# Stochastic dynamics out of equilibrium

12-16 June 2017
IHP
Europe/Paris timezone

## Abstracts and slides

Jinho Baik [slides]
Multi-time distribution of periodic TASEP

The scaling limit of the height function for the models in the KPZ class is expected to be universal. The spatial process at equal time is known to be the Airy process and its variations. However, the temporal process is less well studied. We consider the periodic TASEP with a special initial condition (periodic step initial condition) and evaluate the joint distribution of the height function at multiple locations and times. The limit is given by a multiple integral involving a Fredholm determinant. This is obtained by finding the joint distribution explicitly. The one-point distributions for other initial conditions are also going to discussed. This is a joint work with Zhipeng Liu (NYU).
_______________________________________________________________

Guillaume Barraquand
ASEP on the positive integers with an open boundary and the KPZ equation in a half space

We consider the asymmetric simple exclusion process on the positive integers with a reservoir of particles at the origin. Our main result is a Tracy-Widom GOE limit theorem for the current at the origin, when the boundary parameters are chosen so as to enforce an average density 1/2 near the origin. We will also discuss the weakly asymmetric limit of ASEP height function to the KPZ equation in a half-space with Neumann boundary condition, and we relate the Laplace transform of the solution to the KPZ equation with a multiplicative functional of the GOE point process. The proofs combine several techniques recently introduced in integrable probability, that we all adapt to the half-space case: We study half-line ASEP through a scaling limit of a half-space stochastic six-vertex model, that we relate to a half-space variant of Macdonald processes, whose asymptotics are essentially governed (for the observables of interest here) by the correlation kernel of a certain Pfaffian point process. Joint work with Alexei Borodin, Ivan Corwin and Michael Wheeler.
_______________________________________________________________

Cedric Bernardin [slides]
Diffusion versus Superdiffusion in a stochastic Hamiltonian lattice field model

We consider a Hamiltonian lattice field model with two conserved quantities that we perturb by a conserving stochastic noise. Depending on the form of the noise and of the interaction potential, superdiffusion or diffusion is expected. I will discuss how we can interpolate these different regimes by tuning the parameters of the model.
_______________________________________________________________

Thierry Bodineau [slides]
From hard sphere dynamics to the linearized Boltzmann equation

We consider the evolution of a small perturbation of a dilute gas of hard spheres at equilibrium. We will show that, in the Boltzmann Grad limit, this evolution is described by the linearized Boltzmann equation up to any large time. Joint work with I. Gallagher, L. Saint-Raymond.
_______________________________________________________________

Anna De Masi [slides]
Uphill diffusion in the 2D nearest-neighbor Ising model

See the abstract.
_______________________________________________________________

Frank Den Hollander
Metastability for the Widom-Rowlinson Model

Metastability is the phenomenon where a system, under the influence of a stochastic dynamics, moves between different subregions of its state space on different time scales. In statistical physics, metastability is the dynamical manifestation of a first-order phase transition. In this talk we consider the metastable behaviour of the Widom-Rowlinson model on a two-dimensional torus subject to a Metropolis stochastic dynamics. In this model, particles are randomly created and annihilated inside the torus as if the outside of the torus were an infinite reservoir with a given chemical potential. The particles are viewed as points carrying disks, and the energy of a particle configuration is equal to the volume of the union of the disks, called the halo" of the configuration. Consequently, the interaction between the particles is attractive.

