in ”Salle de conference” located in ”Labora-toire J.A. Dieudonne” (Building denoted by the letter W on the campus map at the endof the document). The campus and the Laboratoire J.A. Dieudonne are open during theday but the access is limited early in the morning or late in the evening.
• Facilities: The conference room is reserved for us during the workshop. Several black-boards are available in the laboratory building (you have to go upstairs to find them).If you need an office (limited possibilities) please ask to the organizers.
• Internet: Wireless connection is available on the campus by using a login and pass-word that will be given during the conference.
• Lunchs: A buffet is provided to participants from Monday to Thursday in the ”Lab-oratoire J.A. Dieudonne” building. The lunch of Friday will hold in the administrativerestaurant of ”Centre des impots” (close to the University but outside of the Campus,follow local people).
• Social dinner: A social dinner will take place on Wednesday at 8.00 pm at ”La Maisonde Marie”, 5, Massena street (close to Massena place, Massena tram station, see themap below). The dinner is free for participants but the registration is mandatory. Themenu is: Stuffed vegetables served with olive butter, Beef stewed in red wine sauce andgnocchi, tart. If you have special dietary requirements, contact the organizers beforeTuesday.
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Social Dinner: Map
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Map of the Campus
Main G
ate
workshop
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