Research interests
Out of equilibrium systems

1/ Exclusion processes
  • Exclusion processes are archetypes of out-of-equilibrium systems. They were among the first examples in which large deviation functions could be calculated explicitely (see below). They consist in a one-dimensional lattice on which pointlike particles hop from site to site. No more than one particle can occupy a given site - hence the name exclusion process.
  • They can also be considered as simple models for transport. Though exact solutions exists for the stationary states, some phenomenological theory called 'domain wall theory' (DWT) has been proposed. It gives a more intuitive understanding of the system; allows to obtain results not only for the stationary but also transient states; and is extended more easily to variants of the exclusion processes.
  • Recently, J. Cividini, with H. Hilhorst and myself, has extended the classical DWT (valid for continuous time dynamics) to deterministic parallel update (a time update more relevant for application to road or pedestrian traffic). The DWT theory can then be made exact even at the microscopic scale.
  • We are also interested in extensions of exclusion processes to multi-lane systems. Indeed, many applications, for example to road traffic or intracellular transport, take place on several parallel tracks.
2/ Large deviations
  • The thermodynamics that we learn at school deals essentially with equilibrium or near to equilibrium systems. It is not clear yet, though great progresses have been made in the field, what would be a thermodynamics of out-of-equilibrium systems.
  • The relevant framework to study out-of-equilibrium systems is the one of large deviations (or equivalently via a Legendre transform, of moments / cumulants generating functions). Actually, when you do equilibrium thermodynamics, you are dealing with large deviations, may be without noticing. This frame can be generalized to out-of-equilibrium systems. For example, for a particle system maintained in a stationnary state, you will be interested in the cumulant generating function of the current over a given time period.
  • Among the attempts to develop out-of-equilibrium thermodynamics, the thermodynamics formalism (introduced first by Ruelle in the frame of dynamical systems) has extended the notion of partition function to out-of-equilibrium systems : the sum over the configurations is replaced by a sum over trajectories, and the inverse temperature β becomes just a parameter that allows to scan the structure of the probability distribution that is under study. Non analyticities of the cumulant generating functions are then called dynamical phase transitions. It is still an open problem to understand the physical meaning of these. It seems that they could play an important role in particular in glassy transitions.
  • This thermodynamic formalism has been extended to Markov processes (discrete time). With M. Ernst and B. Dorfman, we have studied the case of disordered Markov processes (Lorentz gas) and shown how dynamical phase transitions were arising from the structure of frozen disorder.
  • Later, with F. van Wijland and V. Lecomte, we have shown that it was possible to apply this formalism to Markov dynamics, provided a proper interpretation of the definition of the dynamical partition function would be used. Besides, we have shown that the thermodynamic formalism was actually a special case of large deviation function for a specific observable, and that it could be generalized to a whole family of observables.
  • We have applied this approach to various models, and in particular:
    • The (TA)SEP model. It is an archetype of out-of-equilibrium systems.
    • Mean-field equilibrium systems that can be described by a macrocopic description (e.g. magnetization in the Ising model) and for which a detailed balance hold. Then the dominant order for cumulant generating functions (at least for some well chosen observables) can be easily obtained, and some dynamical phase transitions are shown to occur. This allows to study the relation between dynmical and thermodynamical phase transitions.