## Workshop 3

Interacting Particles Systems and Non-equilibrium Dynamics

March 9

^{th}to 13^{th}Participants who intend to present a poster can submit their request to the Poster Session Chair (Dott. Gioia Carinci, gioia.carinci@unimore.it)

Invited speakers:

*C. Bernardin*“3/4 fractional superdiffusion of energy in a harmonic chain with bulk noises”, [abstract]
We consider a harmonic chain perturbed by an energy conserving noise and
show
that after a space-time rescaling the energy-energy correlation function is
given
by the solution of a skew-fractional heat equation with exponent 3/4. We
also investigate
an interpolation microscopic model which makes the bridge between the heat
equation and the skew-fractional heat equation.

*G. Carinci*“Non-equilibrium via current reservoirs”, [abstract]
Stationary non equilibrium states are characterized by the presence of
steady currents flowing through the system and a basic question in
statistical mechanics is to understand their structure. Many papers have
been devoted to the subject in the context of stochastic interacting
particle systems. Usually current density is produced by fixing two
different densities at the boundary. We want instead to implement the mass
transport by introducing current reservoirs that produce a given current by
sending in particles from the left at some rate and taking out particles
from the rightmost occupied site at same rate.
The removal mechanism is therefore of topological rather than metric
nature, since the determination of the rightmost occupied site requires a
knowledge of the entire configuration. This prevents from using correlation
functions techniques. I will discuss recent results obtained in the study
of these topics, whose final purpose is to provide a particle version of a
free boundary-type problem.

*F. Comets*“Cover time of the random walk on the 2-dimensional torus”, [abstract]
The cover time is the time it takes for the simple random walk to visit all
points of the torus of size NxN. Naively
we can hope it compares to the maximum of N^2* independent* i.d. r.v.'s
(with the law of hitting time individual points).
The two agree for some asymptotics but not for all, and dimension 2 is
critical because of long range correlations.
Random interlacements yields the description of the covering process at its
very end.

*I. Corwin*“Stochastic quantum integrable systems”, [abstract]
We will relate recent work at the interface between interacting
particle systems and quantum integrable systems. In particular, we describe
an exactly solvable interacting particle system related to higher spin
U_q(sl2) vertex models and explain how all of the known exactly solvable
models in the KPZ universality class arise from degenerations of this
system.

*B. Derrida*“Universal current fluctuations in non equilibrium systems”, [abstract]
Fluctuations of the current of one dimensional non equilibrium
diffusive systems are well understood. After a short review of the one
dimensional results, the talk will try to show that the statistics of
these fluctuations are exactly the same in higher dimension, for arbitrary
geometries.
This result can be established by a mapping between the variational
problem in finite dimension and in one dimension.

In the particular case of the symmetric exclusion process, the statistical properties of the current are the same as those of disordered quantum wires in the metallic regime, where the distribution of the transmission coefficients of the different channels have been predicted from the theory of random matrices.

In the particular case of the symmetric exclusion process, the statistical properties of the current are the same as those of disordered quantum wires in the metallic regime, where the distribution of the transmission coefficients of the different channels have been predicted from the theory of random matrices.

*Pablo Ferrari*“Phase transition for the dilute clock model.”, [abstract]
We prove that phase transition occurs in the dilute ferromagnetic
nearest-neighbour q-state clock model in Z^d, for every q≥2 and d≥2. This
follows from the fact that the Edwards-Sokal random-cluster representation
of the clock model stochastically dominates a supercritical Bernoulli bond
percolation probability, a technique that has been applied to show phase
transition for the low-temperature Potts model. The domination involves a
combinatorial lemma. Joint work with Inés Armendariz and Nahuel
Soprano-Loto.

*Patrick Ferrari*“Height fluctuations for the stationary KPZ equation”, [abstract]
In this talk I will discuss the height fluctuations of models in
the KPZ (Kardar-Parisi-Zhang) universality class with stationary initial
conditions. In particular, I will present the solution for the KPZ
equation, where the stationary initial height profile is a two-sided
Brownian motion. This is a joint work with Alexei Borodin, Ivan Corwin, and
Bálint Vető (arXiv: 1407.6977).

*G. Giacomin*“Synchronization phenomena and non equilibrium statistical mechanics: the Kuramoto model”, [abstract]
I will present the (stochastic and non stochastic) Kuramoto model
of disordered oscillators, the prototypical model for synchronization
phenomena, from a non equilibrium statistical mechanics viewpoint. I will
then focus on recent results on the relaxation dynamics below the
synchronization threshold in the non stochastic set-up and. The link with
Landau damping in plasma will be discussed.

*G. Jona Lasinio*“Time dependent large deviations and thermodynamic transformations”, [abstract]
We consider a system with a time dependent driving and we analyse
thermodynamic transformations between equilibrium or non-equilibrium
states. We then discuss the role the large deviation rate in the energy
balance of a transformation.

