- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Variational calculation of transport coefficients in diffusive lattice gases

Chikashi Arita1,*, P. L. Krapivsky2,3,†, and Kirone Mallick3,‡

  • 1Theoretische Physik, Universität des Saarlandes, 66041 Saarbrücken, Germany
  • 2Department of Physics, Boston University, Boston, Massachusetts 02215, USA
  • 3Institut de Physique Théorique, IPhT, CEA Saclay and URA 2306, CNRS, 91191 Gif-sur-Yvette Cedex, France

  • *c.arita@physik.uni-saarland.de
  • †pkrapivsky@gmail.com
  • ‡kirone.mallick@cea.fr
PDF

Phys. Rev. E 95, 032121 – Published 9 March, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.032121

Abstract

A diffusive lattice gas is characterized by the diffusion coefficient depending only on the density. The Green-Kubo formula for diffusivity can be represented as a variational formula, but even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient can be computed only in the exceptional situation when the lattice gas is gradient. In the general case, minimization over an infinite-dimensional space is required. We propose an approximation scheme based on minimizing over finite-dimensional subspaces of functions. The procedure is demonstrated for one-dimensional generalized exclusion processes in which each site can accommodate at most two particles. Our analytical predictions provide upper bounds for the diffusivity that are very close to simulation results throughout the entire density range. We also analyze nonequilibrium density profiles for finite chains coupled to reservoirs. The predictions for the profiles are in excellent agreement with simulations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (40)

  1. “As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases [...] Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua,” p. 454 in D. Hilbert, Mathematical problems, international congress of mathematicians at Paris in 1900 [Bull. Am. Math. Soc. 8, 437 (1902)].
  2. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer-Verlag, New York, 1991).
  3. C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems (Springer-Verlag, Berlin, 1999).
  4. T. M. Liggett, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes (Springer, New York, 1999).
  5. S. Katz, J. L. Lebowitz, and H. Spohn, J. Stat. Phys. 34, 497 (1984).
  6. J. L. Lebowitz, E. Presutti, and H. Spohn, J. Stat. Phys. 51, 841 (1988).
  7. B. Derrida, Phys. Rep. 301, 65 (1998).
  8. B. Schmittmann and R. K. P. Zia, Statistical Mechanics of Driven Diffusive Systems, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1995), Vol. 17.
  9. G. M. Schütz, Exactly Solvable Models for Many-Body Systems Far From Equilibrium, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 2000), Vol. 19.
  10. R. A. Blythe and M. R. Evans, J. Phys. A 40, R333 (2007).
  11. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010).
  12. B. Derrida, J. Stat. Mech. (2007) P07023.
  13. T. Bodineau and B. Derrida, C. R. Phys. 8, 540 (2007).
  14. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, J. Stat. Phys. 107, 635 (2002).
  15. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Rev. Mod. Phys. 87, 593 (2015).
  16. H. Spohn, J. Phys. A 16, 4275 (1983).
  17. J. S. Hager, J. Krug, V. Popkov, and G. M. Schütz, Phys. Rev. E 63, 056110 (2001).
  18. Y. Baek, Y. Kafri, and V. Lecomte, Phys. Rev. Lett. 118, 030604 (2017)
  19. P. L. Krapivsky, J. Stat. Mech. (2013) P06012.
  20. J. M. Carlson, J. T. Chayes, E. R. Grannan, and G. H. Swindle, Phys. Rev. Lett. 65, 2547 (1990).
  21. D. Gabrielli and P. L. Krapivsky (unpublished).
  22. U. Bhat and P. L. Krapivsky, Phys. Rev. E 90, 012133 (2014).
  23. C. Kipnis, C. Landim, and S. Olla, Commun. Pure Appl. Math. 47, 1475 (1994).
  24. C. Kipnis, C. Landim, and S. Olla, Ann. Inst. Henri Poincaré Probab. Statist. 31, 191 (1995).
  25. D. Chowdhury, L. Santen, and A. Schadschneider, Phys. Rep. 329, 199 (2000).
  26. T. Chou, K. Mallick, and R. K. P. Zia, Rep. Prog. Phys. 74, 116601 (2011).
  27. By “product measure” we mean that in the equilibrium correlation functions factorize, e.g., the pair correlation function satisfies 〈τiτj〉=〈τi〉〈τj〉 for i≠j.
  28. C. Cocozza-Thivent, Z. Wahrsch. Verw. Geb. 70, 509 (1985).
  29. C. Arita and C. Matsui, Europhys. Lett. 114, 60012 (2016).
  30. T. Seppäläinen, Ann. Probab. 27, 361 (1999).
  31. T. Becker, K. Nelissen, B. Cleuren, B. Partoens, and C. Van den Broeck, Phys. Rev. Lett. 111, 110601 (2013).
  32. T. Antal, P. L. Krapivsky, and K. Mallick, J. Stat. Mech. (2007) P08027.
  33. C. Godrèche, in Ageing and the Glass Transition, Lecture Notes in Physics Vol. 716 (Springer, Berlin, 2007), p. 261.
  34. C. Arita, P. L. Krapivsky, and K. Mallick, Phys. Rev. E 90, 052108 (2014).
  35. T. Becker, K. Nelissen, B. Cleuren, B. Partoens, and C. Van den Broeck, Phys. Rev. E 93, 046101 (2016).
  36. C. Arita, P. L. Krapivsky, and K. Mallick, Phys. Rev. E 94, 016101 (2016).
  37. Reviews of kinetically constrained lattice gases: F. Ritort and P. Sollich, Adv. Phys. 52, 219 (2003); J. P. Garrahan, P. Sollich, and C. Toninelli, in Dynamical Heterogeneities in Glasses, Colloids and Granular Media, edited by L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, and W. van Saarloos (Oxford University Press, Oxford, UK, 2011), Chap. 10.
  38. M. A. Załuska-Kotur and Z. W. Gortel, Phys. Rev. B 76, 245401 (2007); F. Krzyżewski and M. A. Załuska-Kotur, ibid. 78, 235406 (2008); M. Mińkowski and M. A. Załuska-Kotur, ibid. 91, 075411 (2015).
  39. P. Gaspard and T. Gilbert, arXiv:1611.05809.
  40. M. Sasada, arXiv:1611.08866.

Outline

Information

Phys. Rev. E 95, 032121– Published 9 March, 2017
  • Received 14 November 2016

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation