288 research outputs found
Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle
Stochastic lattice gases with degenerate rates, namely conservative particle
systems where the exchange rates vanish for some configurations, have been
introduced as simplified models for glassy dynamics. We introduce two
particular models and consider them in a finite volume of size in
contact with particle reservoirs at the boundary. We prove that, as for
non--degenerate rates, the inverse of the spectral gap and the logarithmic
Sobolev constant grow as . It is also shown how one can obtain, via a
scaling limit from the logarithmic Sobolev inequality, the exponential decay of
a macroscopic entropy associated to a degenerate parabolic differential
equation (porous media equation). We analyze finally the tagged particle
displacement for the stationary process in infinite volume. In dimension larger
than two we prove that, in the diffusive scaling limit, it converges to a
Brownian motion with non--degenerate diffusion coefficient.Comment: 25 pages, 3 figure
Effect of spatial arrangement of presynaptic calcium channels on the calcium current cooperativity of neurotransmitter release
The Continuum Directed Random Polymer
Motivated by discrete directed polymers in one space and one time dimension,
we construct a continuum directed random polymer that is modeled by a
continuous path interacting with a space-time white noise. The strength of the
interaction is determined by an inverse temperature parameter beta, and for a
given beta and realization of the noise the path evolves in a Markovian way.
The transition probabilities are determined by solutions to the one-dimensional
stochastic heat equation. We show that for all beta > 0 and for almost all
realizations of the white noise the path measure has the same Holder continuity
and quadratic variation properties as Brownian motion, but that it is actually
singular with respect to the standard Wiener measure on C([0,1]).Comment: 21 page
Last passage percolation and traveling fronts
We consider a system of N particles with a stochastic dynamics introduced by
Brunet and Derrida. The particles can be interpreted as last passage times in
directed percolation on {1,...,N} of mean-field type. The particles remain
grouped and move like a traveling wave, subject to discretization and driven by
a random noise. As N increases, we obtain estimates for the speed of the front
and its profile, for different laws of the driving noise. The Gumbel
distribution plays a central role for the particle jumps, and we show that the
scaling limit is a L\'evy process in this case. The case of bounded jumps
yields a completely different behavior
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