*A. Kupiainen*“Renormalizing Stochastic PDE's”, [abstract]
I will explain a Renormalization Group approach to the study of
existence and
uniqueness of solutions to stochastic partial differential equations driven
by
space-time white noise. As an example I sketch a proof of well-posedness
and independence of regularization for the $\phi^4$ model in three
dimensions
recently studied by Hairer and by Catellier and Chouk.

*J Kurchan*“Thermal dynamics and Darwinian dynamics”,*H. Lacoin*“Convergence to equilbrium for the exclusion process on the circle”, [abstract]
We consider the exclusion process on a circle with 2n sites and n
particles.
The particles jumps to neighboring sites independently with rate one, but
are forced to respect the exclusion rule:
there cannot be more than one particle per site.
In this talk we will discuss the convergence to equilibrium of this system
in terms of total variation distance to equilibrium, and in particular the
cutoff phenomenon: around a time $C N^2\log N$ the distance to equilibrium
drops abruptly from $1$ to $0$ in a time window of width $N^2$.

*J. Lebowitz*“Lee-Yang Zeros, Central limit Theorems and More”, [abstract]
I will discuss some old and some recent work
(joint with B. Pittel, D. Ruelle and E. Speer) concerning
the relation between the zeros of the grand canonical partition function
and different properties,e.g. (Local) Central Limit Theorems, for
equilibrium systems.
Many results extend to determinantal point processes, such as the
distribution
of eigenvalues of random matrices, and also to graph counting polynomials.

*R. Livi*“Checking energy transport in new models”, [abstract]
The use of simple lattice models with short range interactions,
like the FPU chains or lattice gases, unveiled the presence of anomalous
energy transport in low-dimensional systems. On the other hand, recent
numerical studies seem to predict quite contradictory results, also with
the predictions of fluctuating hydrodynamics, that provided a solid
theoretical ground for the universality of KPZ-scaling characterizing
the divergence of transport coefficients in 1d models. We present some
new results obtained for gas-like models, performed by the
multi-particle-collision (MPC) protocol, that allow to understand the
reason of the
discrepancy observed in lattice models. Moreover, we shall briefly
report on new, partially unexpected scenarions of energy transport
in systems with long-range interactions.

*C. Perez Espigarez*“The spatial fluctuation theorem”, [abstract]
For systems of interacting particles, or for interacting
diffusions in d dimensions, driven out-of- equilibrium by an external
field, a spatial fluctuation relation for the generating function of the
current is derived as a consequence of spatial symmetries. Those symmetries
are in turn associated to spatial transformations on the physical space
that leave invariant the path space measure without driving. This shows
that in dimension d ≥ 2 new fluctuation relations arise beyond the
Gallavotti- Cohen fluctuation theorem related the time-reversal symmetry.

*L. Rolla*“Absorbing-state Phase Transitions: Challenges for Mathematicians”, [abstract]
Modern statistical mechanics offers a large class of
driven-dissipative systems that naturally evolve to a critical state.
Here we consider two infinite-volume systems: the activated random
walks and the stochastic sandpile.

The main goal in this field is to describe the critical behavior, the scaling relations and critical exponents of these systems, and whether their critical density is the same as the long-time limit attained in their driven-dissipative finite-volume version. These questions are however far beyond the reach of current techniques. Due to strong non-locality of correlations and dynamic long-range effects, classical analytic and probabilistic tools fail in most cases of interest, making the rigorous analysis of such systems a major mathematical challenge.

In this talk we will report on the progress obtained in recent years. We will show some novel ideas and techniques which allowed some steps forward in understanding the phase transition in these systems, and discuss some of the open problems.

The main goal in this field is to describe the critical behavior, the scaling relations and critical exponents of these systems, and whether their critical density is the same as the long-time limit attained in their driven-dissipative finite-volume version. These questions are however far beyond the reach of current techniques. Due to strong non-locality of correlations and dynamic long-range effects, classical analytic and probabilistic tools fail in most cases of interest, making the rigorous analysis of such systems a major mathematical challenge.

In this talk we will report on the progress obtained in recent years. We will show some novel ideas and techniques which allowed some steps forward in understanding the phase transition in these systems, and discuss some of the open problems.

*E. Saada*“Supercritical behavior of asymmetric zero-range process with sitewise disorder”, [abstract]
This talk is based on joint works with
C. Bahadoran, T. Mountford, K. Ravishankar.
We establish necessary and sufficient conditions for
weak convergence to the upper invariant measure for
asymmetric nearest neighbour zero range processes
with non homogeneous jump rates.

*T. Sasamoto*“Fluctuations for one-dimensional Brownian motions with oblique reflection”, [abstract]
We consider a system of Brownian motions in one-dimension in which the j-th
particle
is reflected by the (j+1)-th particle with weight p and also by the
(j-1)-th particle with
weight q. The system is defined more precisely using the local time for
positions of
neighboring particles.

