Hydrodynamic limit for a particle system with degenerate rates

P. Gonçalves; C. Landim; C. Toninelli

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 4, page 887-909
  • ISSN: 0246-0203

Abstract

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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 kinetically constrained lattice gases (KCLG) which have been introduced and intensively studied in physical literature as simple models for the liquid/glass transition. Due to the degeneracy of rates there exist 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 of KCLG. 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.

How to cite

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Gonçalves, P., Landim, C., and Toninelli, C.. "Hydrodynamic limit for a particle system with degenerate rates." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 887-909. <http://eudml.org/doc/78060>.

@article{Gonçalves2009,
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 kinetically constrained lattice gases (KCLG) which have been introduced and intensively studied in physical literature as simple models for the liquid/glass transition. Due to the degeneracy of rates there exist 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 of KCLG. 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.},
author = {Gonçalves, P., Landim, C., Toninelli, C.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {hydrodynamic limit; porous medium equation; spectral gap; degenerate rates},
language = {eng},
number = {4},
pages = {887-909},
publisher = {Gauthier-Villars},
title = {Hydrodynamic limit for a particle system with degenerate rates},
url = {http://eudml.org/doc/78060},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Gonçalves, P.
AU - Landim, C.
AU - Toninelli, C.
TI - Hydrodynamic limit for a particle system with degenerate rates
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 887
EP - 909
AB - 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 kinetically constrained lattice gases (KCLG) which have been introduced and intensively studied in physical literature as simple models for the liquid/glass transition. Due to the degeneracy of rates there exist 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 of KCLG. 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.
LA - eng
KW - hydrodynamic limit; porous medium equation; spectral gap; degenerate rates
UR - http://eudml.org/doc/78060
ER -

References

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  6. [6] F. Ritort and P. Sollich. Glassy dynamics of kinetically constrained models. Adv. in Phys. 52 (2003) 219–342. 
  7. [7] H. Spohn. Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer-Verlag, 1991. Zbl0742.76002
  8. [8] C. Toninelli and G. Biroli. Jamming percolation and glassy dynamics. J. Stat. Phys. 126 (2007) 731–763. Zbl1120.82012MR2311884
  9. [9] J. L. Vazquez. An introduction to the mathematical theory of the porous medium equation. In Shape Optimization and Free Boundaries 261–286. M. C. Delfour and G. Sabidussi (Eds). Kluwer, Dordrecht, 1992. Zbl0765.76086MR1260981
  10. [10] H.-T. Yau. Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22 (1991) 63–80. Zbl0725.60120MR1121850

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