Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

A STUDY ON CONDENSATION IN ZERO RANGE PROCESSES

  • Received : 2018.08.15
  • Accepted : 2018.09.17
  • Published : 2018.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

We investigate the condensation transition of a zero range process with jump rate g given by $g(k)=\left\{\frac{M}{k^{\alpha}},\;if\;k{\leq}an\\{\frac{1}{k^{\alpha}}},\;if\;k>an,$ (0.1), where ${\alpha}$ > 0 and a(0 < a < 1/2) is a rational number. We show that for any ${\epsilon}$ > 0, there exists $M^*$ > 0 such that, for any 0<$M{\leq}M^*$, the maximum cluster size is between ($a-{\epsilon}$)n and ($a+{\epsilon}$)n for large n.

Keywords

References

  1. I. Armendariz and M. Loulakis, Thermodynamic Limit for the Invariant Measures in Supercritical Zero Range Processes, Prob. Th. Rel. Fields, 145 (2009), no. 1-2, 175-188. https://doi.org/10.1007/s00440-008-0165-7
  2. I. Armendariz, S. Grosskinsky and M. Loulakis, Metastability in a condensing zero-range process in the thermodynamic limit, Prob. Th. Rel. Fields, 169 (2017), no. 1-2, 105-175. https://doi.org/10.1007/s00440-016-0728-y
  3. J. Beltran and C. Landim, Metastabiltiy of reversible condensed zero range processes on a finite set, Prob. Th. Rel. Fields, 152 (2012), no. 3, 781-807. https://doi.org/10.1007/s00440-010-0337-0
  4. L. Molino, P. Chleboun, and S. Grosskinsky, Condensation in randomly pertubed zero-range processes, J. Phys. A: Math. Theor., 45 (2012), no. 20, 205001. https://doi.org/10.1088/1751-8113/45/20/205001
  5. M.R. Evans, Phase transitions in one-dimensional nonequilibrium systems, Braz. J. Phys. 30 (2000), 42-57. https://doi.org/10.1590/S0103-97332000000100005
  6. M.R. Evans and T. Hanney, Nonequilibrium statistical mechanics of the zero range process and related models, J. Phys. A: Math. Gen., 38 (2005), 195-240. https://doi.org/10.1088/0305-4470/38/1/014
  7. S. Grosskinsky and G. Schutz, Discontinuous condensation transition and nonequivalence of ensembles in a zero-range process, J. Stat. Phys., 132 (2008), no. 1, 77-108. https://doi.org/10.1007/s10955-008-9541-z
  8. S. Grosskinsky, P. Chleboun, and G. Schutz , Instability of condensation in the zero range process with random interaction, Phys. Rev. E:Stat. Nonlin. Soft Matter Phys., 78 (2008), 030101. https://doi.org/10.1103/PhysRevE.78.030101
  9. I. Jeon, Phase transition for perfect condensation and instability under the pertubations on jump rates of the Zero Range Process, J. Phys. A: Math. and Theor., 43 (2010), 235002. https://doi.org/10.1088/1751-8113/43/23/235002
  10. I. Jeon, Condensation in perturbed zero range processes, J. Phys. A: Math. and Theor., 44 (2011), 255002. https://doi.org/10.1088/1751-8113/44/25/255002
  11. I. Jeon, Condensation in density dependent zero range processes, J. KSIAM, 17(4) (2013), 267-278.
  12. I. Jeon and P. March, Condensation transition for zero range invariant measures, In Stochastic models. Proceedings of the International Conference on Stochastic Models in Honor of Professor Donald A. Dawson(Luis G. Gorostiza, B. Gail Ivanoff, eds), 2000, 233-244.
  13. I. Jeon, P. March, and B. Pittel, Size of the largest cluster under zero-range invariant measures, Ann. of Prob., 28 (2000), 1162-1194. https://doi.org/10.1214/aop/1019160330
  14. C. Landim, Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes, Comm. Math. Phys. 330(1), (2014), 132
  15. T. M. Liggett, Interacting Particle Systems, Springer-Verlag.1985
  16. F. Spitzer, Interaction of Markov processes, Adv. Math., 5 (1970), 246-290. https://doi.org/10.1016/0001-8708(70)90034-4