The system with a symmetric (p=q=1/2) reflection, corresponding to independent Brownian motions with ordering maintained, was introduced by Harris in 1965 and has been studied by many authors since then.

In the totally asymmetric (q=1) case, a particle with smaller index has a priority to the one with a larger index; the latter is simply reflected by the former. The finite particle system for this special case was discussed by Warren and others. More recently, there have been some progress for more general initial conditions by by Ferrai, Spohn and Weiss.

In this presentation we consider generic asymmetric case where 0<p<q<1. The large time properties are expected to be similar to the totally asymmetric case, i.e., to belong to the KPZ universality class. But the techniques for the totally asymmetric case do not work for the general case. Our analysis is based on a duality (in fact a self-duality) property for the process, which allows us to obtain a few formulas for quantities related to current and discuss the asymptotics.

The presentation is based on a collaboration with H. Spohn[SS].

[SS] T. Sasamoto, T. Spohn: Point-interacting Brownian motions in the KPZ universality class, arXiv: 1411.3142

The system with a symmetric (p=q=1/2) reflection, corresponding to independent Brownian motions with ordering maintained, was introduced by Harris in 1965 and has been studied by many authors since then.

In the totally asymmetric (q=1) case, a particle with smaller index has a priority to the one with a larger index; the latter is simply reflected by the former. The finite particle system for this special case was discussed by Warren and others. More recently, there have been some progress for more general initial conditions by by Ferrai, Spohn and Weiss.

In this presentation we consider generic asymmetric case where 0<p<q<1. The large time properties are expected to be similar to the totally asymmetric case, i.e., to belong to the KPZ universality class. But the techniques for the totally asymmetric case do not work for the general case. Our analysis is based on a duality (in fact a self-duality) property for the process, which allows us to obtain a few formulas for quantities related to current and discuss the asymptotics.

The presentation is based on a collaboration with H. Spohn[SS].

[SS] T. Sasamoto, T. Spohn: Point-interacting Brownian motions in the KPZ universality class, arXiv: 1411.3142

*H. Spohn*“One-dimensional Kardar-Parisi-Zhang equation with several components”, [abstract]
The scalar KPZ equation is essentially equivalent to a disordered
system of equilibrium statistical mechanics, namely the directed polymer in
a random medium. Interesting physical applications concern several
components.
Then the mapping seems to break down. I will discuss lattice gases with
several
species in one dimension and explain the dramatic differences when compared
to the scalar case.

*J. Tailleur*“What is the Pressure of an Active Particle Fluid?”, [abstract]
Pressure is the mechanical force per unit area that a confined
system exerts on its container. In thermal equilibrium, the pressure
depends only on bulk properties (density, temperature, etc.) through an
equation of state. I will show that in active systems containing
self-propelled particles, the pressure instead can depend on the precise
interactions between the system's contents and its confining walls. Generic
active fluids therefore have no equation of state. I will discuss how one
is recovered in certain limiting cases, which include "active Brownian
spheres", a much-studied simplified model of self-propelled particles. Even
in these cases, the mechanical pressure can exhibit anomalous properties
that defy the familiar thermodynamic description of passive fluid materials.

*C. Toninelli*“Kinetically constrained models: non-equilibrium dynamics”, [abstract]
Kinetically Constrained Models (KCM) are interacting particle systemswhich
have been introduced by physicists to model the liquid/glass transition
and more generally the "glassy behavior" occurringin a large variety of
systems. The key feature is that an elementary move is allowed only if the
configurationverifies a local constraint which specifies the maximal number
of particles in a proper neighborhood. The study of the non equilibrium
behavior of KCM poses very challenging problems. On the one hand numerical
simulations often do not give clear cut answers in this regime due to the
extremely slow dynamics. On the other hand the standard mathematical tools
cannot be used due to the the non attractiveness and the failure of
classic coercive inequalities to analyze relaxation to equilibrium. In this
seminar we will present some results as well as open problems on the non
equilibrium behavior for these models, in particular the so called East
model.

*D. Tsagkarogiannis*“Current reservoirs in the simple exclusion process.”, [abstract]
Stationary non equilibrium states are characterized by the
presence of steady currents flowing through the system as a response to
external forces. We model this process considering the simple exclusion
process in one space dimension with appropriate boundary mechanisms which
create particles on the one side and kill particles on the other. The
system is designed to model Fick's law which relates the current to the
density gradient. In this talk we review some results obtained
in collaboration with Anna De Masi, Errico Presutti and Maria Eulalia
Vares. For this process, the hydrodynamic limit is given by the linear heat
equation with Dirichlet boundary conditions obtained by solving a
non-linear equation which fixes the values of the density at the boundary.
The rescaled limiting density profile of the (unique) invariant measure of
the process coincides with the unique stationary solution of the
hydrodynamic equation. Last, we show a spectral gap estimate for the (non
equilibrium) stationary process uniformly on the system size.