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We study the one-dimensional asymmetric simple exclusion process on the lattice $\{1, \dots,N\}$ with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale $N^{-a}$ with $a>0$. We prove that at the time scale $N^{1+a}$ the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic boundary rates. We consider two different types of boundary rates: the "Liggett boundaries" that correspond to the projection of the infinite dynamics, and the reversible boundaries, that correspond to the contact with particle reservoirs in equilibrium. The proof is based on the control of the Lax boundary entropy--entropy flux pairs and a coupling argument., final version

We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the onedimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass.

We investigate the properties of a harmonic chain in contact with a thermal bath at one end and subjected, at its other end, to a periodic force. The particles also undergo a random velocity reversal action, which results in a finite heat conductivity of the system. We prove the approach of the system to a time periodic state and compute the heat current, equal to the time averaged work done on the system, in that state. This work approaches a finite positive value as the length of the chain increases. Rescaling space, the strength and/or the period of the force leads to a macroscopic temperature profile corresponding to the stationary solution of a continuum heat equation with Dirichlet-Neumann boundary conditions.

We derive a macroscopic heat equation for the temperature of a pinned harmonic chain subject to a periodic force at its right side and in contact with a heat bath at its left side. The microscopic dynamics in the bulk is given by the Hamiltonian equation of motion plus a reversal of the velocity of a particle occurring independently for each particle at exponential times, with rate $$\gamma $$ γ . The latter produces a finite heat conductivity. Starting with an initial probability distribution for a chain of n particles we compute the current and the local temperature given by the expected value of the local energy. Scaling space and time diffusively yields, in the $$n\rightarrow +\infty $$ n → + ∞ limit, the heat equation for the macroscopic temperature profile T(t, u), $$t>0$$ t > 0 , $$u \in [0,1]$$ u ∈ [ 0 , 1 ] . It is to be solved for initial conditions T(0, u) and specified $$T(t,0)=T_-$$ T ( t , 0 ) = T - , the temperature of the left heat reservoir and a fixed heat flux J, entering the system at $$u=1$$ u = 1 . |J| equals the work done by the periodic force which is computed explicitly for each n.

International audience; We prove the hydrodynamic limit for an harmonic chain with a random exchange of momentum that conserves the kinetic energy but not the momentum. The system is open and subject to two thermostats at the boundaries and to external tension. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities, the volume stretch and the energy, converge to the solution of a non-linear diffusive systems of conservative partial differential equations.

International audience; We characterize the non equilibrium stationary states in two classes of systems where phase transitions are present. We prove that the interface in the limit is a plane which separates the two phases.

We consider an infinite chain of coupled harmonic oscillators with a Poisson thermostat at the origin. In the high frequency limit, we establish the reflection-transmission-scattering coefficients for the wave energy scattered off the thermostat. Unlike the case of the Langevin thermostat [5], in the macroscopic limit the Poissonian thermostat scattering generates a continuous cloud of waves of frequencies different from that of the incident wave.

We study the quasi-static limit for the $L^\infty$ entropy weak solution of scalar one-dimensional hyperbolic equations with strictly concave or convex flux and time dependent boundary conditions. The quasi-stationary profile evolves with the quasi-static equation, whose entropy solution is determined by the stationary profile corresponding to the boundary data at a given time., final version

To appear in the Springer INdAM Series; International audience; We investigate how a thermal boundary, modelled by a Langevin dynamics, affect the macroscopic evolution of the energy at different space-time scales.

International audience; We consider the limit of a linear kinetic equation, with reflection-transmission-absorption at an interface, with a degenerate scattering kernel. The equation arise from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation, with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected, or transmitted upon crossing the interface.

International audience; We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. In the high frequency limit, we establish the reflection-transmission coefficients for the wave energy for the scattering of the thermostat. To our surprise, even though the thermostat fluctuations are time-dependent, the scattering does not couple wave energy at various frequencies.

This is an improved version of hal-01852377, including boundary conditions and larger time scales.; International audience; We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random terms conservative of such quantities. We prove that these fluctuations evolve macroscopically following the linearized Euler equations with the corresponding boundary conditions, even in some time scales larger than the hyperbolic one.

