Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Couplings, attractiveness and hydrodynamics for conservative particle systems

Thierry Gobron and Ellen Saada

Full-text: Open access

Abstract

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived Markovian coupled process (ξt, ζt)t≥0 satisfies:

(A) if ξ0ζ0 (coordinate-wise), then for all t≥0, ξtζt a.s.

In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y, with k≥1. These models include simple exclusion for which k=1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the rates to insure attractiveness; we construct a Markovian coupled process which both satisfies (A) and makes discrepancies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions. We apply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k≤2) which arises from a solid-on-solid interface dynamics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley–Aldous–Diaconis model. We derive the hydrodynamic limit of these two one-dimensional models.

Résumé

L’attractivité est un outil fondamental d’étude des systèmes à une infinité de particules en interaction; la construction du couplage de base est la méthode habituelle pour démontrer cette propriété (par exemple pour l’exclusion simple). Le processus couplé markovien (ξt, ζt)t≥0 obtenu vérifie:

(A) si ξ0ζ0 (coordonnée par coordonnée), alors pour tout t≥0, ξtζt p.s.

Nous considérons dans cet article des systèmes de particules conservatifs sur ℤd qui généralisent le processus des misanthropes en ce que, à chaque transition, k particules peuvent sauter d’un site x vers un autre site y, avec k≥1. Ces modèles incluent l’exclusion simple, où k=1, mais, au-delà de cette valeur, le couplage de base n’est plus valide et il faut une autre construction. Nous obtenons des conditions nécessaires et suffisantes pour l’attractivité sur les taux de transition; nous construisons un processus couplé markovien qui à la fois satisfait (A), et fait décroitre les discrépances entre ses deux marginales. Nous déterminons les probabilités invariantes et invariantes par translation extrémales sous des conditions générales d’irréductibilité. Nous appliquons nos résultats à des exemples incluant un modèle d’exclusion asymétrique à deux espèces avec conservation de la charge (où k≤2) issu d’une dynamique d’interfaces ‘solid-on-solid’, et un modèle de batons (où k n’est pas borné) en correspondance avec un processus de Hammersley–Aldous–Diaconis discret généralisé. Nous obtenons la limite hydrodynamique de ces deux modèles unidimensionnels.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 4 (2010), 1132-1177.

Dates
First available in Project Euclid: 4 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1288878341

Digital Object Identifier
doi:10.1214/09-AIHP347

Mathematical Reviews number (MathSciNet)
MR2744889

Zentralblatt MATH identifier
1252.60093

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Keywords
Conservative particle systems Attractiveness Couplings Discrepancies Macroscopic stability Hydrodynamic limit Misanthrope process Discrete Hammersley–Aldous–Diaconis process Stick process Solid-on-solid interface dynamics Two-species exclusion model

Citation

Gobron, Thierry; Saada, Ellen. Couplings, attractiveness and hydrodynamics for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 1132--1177. doi:10.1214/09-AIHP347. https://projecteuclid.org/euclid.aihp/1288878341


Export citation

References

  • [1] E. D. Andjel. Invariant measures for the zero range process. Ann. Probab. 10 (1982) 525–547.
  • [2] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. A constructive approach to Euler hydrodynamics for attractive processes. Application to k-step exclusion. Stochastic Process. Appl. 99 (2002) 1–30.
  • [3] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34 (2006) 1339–1369.
  • [4] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Strong hydrodynamic limit for attractive particle systems on ℤ. Electron. J. Probab. 15 (2010) 1–43.
  • [5] M. Balázs. Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 639–685.
  • [6] M. Balázs, F. Rassoul-Agha, T. Seppäläinen and S. Sethuraman. Existence of the zero range process and a deposition model with superlinear growth rates. Ann. Probab. 35 (2007) 1201–1249.
  • [7] M. Bramson and T. Mountford. Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 (2002) 1082–1130.
  • [8] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verv. Gebiete 70 (1985) 509–523.
  • [9] P. Collet, F. Dunlop, D. Foster and T. Gobron. Product measures and dynamics for solid-on-solid interfaces. J. Stat. Phys. 89 (1997) 509–536.
  • [10] P. A. Ferrari and J. B. Martin. Multi-class processes, dual points and M/M/1 queues. Markov Process. Related Fields 12 (2006) 175–201.
  • [11] J. Fritz and K. Nagy. On uniqueness of the Euler limit of one-component lattice gas models. Alea 1 (2006) 367–392.
  • [12] J. Fritz and B. Tóth. Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas. Comm. Math. Phys. 249 (2004) 1–27.
  • [13] B. M. Kirstein. Monotonicity and comparability of time-homogeneous Markov processes with discrete state space. Math. Oper. Stat. 7 (1976) 151–168.
  • [14] T. Kamae and U. Krengel. Stochastic partial ordering. Ann. Probab. 6 (1978) 1044–1049.
  • [15] T. Kamae, U. Krengel and G. L. O’Brien. Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 (1977) 899–912.
  • [16] C. Kipnis and C. Landim. Scaling Limits for Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin, 1999.
  • [17] T. M. Liggett. Coupling the simple exclusion process. Ann. Probab. 4 (1976) 339–356.
  • [18] T. M. Liggett. Interacting Particle Systems. Springer, New York, 2005.
  • [19] A. W. Massey. Stochastic orderings for Markov processes on partially ordered spaces. Math. Oper. Res. 12 (1987) 350–367.
  • [20] T. Seppäläinen. A microscopic model for the Burgers equation and longest increasing subsequences. Electron. J. Probab. 1 (1996), approx. 51 pp. (electronic).
  • [21] D. Stoyan. Comparison Methods for Queues and Other Stochastic Models. Wiley, New York, 1983.
  • [22] F. Tabatabaei and G. M. Schütz. Shocks in the asymmetric simple exclusion process with internal degree of freedom. Phys. Rev. E 74 (2006) 051108.
  • [23] F. Tabatabaei and G. M. Schütz. Nonequilibrium field-induced phase separation in single-file diffusion. Diffusion Fundamentals 4 (2006) 5.1–5.38.
  • [24] S. R. S. Varadhan. Lectures on hydrodynamic scaling. In Hydrodynamic Limits and Related Topics (Toronto, ON, 1998). Fields Inst. Commun. 27 3–40. Amer. Math. Soc., Providence, RI, 2000.