# Large Scale Stochastic Dynamics

### Claudio Landim

Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil### Stefano Olla

Université de Paris Dauphine, France### Herbert Spohn

TU München, Garching bei München, Germany

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## Abstract

``Large Scale Stochastic Dynamics'' is at the crossroad of probability theory and statistical physics. One central theme of statistical physics is the emergent behavior resulting from the interaction of many identical components, the paradigm being a fluid or a gas. On the atomistic scale they consist of a huge number of identical molecules. Their motion is governed by Newton's equation of classical mechanics (ignoring quantum effects). The emergent description, valid only for particular initial states and on a sufficiently coarse space-time scale, are the compressible Navier-Stokes equations of fluid dynamics. Roland Dobrushin (1929-1995) and Frank Spitzer (1926-1992) had the vision that in the context of stochastic dynamics with many identical components the issue of emergent behavior is both mathematically challenging and important in modeling applications. The latter judgment turned out to be more than true. Stochastic algorithms, such as kinetic Monte Carlo, importance sampling, Monte Carlo Markov chains, Glauber dynamics, and others, are daily practice. Their mathematical vision has evolved over the past twenty years into a rich, multifaceted research program. Our workshop is like a snap-shot of the current activities, in fact quite distinct from the snap-shot taken exactly three years ago. A partial list of topics reads\\[-2mm] \begin{itemize} \item energy transport in anharmonic lattices \item large deviations for diffusive systems \item motion of tracer particles \item nonequilibrium phase transitions \item superdiffusivity and the Airy processes \item hydrodynamic limits \item random polymers \end{itemize}\bigskip We had 44 participants from 11 countries, mostly probabilists, but also experts from partial differential equations and statistical physics. They all enjoyed tremendously the unique and stimulating atmosphere at the Mathematische Forschungsinstitut Oberwolfach and hope to return some day.

## Cite this article

Claudio Landim, Stefano Olla, Herbert Spohn, Large Scale Stochastic Dynamics. Oberwolfach Rep. 4 (2007), no. 3, pp. 2443–2530

DOI 10.4171/OWR/2007/42