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Hydrodynamic Limit for a Particle System with degenerate rates | | Patricia Goncalves
; Claudio Landim
; Cristina Toninelli
; | | Date: |
18 Apr 2007 | | Subject: | Probability (math.PR) | | Abstract: | We study the hydrodynamic limit for some conservative particle systems with
degenerate rates, namely with nearest neighbor exchange rates which vanish for
certain configurations. These models belong to the class of {sl kinetically
constrained lattice gases} (KCLG) which have been introduced and intensively
studied in physics literature as simple models for the liquid/glass transition.
Due to the degeneracy of rates for KCLG there exists {sl blocked
configurations} which do not evolve under the dynamics and in general the
hyperplanes of configurations with a fixed number of particles can be
decomposed into different irreducible sets. As a consequence, both the Entropy
and Relative Entropy method cannot be straightforwardly applied to prove the
hydrodynamic limit. In particular, some care should be put when proving the One
and Two block Lemmas which guarantee local convergence to equilibrium. We show
that, for initial profiles smooth enough and bounded away from zero and one,
the macroscopic density profile for our KCLG evolves under the diffusive time
scaling according to the porous medium equation. Then we prove the same result
for more general profiles for a slightly perturbed dynamics obtained by adding
jumps of the Symmetric Simple Exclusion. The role of the latter is to remove
the degeneracy of rates and at the same time they are properly slowed down in
order not to change the macroscopic behavior. The equilibrium fluctuations and
the magnitude of the spectral gap for this perturbed model are also obtained. | | Source: | arXiv, arxiv.0704.2242 | | Services: | Forum | Review | PDF | Favorites | |
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