We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite., 22 pages, 2 figures, to appear in Journal of Statistical Physics (JSP) v1-->v2: Lemma in section 3.1 expanded, otherwise only minor changes

We prove existence of $L^2$-weak solutions of a quasilinear wave equation with boundary conditions. This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The $L^2$-valued solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain of anharmonic springs connected to a thermal bath. The proof of the existence is done using a vanishing viscosity approximation with extra Neumann boundary conditions added. In this setting we obtain a uniform a priori estimate in $L^2$, allowing us to use $L^2$ Young measures, together with the classical tools of compensated compactness. We then prove that the viscous solutions converge to weak solutions of the quasilinear wave equation strongly in $L^p$, for any $p \in [1,2)$, that satisfy, in a weak sense, the boundary conditions. Furthermore, these solutions satisfy the Clausius inequality: the change of the free energy is bounded by the work done by the boundary tension. In this sense they are the correct thermodynamic solutions, and we conjecture their uniqueness., Comment: Added section 6 on Lax entropy condition and improved proof of regularity in section 4

International audience; We consider a one-dimensional chain of coupled oscillators in contact at both ends with heat baths at different temperatures, and subject to an external force at one end. The Hamiltonian dynamics in the bulk is perturbed by random exchanges of the neighbouring momenta such that the energy is locally conserved. We prove that in the stationary state the energy and the volume stretch profiles, in large scale limit, converge to the solutions of a diffusive system with Dirichlet boundary conditions. As a consequence the macroscopic temperature stationary profile presents a maximum inside the chain higher than the thermostats temperatures, as well as the possibility of uphill diffusion (energy current against the temperature gradient). Finally, we are also able to derive the non-stationary macroscopic coupled diffusive equations followed by the energy and volume stretch profiles.

International audience; We study the equivalence of microcanonical and canonical ensembles in continuous systems, in the sense of the convergence of the corresponding Gibbs measures and the first order corrections. We are particularly interested in extensive observables, like the total kinetic energy. This result is obtained by proving an Edgeworth expansion for the local central limit theorem for the energy in the canonical measure, and a corresponding local large deviations expansion. As an application we prove a formula due to Lebowitz-Percus-Verlet that express the asymptotic microcanonical variance of the kinetic energy in terms of the heat capacity.

We consider a chain of particles connected by an-harmonic springs, with a boundary force (tension) acting on the last particle, while the first particle is kept pinned at a point. The particles are in contact with stochastic heat baths, whose action on the dynamics conserves the volume and the momentum, while energy is exchanged with the heat baths in such way that, in equilibrium, the system is at a given temperature $T$. We study the space empirical profiles of volume stretch and momentum under hyperbolic rescaling of space and time as the size of the system growth to be infinite, with the boundary tension changing slowly in the macroscopic time scale. We prove that the probability distributions of these profiles concentrate on $L^2$-valued weak solutions of the isothermal Euler equations (i.e. the non-linear wave equation, also called p-system), satisfying the boundary conditions imposed by the microscopic dynamics. Furthermore, the weak solutions obtained satisfy the Clausius inequality between the work done by the boundary force and the change of the total free energy in the system. This result includes the shock regime of the system.

We show that a quantum particle in $\mathbb{R}^d$, for $d \geq 1$, subject to a white-noise potential, moves super-ballistically in the sense that the mean square displacement $\int \|x\|^2 \langle \rho(x,x,t) \rangle ~dx$ grows like $t^{3}$ in any dimension. The white noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. This is a known result in one dimension (see refs. Fischer, Leschke, M\"uller and Javannar, Kumar}. The energy of the system is also shown to increase linearly in time. We also prove that for the same white-noise potential model on the lattice $\mathbb{Z}^d$, for $d \geq 1$, the mean square displacement is diffusive growing like $t^{1}$. This behavior on the lattice is consistent with the diffusive behavior observed for similar models in the lattice $\mathbb{Z}^d$ with a time-dependent Markovian potential (see ref. Kang, Schenker)., Comment: 18 pages

International audience We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap.