We are interested in the metastable behaviour at low temperature when the chemical potential is supercritical. In particular, we start with the empty torus and are interested in the first time when we reach the full torus, i.e., the torus is fully covered by disks. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet of overlapping disks, which plays the role of a critical droplet'' that triggers the crossover. In the limit as the temperature tends to zero, we compute the asymptotic scaling of the average crossover time, show that the crossover time divided by its average is exponentially distributed, and identify the size and the shape of the critical droplet. It turns out that the critical droplet exhibits surface fluctuations", which need to be understood in order to obtain a fine estimate of the crossover time. Joint work with Sabine Jansen, Roman Kotecky and Elena Pulvirenti.
_______________________________________________________________

Alessandra Faggionato [slides]
1d Mott variable range hopping

Mott variable range hopping is a basic mechanism of electron transport in strongly disordered solids. In a mean field approximation,  the mathematical model is given by a  random walk on a simple point process of R^d with points marked by energy random variables. Jumps can be  arbitrarily large, while the jump rates decay exponentially in the  jump length and depend on the energy marks by a Boltzmann-like factor.  We will focus on  the 1d case. We  recall some previous results, and discuss more in detail the effect of applying an external uniform force field, by presenting results recently obtained in collaboration with N. Gantert and M. Salvi. In particular, we  present  conditions assuring ballisticity and sub-ballisticity, which reduce to a full characterization  in the case of a renewal simple point process. Moreover, we  derive the Einstein relation and discuss the linear response for a suitable class of observables.
_______________________________________________________________

Pablo Ferrari [slides]
Hydrodynamics of N Branching Brownian motions with selection

The BBM is a system of particles performing Brownian motions such that at rate 1 each particle branches creating a new particle at its current site. Particle motions are independent. In the N-BBM, only N particles are kept by erasing the leftmost particle and at each branching event. This process was proposed by Brunet and Derrida in the 90's and studied recently by Maillard. We show that the empirical measure of the particles at time t converges as N goes to infinity to a measure with density u(r,t), the solution of a pde with free boundaries. Joint work with Anna De Masi, Errico Presutti and Nahuel Soprano Loto.
_______________________________________________________________

Invariant measures in coupled KPZ equations

In a joint work with M. Hoshino (JFA '17), we studied invariant measures and global well-posedness of coupled KPZ equation under the symmetry condition (called trilinear condition) on the coupling constants appearing in the nonlinear term.  For coupled KPZ equation, the Cole-Hopf transform does not work in general, whereas Ertas-Kardar '92 gave an example for which Cole-Hopf transform does work componentwisely, so that one can show the existence of an invariant measure. However, the coupling constants in their equation do not satisfy the trilinear condition. We will give some results on the invariant measure of such equation.
_______________________________________________________________

Giovanni Gallavotti [slides]
Reversibility and equivalence of non equilibrium ensembles

The possibility of reversible evolution equations for fluids and particles systems is discussed with reference the Navier-Stokes or the Lorenz-96 system and their complete equivalence of their stationary states (regarded as ensembles) with the ones of the corresponding irreversible evolution equations.
_______________________________________________________________

Massimiliano Gubinelli
Weak universality of fluctuations and singular stochastic PDEs

Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. Due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. In this talk I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality.
_______________________________________________________________

Milton Jara
Non-equilibrium fluctuations of interacting particle systems

Using a reaction-diffusion model as a test bed, we show that the entropy production rate with respect to local equilibrium reference measures is of order O(n^(d-2)), improving on the bound o(n^d) given by Yau's relative entropy method. This bound allows to prove a central limit theorem around its hydrodynamic limit for the density of particles in dimensions d = 1,2,3. The proof does not require explicit knowledge of the invariant measures of the systems, and therefore it is suitable to tackle various problems of interest of non-equilibrium statistical mechanics. Joint work with Otávio Menezes (IMPA).
_______________________________________________________________

Gianni Jona-Lasinio
Finite time thermodynamics and quantitative analysis of Clausius inequality

In the context of driven diffusive systems, for thermodynamic transformations over a large but finite time window, we consider an expansion of the energy balance around the quasi static limit. In particular, we characterize the transformations which minimize the energy dissipation and describe the optimal correction to the quasi-static limit. Surprisingly, in the case of transformations between homogeneous equilibrium states of an ideal gas, the optimal transformation is a sequence of inhomogeneous equilibrium states.
_______________________________________________________________

