Abstract
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 ℓ 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 ℓ2. It is also shown how one can obtain, via a scaling limit from the logarithmic Sobolev inequality, the exponential decay of a Lyapunov functional for 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
D. Aldous and P. Diaconis, The asymmetric one-dimensional constrained Ising model: rigorous results, J.Statist.Phys. 107:945–975 (2002).
C. An´e, S. Blachère, D. Chafa¨, P. Fougàres, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les in´egalit´es de Sobolev logarithmiques. Panoramas et Synthèses, 10 (Soci´et´e Math´ematique de France, Paris, 2000).
R. Arratia, The motion of a tagged particle in the simple symmetric exclusion system on Z., Ann.Probab. 11:362–373 (1983).
A. Barrat, J. Kurchan, V. Loreto and M. Sellitto, Edwards measure for powders and glasses, Phys.Rev.Lett. 85:5034–5038 (2000).
L. Bertini and B. Zegarlinski, Coercive inequalities for Kawasaki dynamics. The product case, Markov Processes Relat.Fields 5:125–162 (1999).
N. Cancrini and F. Martinelli, On the spectral gap of Kawasaki dynamics under a mixing condition revisited, J.Math.Phys. 41:1391–1423 (2000).
N. Cancrini, F. Martinelli and C. Roberto, The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited, Ann.Inst.H.Poincar´e Probab.Statist. 38:385–436 (2002).
L. F. Cugliandolo, Dynamics of Glassy Systems. arXiv: cond-mat/0210312v2.
P. Dai Pra and G. Posta, Logarithmic Sobolev Inequality for Zero-Range Dynamics. Preprint 2004. arXiv math.PR/0401248.
P. G. De Benedetti, Metastable Liquids (Princeton University Press, Princeton, 1997).
J. D. Deuschel, Algebraic L2 decay of attractive critical processes on the lattice, Ann.Probab. 22:264–283 (1994).
S. Feng, I. Iscoe and T. Sepp¨al¨ainen, A microscopic mechanism for the porous medium equation, Stochastic Process.Appl. 66: 147–182 (1997).
W. Gotze in Liquids Freezing and the Glass Transition, Hansen, D. Levesque, J. Zinn-Justin Z., eds., (North-Holland, Amsterdam, 1991) pp. 287–503.
H. M. Jaeger, J. B. Knight and R. P. Behringer, Granular solids, liquids, and gases, Rev.Mod.Phys. 68:1259–1273 (1996).
E. Janvresse, C. Landim, J. Quastel and H. T. Yau, Relaxation to equilibrium of conservative dynamics. I. Zero-range processes, Ann.Probab. 27:325–360 (1999).
C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm.Math.Phys. 104:1–19 (1986).
T. R. Kirkpatrick and D. Thirumalai, Dynamics of the structural glass transition and the ip-spin-interaction spin-glass model, Phys.Rev.Lett. 58:2091–2094 (1987).
T. R. Kirkpatrick, D. Thirumalai and P. G. Wolynes, Scaling concepts for the dynamics of viscous liquids near an ideal glassy state, Phys.Rev.A 40:1045–1054 (1989).
W. Kob and H. C. Andersen, Kinetic lattice-gas model of cage effects in high density liquids and a test of mode-coupling theory of the ideal glass transition, Phys.Rev.E 48:4364–4377 (1993).
J. Kurchan, L. Peliti and M. Sellitto, Aging in lattice-gas models with constrained dynamics, Europhys.Lett. 39:365–370 (1997).
A.J. Liu, S.R. Nagel, eds., Jamming and rheology: constrained dynamics on microscopic and macroscopic scales(Taylor and Francis, London, 2001).
S. L. Lu and H. T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm.Math.Phys. 156:399–433 (1993).
M. Mezard, Statistical physics of the glass phase, Physica A 306:25–38 (2002).
J. Quastel, Diffusion of color in the simple exclusion process, Comm.Pure Appl.Math. 45:623–679 (1992).
F. Ritort and P. Sollich, Glassy dynamics of kinetically constrained models, Adv.Phys. 52: 219 (2003).
M. Sellitto and J. J. Arenzon, Free-volume kinetic models of granular matter, Phys.Rev.E 62:7793–7796 (2000).
H. Spohn, Tracer diffusion in lattice gases, J.Statist.Phys. 59:1227–1239 (1990).
H. Spohn, Large Scale Dynamics of Interacting Particles Springer, Berlin, (1991).
L. C. E. Struick, Physical aging in amorphous polymers and other materials (Elsevier, Houston, 1976).
C. Toninelli, G. Biroli and D. S. Fisher, Spatial structures and dynamics of kinetically constrained models of glasses Phys.Rev.Lett. 18: 185504 (2004).
C. Toninelli, G. Biroli, Dynamical arrest, tracer diffusion and kinetically constrained lattice gases, to be published in J.Stat.Phys.
H.-T. Yau, Logarithmic Sobolev inequality for lattice gases with mixing conditions, Comm.Math.Phys. 181:367–408 (1996).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bertini, L., Toninelli, C. Exclusion Processes with Degenerate Rates: Convergence to Equilibrium and Tagged Particle. Journal of Statistical Physics 117, 549–580 (2004). https://doi.org/10.1007/s10955-004-3453-3
Issue Date:
DOI: https://doi.org/10.1007/s10955-004-3453-3