We consider a one-dimensional unpinned chain of harmonic oscillators with random masses. We prove that after hyperbolic scaling of space and time the distributions of the elongation, momentum and energy converge to the solution of the Euler equations. Anderson localization decouples the mechanical modes from the thermal modes, allowing the closure of the energy conservation equation even out of thermal equilibrium. This example shows that the derivation of Euler equations rests primarily on scales separation and not on ergodicity. Furthermore, it follows from our proof that the temperature profile does not evolve in any space-time scale., 20 pages

Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space-time scale. In these situations the Fourier law depends also from the gradient of the other conserved quantities. The rotor chain is a classical example of such systems, where energy and angular momentum are conserved. We review here some recent mathematical results about diffusive transport of energy and other conserved quantities, in particular for systems where the bulk Hamiltonian dynamics is perturbed by conservative stochastic terms. The presence of the stochastic dynamics allows to define the transport coefficients (thermal conductivity) and in some cases to prove the local equilibrium and the linear response argument necessary to obtain the diffusive equations governing the macroscopic evolution of the conserved quantities. Temperature profiles and other conserved quantities profiles in the non-equilibrium stationary states can be then understood from the non-stationary diffusive behaviour. We also review some results and open problems on the two step approach (by weak coupling or kinetic limits) to the heat equation, starting from mechanical models with only energy conserved., Review Article for the CRAS-Physique, final version

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature \begin{document}$ T $\end{document} in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution \begin{document}$ \rho(t, y) $\end{document} of a heat equation with the boundary condition \begin{document}$ \rho(t, 0)\equiv T $\end{document} .

We study the 2d stationary fluctuations of the interface in the SOS approximation of the non equilibrium stationary state found in De Masi et al. (J Stat Phys 175:203–221, 2019). We prove that the interface fluctuations are of order $$N^{1/4}$$ , N the size of the system. We also prove that the scaling limit is a stationary Ornstein–Uhlenbeck process.

Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h}(t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h}_N(t,\xi) := \frac{1}{N}\mathsf{h}(Nt,N\xi) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp(-O(N)) $, referred to as speed-$ N $; while the other with probability $ \exp(-O(N^{2})) $, referred to as speed-$ N^2 $. In this work we study the speed-$ N^2 $ functional LDP of the TASEP, and establishes (non-matching) large deviation upper and lower bounds., Comment: 13 figures, 59 pages; updated to match the version to be published

We consider the asymmetric simple exclusion process in dimension d 3. We review some results concerning the equilibrium bulk uctuations and the asymptotic behaviour of a second class particle. x1. Introduction The asymmetric simple exclusion process is the simplest model of a driven lattice gas. This model is given by the dynamics of innitely many particles moving on Z d as asymmetric random walks with an exclusion rule: when a particle attempts to jump on a site occupied by another particle the jump is suppressed. Of course we consider initial congurations where there is at most one particle per site. We denote the congurations by 2 f0; 1g Z d so that (x) = 1 if site x is occupied for the conguration and (x) = 0 if site x is empty. The number of particles is conserved and the Bernoulli product measures f ; 2 [0; 1]g are the ergodic invariant measures. Rezakhanlou, in [11], proved that the empirical eld of particles, after a hyperbolic rescaling of space and time by a parameter

International audience; We consider a non acoustic chain of harmonic oscil-lators with the dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. The macroscopic limits of the energy density, momentum and the curvature (or bending) of the chain satisfy a system of evolution equations. We prove that, in a diffusive space-time scaling, the curvature and momentum evolve following a linear system that corresponds to a damped Euler-Bernoulli beam equation. The macro-scopic energy density evolves following a non linear diffusive equation. In particular the energy transfer is diffusive in this dynamics. This provides a first rigorous example of a normal diffusion of energy in a one dimensional dynamics that conserves the momentum.