Hubert Lacoin
Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling

We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times, thus showing that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunction, stochastic comparisons
and concentration inequalities. Joint work with Cyril Labbé (CEREMADE).
_______________________________________________________________

Claudio Landim [slides]
Stationary states of non-reversible dynamics

We review recent results on the stationary states of non-reversible diffusions and of a superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions.
_______________________________________________________________

Joel Lebowitz [Human Rights session]
Statistical Mechanics and Thermodynamics of Large and Small Systems

Thermodynamics makes almost sure predictions about the thermal behavior of macroscopic systems. Statistical mechanics describes and explains this behavior in terms of the dynamics and statistics, classical or quantum, of the large number of microscopic degrees of freedom of the constituents of macroscopic systems. The extension of these concepts and considerations to small systems, isolated or in contact with reservoirs, is a subject of current work, both theoretical and experimental. I will discuss some aspects of this work.
_______________________________________________________________

Carlangelo Liverani
Deterministic fast-slow systems: beyond averaging

The behaviour of a fast-slow dynamical system can be often described, in first approximation, by some averaged system. However, often in applications such an approximation is inadequate: informations on smaller scales or longer times are needed. I will describe a very simple, but highly non trivial, model in which one can test the possibility to obtain results beyond averaging. This is joint work De Simoi, Poquet, Volk.
_______________________________________________________________

Jonathan Mattingly
Interactions between noise and instabilities (or Exercises in building optimal Lyapunov functions)

I wil consider two examples.

• The first is a two dimensional ODE which is unstable for some initial data. Adding noise produces stable dynamics which converge exponentially to equilibrium. This equilibrium will have quite heavy tails which are determined by the interaction between noise and then nonlinearity even though the density solves a uniformly elliptic PDE. The equilibrium is not irreversible with a nontrivial current. The proof will involve building an optimal Lyapunov function.
• The second example will be a simple Hamiltonian particle interacting with the origin via a Lennard-Jones like potential. We follow a a similar path to the first example. Along this way we will gain some novel insight into an example which motivated the theory of Hypocercivity.

_______________________________________________________________

Tom Mountford
Convergence to upper equilibrium for kinetic gas models

We consider work by Blondel, Cancrini, Martinelli, and Toninelli concerning convergence of the Frederickson-Andersen model.  Our class of graphs is essentially the same (but slightly larger) but we can prove exponential convergence to equilibrium.  The price to be paid is a large value of the parameter q (or the bias towards the occupied sites). Joint work with Glauco Valle.
_______________________________________________________________

Frank Redig [slides]
The asymmetric KMP model

The KMP (Kipnis, Marchioro, Presutti) model is a well-known stochastic model of heat conduction which has been analyzed via a discrete dual. The question motivating the talk will be how to find the correct’’ asymmetric KMP?’’ By correct we mean here that the asymmetric model should have duality properties similar to those of the asymmetric exclusion process, i.e., coming from a coproduct of a q-deformed Lie algebra. We show how this answer to this question leads to a natural class of asymmetric models associated to the q-deformation of SU(1,1). This construction gives automatically self-dualities, allowing e.g. to compute special exponential moments of the current. Based on joint work with G. Carinci, C. Giardina, T. Sasamoto.
_______________________________________________________________

Thermal conductivity for a chain of harmonic oscillators in a magnetic field

We consider a system of harmonic oscillators in a magnetic field perturbed by a stochastic dynamics conserving generalized momentum and energy. The magnetic field destroys the conservation of the momentum, but the thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. Without the magnetic field, it is known that the time correlation function of the energy current C(t) behaves for large time, like t^{-d/2} and the thermal conductivity for the system of size N^d behaves as N^{1/2} for d=1 and log N for d =2. The relation between the thermal conductivity and  the time correlation function of the energy current can be understood via the notion of the sound velocity. For our model, the sound velocity can be $0$. We showed that the time correlation function of the energy current C(t) behaves for large time, like  t^{-d/4-1/2} for the case with a uniform magnetic field and t^{-d/2} for the case with an alternative magnetic field . Also, by the numerical simulation, we see that the thermal conductivity for the system with size N with periodic boundary condition in one-dimensional system behave as N^{3/8} for the case with a uniform magnetic field and N^{1/2} for the case with an alternative magnetic field. The analytical derivation of the relation between these indexes is an important open problem. The talk is based on the joint work with Keiji Saito and Shuji Tamaki.
_______________________________________________________________

