|| Research interests |
Out of equilibrium systems
1/ Exclusion processes
2/ Large deviations
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
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
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.
- 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.