We consider an infinite chain of coupled harmonic oscillators whose Hamiltonian dynamics is perturbed by a random exchange of momentum between particles such that total energy and momentum are conserved, modeling collision between atoms. This random exchange is rarefied in the limit, that corresponds to the hypothesis that in the macroscopic unit time only a finite number of collisions takes place (the Boltzmann-Grad limit). Furthermore, the system is in contact with a Langevin thermostat at temperature T through a single particle. We prove that, after the hyperbolic space-time rescaling, the Wigner distribution, describing the energy density of phonons in space-frequency domain, converges to a positive energy density function W ( t , y , k ) that evolves according to a linear kinetic equation, with the interface condition at y = 0 that corresponds to reflection, transmission and absorption of phonons caused by the presence of the thermostat. The paper extends the results of [15] , where a harmonic chain (with no inter-particle scattering) in contact with a Langevin thermostat has been considered.

Wealth inequality is an important matter for economic theory and policy. Ongoing debates have been discussing recent rise in wealth inequality in connection with recent development of active financial markets around the world. Existing literature on wealth distribution connects the origins of wealth inequality with a variety of drivers. Our approach develops a minimalist modelling strategy that combines three featuring mechanisms: active financial markets; individual wealth accumulation; and compound interest structure. We provide mathematical proof that accumulated financial investment returns involve ever-increasing wealth concentration and inequality across individual investors through time. This cumulative effect through space and time depends on the financial accumulation process and holds also under efficient financial markets, which generate some fair investment game that individual investors do repeatedly play through time.

We study the hydrodynamic limit for the isothermal dynamics of an anharmonic chain under hyperbolic space-time scaling under varying tension. The temperature is kept constant by a contact with a heat bath, realised via a stochastic momentum-preserving noise added to the dynamics. The noise is designed to be large at the microscopic level, but vanishing in the macroscopic scale. Boundary conditions are also considered: one end of the chain is kept fixed, while a time-varying tension is applied to the other end. We show that the volume stretch and momentum converge (in an appropriate sense) to a weak solution of a system of hyperbolic conservation laws (isothermal Euler equations in Lagrangian coordinates) with boundary conditions. This result includes the shock regime of the system. This is proven by adapting the theory of compensated compactness to a stochastic setting, as developed by Fritz (2011 Arch. Ration. Mech. Anal. 201 209?49) for the same model without boundary conditions. Finally, changing the external tension allows us to define thermodynamic isothermal transformations between equilibrium states. We use this to deduce the first and the second principle of thermodynamics for our model.

International audience; We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation.

Let modele de Atlas demi-infini est un systeme unidimensional des particules Browniens, ou seulement la particule plus a gauche a une vitesse positive. Le processus d’increments correspondant a une infinite des mesures invariantes, avec une mesure distinguee, reversible et invariante par translations. On montre que cette mesure attire une grande classe des configurations initiales avec densite bornee ou a croissance moderee. Central a notre preuve est une nouvelle estimation de la vitesse de convergence vers l’equilibre du processus d’increments dans un tableau triangulaire de systemes finis et de grande taille.

We consider a chain of anharmonic oscillators immersed in a heat bath with a temperature gradient and a time varying tension applied to one end of the chain while the other side is fixed to a point. We prove that under diffusive space-time rescaling the volume strain distribution of the chain evolves following a non-linear diffusive equation. The stationary states of the dynamics are of non-equilibrium and have a positive entropy production, so the classical relative entropy methods cannot be used. We develop new estimates based on entropic hypocoercivity, that allows to control the distribution of the positions configurations of the chain. The macroscopic limit can be used to model isothermal thermodynamic transformations between non-equilibrium stationary states., New version, for the publications in Annals of Probability

Starting from the overdamped Langevin dynamics in R n , d X t = − ∇ V ( X t ) d t + 2 β − 1 d W t , we consider a scalar Markov process ξ t which approximates the dynamics of the first component X t 1 . In the previous work (Legoll and Lelievre, 2010), the fact that ( ξ t ) t ≥ 0 is a good approximation of ( X t 1 ) t ≥ 0 is proven in terms of time marginals, under assumptions quantifying the timescale separation between the first component and the other components of X t . Here, we prove an upper bound on the trajectorial error E ( sup 0 ≤ t ≤ T | X t 1 − ξ t | ) for any T > 0 , under a similar set of assumptions. We also show that the technique of proof can be used to obtain quantitative averaging results.

