The general themes of my research are** mathematical equilibrium and non-equilibrium statistical
physics, and interacting particle systems.**

Statistical physics studies the emergence of macroscopic

phenomena such as phase transitions and transport phenomena from the microscopic

world of interacting particles (atoms, molecules). It is intimitaly related to probability theory.

Understanding the** transition from micro to macro** is important both in physics and in biology, e.g. in

1. Describing **phase transitions** in equilibrium statistical mechanics.

2. Rigorously **deriving macroscopic equations** such as the diffusion equation from micro dynamics. This is the field of ``hydrodynamic limits''. Understanding the deviations from these equations for finite but large systems (**large deviations**).

3. Understanding **non-equilibrium steady states**, i.e., the distribution of microscopic degrees of freedom in a system driven away from equilibrium, e.g. in contact with two heat reservoirs at different temperatures. Typically such non-equilibrium states have long-range correlations.

4. Understanding the dynamics of traits of a population on mesoscopic time scales in evolution (**adaptive dynamics**).

Below I describe some current research themes in more detail.

Preprints of most of my papers can be found on the archive

in the section mathematical physics.

Published material can be found on http://www.ams.org/mathscinet/

**Current research themes:**

**1. Duality in interacting particle systems.**

Duality is a useful tool by which two Markov processes can be linked. The way in which the two processes are linked is via a so-called duality function. In this paper we show that duality is related with an algebra of ``creation and annihilation'' operators that is naturally associated with the generator. Different representations of this algebra can give rise to different Markov processes which are then dual with each other and the duality functions correspond to ``the change of basis'' to go from one representation to the other. In that sense processes which are dual to eachother have the same ``abstract generator''. Examples include diffusion processes dual to particle systems, deterministic dynamics (system of ODE) dual to random walkers. Models of this type are useful in e.g. non-equilibrium statistical physics (heat and mass transport) and population dynamics.

The steps of the abstract method are roughly the following (more details e.g. in section 2 of this paper)

1. Identify the generator of the Markov process as an element of a Lie-algebra.

2. By passing to another representation of this Lie algebra one can find another operator that (if one is lucky) is again a generator. This then automatically gives a dual process and the intertwiner is the duality function.

3. To find self-duality functions, one has to look for non-trivial operators that commute with the generator.

The method can applied constructively by starting from a ``nice'' (i.e., commuting with many elements) element of a Lie algebra such as the Casimir operator and constructing a Markov process from it by a similarity (ground-state) transformation.

This is in particular useful if one want to construct asymmetric models with self-duality properties, via q-deformations of the Lie algebras of the corresponding symmetric processes. For the q-deformation of SU(2) this leads to generalized asymmetric exclusion processes (see this paper), and for q-deformation of SU(1,1) to a asymmetric version of the KMP model and the Symmetric Inclusion Process (see this paper). In the context of population dynamics, this corresponds to diffusion processes with selection.

Self-duality is also a very useful property especially for infinite particle systems such as the exclusion process, because it expresses time dependent expectations of special functions (the self-duality functions) in terms of the evolution of only a finite number of particles. Self-duality functions are obtained from an operator that commutes with the generator.

A new process that comes out of this analysis is the so-called ``symmetric inclusion process'' (SIP) which is a natural analogue of the symmetric exclusion process (SEP), where instead of forbidding particles to be at the same place, we stimulate them to be at the same place. Some properties of this process, such as self-duality and propagation of positive correlations for a class of initial measures are discussed in SIP.

Models such as the SIP and its associated interacting diffusion process are ``exactly solvable'' and can be used to model non-equilibrium phenomena such as heat conduction and particle transport. See e.g. this paper.

This work is in collaboration with G. Carinci (Modena), C. Giardina (Modena), J. Kurchan (Paris), K. Vafayi (Leiden), T. Sasamoto (Tokyo)

**2. Transformations and time-evolution of Gibbs measures**.

Gibbs measures are a probabilistic model for systems in equilibrium. More mathematically speaking, Gibbs measures are a particular class of joint distribution of random variables defined on the vertices of a lattice, inspired by the Boltzmann-Gibbs weights of statistical mechanics. Originally, the formalism of Gibbs measures, developed by Dobrushin, Lanford and Ruelle (DLR) is a mathematically rigorous way to give meaning to the Boltzmann-Gibbs weights of statistical mechanics for** infinite systems** and thus to define **equilibrium**. This is done by starting from a collection of local interactions which give a consistent collection of conditional probabilities in Boltzmann-Gibbs form. A Gibbs measure is then a probability measure on configurations of the infinite system that has precisely these prescribed conditional probabilities. Since there can be more such measures of the infinite systems, this formalism captures well the phenomenon of first order phase transitions.

