While dissipative evolution equations, both with and without randomness, have been widely studied in the mathematics community, it is much less understood how they arise from accepted physical models on a finer scale. We think, however, that such information is crucial for determining properties of the models on a coarser scale, and that this has strong implications, particularly for nonequilibrium problems and on how different systems can be coupled, thus creating interesting challenges for pure and applied mathematics alike.

The probability assumptions currently used in non-equilibrium statistical mechanics were inherited from the nineteenth and early twentieth centuries. They work very well provided either that the system is either very close to equilibrium or that the time interval between observations is not too small -- for example, in a gas this time interval is, in effect, assumed to be large compared with the time necessary to execute a collision. The probability assumptions that are used for well enough separated observations have the implication that the successive observational states can be treated as a Markov process. But physical instrumentation has moved on, and in the twenty-first century we have to reckon with the possibility of much shorter time intervals between observations, for which the assumption of a Markov process is not tenable. This talk is about how to develop a general framework of statistical mechanics which does not require such an assumption.

We discuss mechanisms which generate stochastic behavior in dynamical systems. In systems with impacts (billiards) there are two such mechanisms, dispersing and defocusing. It'll be explained why apparently there are no other such mechanisms, why defocusing is a crucial mechanism in systems with coexistence of chaotic and regular dynamics and why focusing components of the boundary in chaotic billiards should be absolutely focusing.

I will discuss a program to investigate the Fourier Law in a lattice of weakly coupled strongly chaotic dynamical systems. I will then illustrate the small part of such a program that has been carried out as of today.

I will sketch a plan for a way to derive a Langevin equation for the slow degrees of freedom of a Hamiltonian whose fast ones are mixing Anosov. It uses the Anosov-Kasuga adiabatic invariant, martingale theory and Ruelle's formula for weakly non-autonomous SRB measures.

After a short introduction on the microscopic origin of randomness in macroscopic equations and the role of noise in the analysis of phase transitions, I will give an outline on what known about phase transitions for particle systems in the continuum and present some open problems on which I am working.

I will review some recent work on the macroscopic non-equilibrium behavior of the Hamiltonian dynamics of chains of oscillators perturbed by a conservative noise. The stochastic perturbation is given by hypoelliptic diffusions on the momentum of particles, such that total energy, and eventually total momentum, is conserved. I will discuss hydrodynamic, kinetic, and weak coupling limits for these models. These are joint works with Giada Basile, Cedric Bernardin, Milton Jara, Carlangelo Liverani and Herbert Spohn.

We consider a parabolic stochastic differential equation with a (nonlinear) transport term perturbed by a conservative noise and discuss the limit in which both the viscosity and the noise strength vanish. The lack of uniqueness for the formal limiting equation implies the existence of two different large deviations scales. The coarser one is analyzed in a Young measure setting and the rate functional vanishes on measure valued solutions to the conservation law. In the finer scale the rate functional is finite only on weak solutions and it is given by the entropy production corresponding to a specific entropy function determined by an Einstein relation.

Mechanical systems with so-called wiggly energies have many local equilibria. Under slow external forcing a gradient system with such an energy may display rate-independent behavior. In this talk we discuss the mathematical justification of this limit. This is joint work with Lev Truskinovsky.

We consider weakly perturbed discrete wave-equations in a kinetic scaling limit: space and time are both scaled by a square of the coupling constant which is then taken to zero. As shown in [J. Lukkarinen and H. Spohn, Arch. Ration. Mech. Anal. 183 (2007) 93-162], in three dimensions and with a random perturbation of the masses of the particles, the kinetic limit of the disorder-averaged Wigner transform of the solution satisfies a certain linear phonon Boltzmann equation. We also briefly discuss how a similar scheme can be applied for non-linear perturbations with suitable random initial data. As the resulting Boltzmann equations are irreversible, this provides an example how non-Hamiltonian energy transport can arise in such scaling limits of Hamiltonian systems.

We present a new approach to give a rigorous justification of the nonlinear Boltzmann equation as a scaling limit of discrete deterministic evolution for large times which avoids the convergence problems of the BBGKY hierarchy. The method is based on a careful analysis of the distribution of collision trees which are extracted from the dynamics. At the moment the proof is complete in the case of kinetic annihilation, in the case of collisional dynamics some estimates are still missing. New sharp conditions on the regularity of the velocity distribution have been discovered, and examples for the failure of the Boltzmann equation will be given.

I discuss recent experimental data concerning ferromagnetism, a metal-insulator transition and colossal negative magnetoresistance in Eu_x Ca_{1-x} B_6 collected in the group of my colleague H.-R. Ott at ETH Zurich. I then propose a phenomenological model enabling me to interpret those data. In connection with this model, I discuss mathematical problems related to itinerant ferromagnetism (Zener's mechanism) and Anderson localization caused by random exchange interactions between electrons in a conduction band and localized spins. I indicate solutions to these problems and show how such solutions qualitatively explain the experimental data.

The dynamics of a single particle in a weak random environment is considered. We show that in a suitable scaling limit the electron density satisfies a heat equation. Related classical and quantum problems will be presented. We explain the key difference between the lattice and continuum models.

We study the problem of parameter estimation for time-series possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized single--scale model to such multiscale data. We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified. We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly.

It is a common understanding in Physics that dynamics at a given scale originate from dynamics on a finer scale. In this talk I will demonstrate how this works for the case of low energy solutions of the nonlinear Schrödinger and Hartree equations with external potentials. We show how the description of such solutions can be reduced to dynamics of rigidly moving well localized structures - solitons (ground states). I will review some recent results on and state open problems in this subject.

The FPU chain is a 1D chain of particles evolving via conservative, Hamiltonian dynamics. It has a huge physics literature starting with the work of F, P and U in 1947 to scrutinize some predictions of statistical mechanics, and many interesting scaling regimes connecting it to both hyperbolic conservation laws and dispersive dynamics depending on length and time scale. I will discuss rigorous results including the radiative decay of coherent initial data under linear dynamics and the existence of open sets for which coherent initial data survive for all times under nonlinear dynamics.

We review some recent work on averaging of motion in periodic potentials perturbed by slowly varying ones. We show that for a "critical" scaling, provided the trajectories are local minimizers of the action at an appropriate "mesoscale", the solutions converge to those described by the effective Lagrangians and effective Hamiltonians. We discuss a relations of the microscopic part to "temperature", and some other related issues.

In this talk we present a method to construct higher order Monte Caro numerical schemes for the coarse-graining of stochastic lattice systems with short and long range interactions. The main tool is the cluster expansion of the partition function of the conditioned (to the coarse-grained variables) measure. We also discuss a strategy on how to recover microscopic information from the coarse-grained one.

In this talk we show that the "cold" FPU chain with double-well energy will thermalize even if driven infinitely slow. The resulting thermodynamic behavior, which can be qualified as rubber elasticity, can be described fully analytically in the case when the energy is bi-quadratic. As microscopic dissipation is added, the elastic behavior is gradually replaced by ideal plasticity.