Abstracts
Richard Blythe (Edinburgh) Fixation and coarsening in Moran-type processes
In this talk I will introduce stochastic birth-death processes of the Moran type, in which a population evolves step-by-step through a single individual being replaced by an offspring of another individual in the population. In a closed system, all but one species goes extinct, resulting in all individuals being of the same type (a trivial type of condensation). In a system open to migration, there can be long periods during which most individuals are of the same type (a transient condensate-like object). In this talk I will demonstrate that one multiple Moran-type processes are coupled together, regions in which one species dominates grow in size (coarsen), and that the way in which the system coarsens depends on the local noise strength and in particular whether a given site is typically near the state of fixation.
Martin Evans (Edinburgh) Explosive condensation in a mass transport model
We study a far-from-equilibrium system of interacting particles, hopping between sites of a 1D lattice with a rate which increases with the number of particles at interacting sites. We find that clusters of particles, which initially spontaneously form in the system, begin to move at increasing speed as they gain particles. Ultimately, they produce a moving condensate which comprises a finite fraction of the mass in the system. We show that, in contrast with previously studied models of condensation, the relaxation time to steady state decreases as an inverse power of lnL with system size L and that condensation is instantaneous for L→∞.
This is joint work with Bartek Waclaw (Edinburgh).
Thierry Gobron (Cergy-Pontoise) Couplings and attractive particle systems
Attractiveness is a fundamental property which can be turned into an efficient tool for the study of infinite interacting particle systems. It means that for a given infinitesimal generator, there exists a coupled process $(\xi_t,\zeta_t)_{t\ge 0}$ such that if
$ \xi_0 \le \zeta_0 $ (coordinate-wise) , then for all $t\ge 0$, $ \xi_t \le \zeta_t $ a.s.
Under irreducibility requirements, this property allows for the determination of the set of all extremal invariant and translational invariant measures, and provides a route to construct the hydrodynamical limit of the particle system.
We consider two types of models for which the basic coupling construction is not possible under the necessary and sufficient conditions for attractiveness. First, a class of generalized misanthrope processes on $Z^d$, for which at each transition, $k$ particles may jump together from a site $x$ to another site $y$, with $k>1$. Then we consider exclusion processes with speed change, where jump rate from a site $x$ to an empty site $y$ depends on the occupation on other sites. In both cases, we construct an increasing coupling under necessary and sufficient conditions for attractiveness, and show that the ''discrepancies'' between the two marginals can be also controlled, even when they are not initially ordered. We apply these results to simple one dimensional examples that illustrate the interest of this approach.
Joint work with Ellen Saada (Université Paris 5).
Richard Kraaij (Delft) Stationary product measures for conservative particle systems and ergodicity criteria
We study conservative particle systems on W^S, where S is countable and W = {0, ..., N} or the natural numbers. The rate of a particle moving from site x to site y is given by p(x,y) b(eta_x, eta_y), where eta_z is the number of particles at site z. Under assumptions on b and the assumption that p is finite range, which allow for the exclusion, zero range and misanthrope processes, we show exactly what the stationary product measures are.
Furthermore we show that a stationary measure mu is ergodic if and only if the tail sigma algebra of the partial sums is trivial under mu. This is a consequence of a more general result on interacting particle systems that shows that a stationary measure is ergodic if and only if the sigma algebra of sets invariant under the transformations of the process is trivial. We apply this result combined with a coupling argument on the stationary product measures to determine which product measures are ergodic. For the case that W is finite this gives a complete characterisation.
For the case that W is the set of natural numbers we show that for nearly all functions b a stationary product measure is ergodic if and only if it is supported by configurations with an infinite amount of particles. We show that this picture is not complete, we give an example of a system where b is such that there is a stationary product measure which is not ergodic, even though it concentrates on configurations with an infinite number of particles.
Frank Redig (Delft) Dynamics of condensation in the inclusion process
The inclusion process is an interacting particle system where particles perform random walks on a finite (or countable) set and have an attractive interaction.It is a natural bosonic counterpart of the extensively studied exclusion process. Exactly as for the exlcusion process, this model is exactly solvable by a self-duality property. The stationary distributions are products of discrete gamma distributions.If the random walk rate becomes small w.r.t. the attractiveinteraction and many particles are in the system, condensates -i.e. huge piles of particles- start to form. We analyse the limiting dynamics of these condensates. On an appropriate time scale, this dynamics consistof a mixture of diffusive merging of condensatesand random walk of individual condensates, until a single condensate is formed, which then continues to move as a random walk.
This is joint work with S. Grosskinsky and K. Vafayi.
Ellen Saada (Paris) Hydrodynamic behavior of conservative attractive particle systems in random environment; application to misanthropes and traffic models
We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on Z in random ergodic environment. The limit is given by the entropy solution of a scalar conservation law with a Lipschitz macroscopic flux function. Our result is a strong law of large numbers, that we derive by a constructive method.
We illustrate our result on various examples, such as generalizations of misanthropes processes and k-step K-exclusion, with different types of random environments. We will detail a traffic model and a queuing model.
This is based on joint works with C. Bahadoran (Clermont-Ferrand), H. Guiol (Grenoble), T. Mountford (Lausanne), K. Ravishankar (SUNY, New Paltz).