We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is perturbed by a random exchange of velocities, such that the energy and momentum of the chain are conserved. Consequently, the evolution of the system has only three conserved quantities: volume, momentum and energy. We show the existence of two space--time scales on which the en- ergy of the system evolves. On the hyperbolic scale (t$\epsilon$--1,x$\epsilon$--1) the limits of the conserved quantities satisfy a Euler system of equa- tions, while the thermal part of the energy macroscopic profile re- mains stationary. Thermal energy starts evolving at a longer time scale, corresponding to the superdiffusive scaling (t$\epsilon$--3/2, x$\epsilon$--1) and follows a fractional heat equation. We also prove the diffusive scal- ing limit of the Riemann invariants - the so called normal modes, corresponding to the linear hyperbolic propagation., Comment: Accepted for the publication in 'Nonlinearity'

International audience; We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.

We consider lattice dynamics with a small stochastic perturbation of order ε and prove that for a space-time scale of order \varepsilon\^-1 the local spectral density (Wigner function) evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain the transport equation predicts a slow decay, as 1/\sqrt{t}, for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method., Comment: Changed title and introduction

Equilibrium statistical mechanics studies random fields distrib- uted according to a Gibbs probability measure. Such random fields can be equipped with a stochastic dynamics given by a Markov process with the correspondingly high-dimensional state space. One particular case are sto- chastic partial differential equations suitably regularized. Another common version is to consider the evolution of random fields taking only values 0 or 1. The workshop was concerned with an understanding of qualitative properties of such high-dimensional Markov processes. Of particular interest are non- reversible dynamics for which the stationary measures are determined only through the dynamics and not given a priori (as would be the case for re- versible dynamics). As a general observation, properties on a large scale do not depend on the precise details of the local updating rules. Such kind of universality was a guiding theme of our workshop.

We obtain macroscopic adiabatic thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment. The microscopic dynamics is given by a chain of oscillators subject to a varying tension (external force) and to collisions with external independent particles of "infinite mass". The effect of each collision is to change the sign of the velocity without changing the modulus. This way the energy is conserved by the resulting dynamics. After a diffusive space-time scaling and cross-graining, the profiles of volume and energy converge to the solution of a deterministic diffusive system of equations with boundary conditions given by the applied tension. This defines an irreversible thermodynamic transformation from an initial equilibrium to a new equilibrium given by the final tension applied. Quasi-static reversible adiabatic transformations are then obtained by a further time scaling. Then we prove that the relations between the limit work, internal energy and thermodynamic entropy agree with the first and second principle of thermodynamics., New version accepted for the publication in Brazilian Journal of Probability and Statistics

We study the Green-Kubo formula \({\kappa (\varepsilon, \varsigma)}\) for the heat conductivity of an infinite chain of d-dimensional finite systems (cells) coupled by a smooth nearest neighbor potential \({\varepsilon V}\) . The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength \({\varsigma}\) . Noting that \({\kappa (\varepsilon, \varsigma)}\) exists and is finite whenever \({\varsigma > 0}\) , we are interested in what happens when the strength of the noise \({\varsigma \to 0}\) . For this, we start in this work by formally expanding \({\kappa (\varepsilon, \varsigma)}\) in a power series in \({\varepsilon}\) , \({\kappa (\varepsilon, \varsigma) = \varepsilon^2 \sum_{n\geq 2} \varepsilon^{n-2} \kappa_n (\varsigma)}\) and investigating the (formal) equations satisfied by \({\kappa_n (\varsigma)}\) . We show in particular that \({\kappa_2 (\varsigma)}\) is well defined when no pinning potential is present, and coincides formally with the heat conductivity obtained in the weak coupling (van Hove) limit, where time is rescaled as \({\varepsilon^{-2}t}\) , for the cases where the latter has been established (Liverani and Olla, in JAMS 25:555–583, 2012; Dolgopyat and Liverani, in Commun Math Phys 308:201–225, 2011). For one-dimensional systems, we investigate \({\kappa_2 (\varsigma)}\) as \({\varsigma \to 0}\) in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly an harmonic oscillators. Moreover, we formally identify \({\kappa_2 (\varsigma)}\) with the conductivity obtained by having the chain between two reservoirs at temperature T and \({T+\delta T}\) , in the limit \({\delta T \to 0}\) , \({N \to \infty}\) , \({\varepsilon \to 0}\) .

