Preprint A521/2007
Hydrodynamic Limit for a Particle System with degenerate rates

Cristina Toninelli | GonÃ§alves, Patricia | Landim, Claudio

**Keywords: **
Hydrodynamic limit | Porous medium equation | Spectral Gap | Degenerate Rates

We study the hydrodynamic limit for some conservative particle
systems with degenerate rates, namely with nearest neighbour
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.