Timo Seppalainen
Variational formulas and geodesics for percolation models

We discuss recently found variational descriptions of the limit shape of first-passage percolation and some related ideas, such as a convex duality between the mean edge weight and the Euclidean length of the geodesic. Based on joint work with Arjun Krishnan and Firas Rassoul-Agha.
_______________________________________________________________

Fabio Toninelli [slides]
Discrete interface dynamics and hydrodynamic limits

Dimer models provide natural models of (2+1)-dimensional random discrete interfaces and of stochastic interface dynamics. I will discuss two examples of such dynamics, a reversible one and a driven one (growth process). The latter is conjectured to belong to the so-called "anisotropic KPZ universality class". In both cases we can prove the convergence of the stochastic interface evolution to a deterministic PDE after suitable (diffusive or hyperbolic respectively in the two cases) space-time rescaling. For the growth model, in general the solution  has to be interpreted in the viscosity sense. Joint work with B. Laslier (arXiv:1701:05100) and M. Legras (arXiv:1704.06581)
_______________________________________________________________

A closer look at the Polaron measure

In this joint work with Chiranjib Mukherjee, we examine the behavior of the Polaron measure dQ=?frac{1}{Z{T}exp[int_0^Tint_0^T ?frac{e^{-|t-s|}}{|x(t)-x(s)|} dt ds] dP where P is the Wiener measure. We are interested in the behavior as T tends to infinity as well as when the exponential e^{-|t-s|} is replaced by lambda e^{-lambda |t-s|}.

_______________________________________________________________

Maria Eulalia Vares [slides]
Some results on two-dimensional anisotropic Ising spin systems and percolation

In this talk I will discuss a few results about a class of Ising spin systems and percolation on Z^2. The older results (2015) have to do with spins subject to highly anisotropic ferromagnetic interactions: along each horizontal layer we have a Kac potential while the vertical interaction is of nearest neighbor type. We focus on the spontaneous magnetization at the mean field critical temperature for a small but fixed strength of the vertical interaction. This is based on a joint work in collaboration with L. R. Fontes, D. Marchetti, I. Merola, and E. Presutti. I also hope to discuss partial results (ongoing collaboration with T. Mountford) regarding the case of the corresponding anisotropic percolation but when the nearest neighbor vertical edges are open with a very small probability that suitably tends to zero as a power of the inverse of the range of the horizontal spread out percolation.

_______________________________________________________________

Public lecture by Isabelle Gallagher (Université Paris Diderot), Wednesday June 14th
Probabilités, Irréversibilité et Propagation du chaos

Suivant l'échelle à laquelle on observe un objet physique, on peut en faire une description très différente. Par exemple l'air est constitué d'un nombre gigantesque de molécules qui s'agitent en permanence en suivant les lois de la mécanique classique (comme des boules de billard), mais cette description microscopique est souvent moins utile d'un point de vue pratique que la description macroscopique de l'évolution dans le temps et dans l'espace de sa température, sa vitesse etc. La question de concilier ces deux descriptions, microscopique et macroscopique, est un problème mathématique identifié comme une question fondamentale depuis le début du 20ème siècle par le mathématicien D. Hilbert. Liée à cette question est la compréhension de l'apparition de l'irréversibilité : chaque molécule a un mouvement parfaitement réversible dans le temps, mais observé à grande échelle le comportement d'un gaz ne l'est pas en général. Nous verrons que ce paradoxe apparent est lié à ce que notre description des objets physiques qui nous entourent est en fait de nature probabiliste.