We consider hydrodynamic limits of interacting particles systems with open boundaries, where the exterior parameters change in a time scale slower than the typical relaxation time scale. The limit deterministic profiles evolve quasi-statically. These limits define rigorously the thermodynamic quasi static transformations also for transition between non-equilibrium stationary states. We study first the case of the symmetric simple exclusion, where duality can be used, and then we use relative entropy methods to extend to other models like zero range systems. Finally we consider a chain of anharmonic oscillators in contact with a thermal Langevin bath with a temperature gradient and a slowly varying tension applied to one end., Comment: Accepted for teh publication in Journal of Statistical Physics

We study the Green-Kubo (GK) formula $\kappa (\varepsilon, \xi)$ for the heat conductivity of an infinite chain of $d$-dimensional finite systems (cells) coupled by a smooth nearest neighbour potential $\varepsilon V$. The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength $\xi$. Noting that $\kappa (\varepsilon, \xi)$ exists and is finite whenever $\xi> 0$, we are interested in what happens when the strength of the noise $\xi \to 0$. For this, we start in this work by formally expanding $\kappa (\varepsilon, \xi)$ in a power series in $\varepsilon$, $\kappa (\varepsilon, \xi) = \varepsilon^2 \sum_{n\ge 2} \varepsilon^{n-2} \kappa_n (\xi)$ and investigating the (formal) equations satisfied by $\kappa_n (\xi$. We show in particular that $\kappa_2 (\xi)$ is well defined when no pinning potential is present, and coincides formally with the heat conductivity obtained in the weak coupling (van Hove) limit, where time is rescaled as $\varepsilon^{-2}t$, for the cases where the latter has been established \cite{LO, DL}. For one-dimensional systems, we investigate $\kappa_2 (\xi)$ as $\xi\to 0$ in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly anharmonic oscillators. Moreover, we formally identify $\kappa_2 (\xi)$ with the conductivity obtained by having the chain between two reservoirs at temperature $T$ and $T+\delta T$, in the limit $\delta T\to 0$, $N \to \infty$, $\varepsilon \to 0$., Comment: New version with many improvements

We investigate the harmonic chain forced by a multiplicative noise, the evolution is given by an infinite system of stochastic differential equations. Total energy and deformation are preserved, the conservation of momentum is destroyed by the noise. Gaussian product measures are the extremal stationary states of this model. Equilibrium fluctuations of the conserved fields at a diffusive scaling are described by a couple of generalized Ornstein-Uhlenbeck processes.

We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove that the stationary state, in the limit as N→ ∞, satisfies Fourier’s law and the linear profile for the energy average

We consider a tracer particle performing a nearest neighbor random walk on Z d in dimension d ⩾ 3 with random jump rates. This kind of a walk models the motion of a charged particle under a constant external electric field. We assume that the jump rates admit only two values 0 γ - γ + + ∞ , representing the lower and upper conductivities. We prove the existence of the mobility coefficient and that it equals to the diffusivity coefficient of the particle in zero external field.

In this paper we rigorously establish the existence of the mobility coefficient for a tagged particle in a simple symmetric exclusion process with adsorption/desorption of particles, in a presence of an external force field interacting with the particle. The proof is obtained using a perturbative argument. In addition, we show that, for a constant external field, the mobility of a particle equals to the self-diffusivity coefficient, the so-called Einstein relation. The method can be applied to any system where the environment has a Markovian evolution with a fast convergence to equilibrium (spectral gap property). In this context we find a necessary relation between forward and backward velocity for the validity of the Einstein relation. This relation is always satisfied by reversible systems. We provide an example of a non-reversible system, where the Einstein relation is valid.