Alternatively, one can think of Gibbs measures as a natural extension of Markov random fields. The condition for a probability measure to be Gibbs means intuitively that the conditional distribution of the random variable at one vertex, given all the others, **depends** **uniformly weakly on the variables on vertices very far away**: this is the so-called** quasi-locality** property. Depending on the interpretation of the random variables associated to the vertices, Gibbs measures can model many models with interacting components such as magnetic systems (Ising spins, continuous spins), lattice gases, graphical models and images.

A natural question is whether the quasi-locality property behaves well under (stochastic) transformations, and/or application of a Markovian dynamics such as Glauber or Kawasaki dynamics. Applying a dynamics to a Gibbs measure models processes as heating and cooling or transport/diffusion. Tranformations are natural e.g. in the context of image analysis, where one considers noisy images or in information theory where one considers distorted messages.

The question is then whether the states at a given time can be described as equilibrium states with some ``reasonable'' interaction. At first sight surprisingly, this turns out not to be the case: the quasilocality property can be lost in the course of time, and sometimes also be recovered. Recently we gave a large deviation interpretation of this phenomenon, which is related to `**`finding the optimal history that leads to an exceptional state at a given time T''**. **Uniqueness or non-uniqueness of these trajectories** is connected with the preservation or loss of quasilocality. Here and here are two examples of papers related to this subject.

Here is an illustration of heating up or cooling down in the Ising model.

The work around this theme has been done in collaboration with

A. van Enter (Groningen), R. Fernandez (Utrecht), F. den Hollander (Leiden), C. Kuelske (Bochum), A. Le Ny (Paris), S. Roelly (Potsdam), W. Ruszel (Eindhoven), F. Wang (Leiden)

**3. Self-organized criticality and abelian sandpiles.**

Since its introduction in 1987, the Bak-Tang-Wiesenfeld (BTW) sandpile-model of self-organized criticality has attracted a lot of interest.Â In the combinatorics community it is known under the name ``chip-firing". The model is very easy to describe: consider a finite subset V of e.g. the two-dimensional lattice Z^2. To each site x we associate a height h,Â an integer between 1 and 4 (number

of grains at site x). We pick a random site and add one grain to that site. If h+1 > 4, then the site ``topples", giving one grain to each neighbor inside V, and loosing itself four grains. A site on the boundary looses grains when it topples. If we continue this process, then the system will end up in a stationary state, which is the uniform measure on the unique recurrent class. Recurrent configurations can be caracterized by the so-called ``burning algorithm" invented by Dhar. They are in one-to-one correspondence with rooted spanning trees. Pointwise addition and relaxation of recurrent configurations defines an abelian group, the so-called ``sandpile group of the graph" (in this case Z^2). Numerical studies and exact results indicate that in the stationary state of this model, the correlation between height variables decays as a power, just as for an equilibrium system at the critical point. Therefore BTW called this phenomenon** ``self-organized criticality". **

Our research on this subject tries to answer the following questions:

1) Do the stationary measures in finite volume converge (in the thermodynamic limit)

to a unique measure on infinite volume height configurations ?

2) Can we define a Markov process on infinite volume height-configurations with this

measure as a unique stationary measure?

3) What are the ergodic properties of this Markov process?

4) Can we say something about finiteness or -more ambitiously- statistics of avalanches ?

5) What can we say about the abelian group structure in infinite volume ?

Some answers to these questions can be found in Les Houches lecture notes 2005 and references therein.

Recent papers include the dissipative model with arbitrary small dissipation (giving some non-trivial lower bounds on the avalanches of the critical model), and the model on an infinite random binary tree.

A very nice computer experiment for the BTW model can be found on the

website:

http://www.cmth.bnl.gov/~maslov/Sandpile.htm

Many recent developments on this subject can also be found on the home page of Anne Fey.

The work around this theme has been done in collaboration with

A. Fey (Delft), A. Jarai (Bath), C. Maes (Leuven), R. Meester (Amsterdam), E. Saada (Paris), W. Ruszel (Eindhoven).

**4. Other themes of research**.

-Large deviations for multiple ergodic averages.

- Concentration inequalities and their applications for Markov processes, interacting particle systems, speed of relaxation to equilibrium.

- Random walk in dynamic random environment.

- Condensation in the inclusion process.

-Wealth Distribution models (1,2)

- Adaptive dynamics: derivation of diffusion limits from microscopic individual-based models.