Physics of Particles and Nuclei Letters

, Volume 14, Issue 6, pp 944–952 | Cite as

Active-to-absorbing phase transition subjected to the velocity fluctuations in the frozen limit case

  • N. V. AntonovEmail author
  • M. Hnatich
  • A. S. Kapustin
  • T. Lučivjanský
  • L. Mižišin
Physics of Solid State and Condensed Matter


The directed bond percolation process is studied in the presence of compressible velocity fluctuations with long-range correlations. We discuss a construction of a field theoretic action and a way of obtaining its large scale properties using the perturbative renormalization group. The most interesting results for the frozen velocity limit are given.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge Univ. Press, New York, 2014).Google Scholar
  2. 2.
    M. Henkel, H. Hinrichsen, and S. Lübeck, Non-Equilibrium Phase Transitions, Vol. 1: Absorbing Phase Transitions (Springer, Dordrecht, 2008).zbMATHGoogle Scholar
  3. 3.
    H. K. Janssen and U. C. Täuber, “The field theory approach to percolation processes,” Ann. Phys. (N.Y.) 315, 147–192 (2005); arXiv:cond-mat/0409670.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    H. K. Janssen, “On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state,” Z. Phys. B: Condens. Matter 42, 151–154 (1981).ADSCrossRefGoogle Scholar
  5. 5.
    H. K. Janssen, “Renormalized field theory of the Gribov process with quenched disorder,” Phys. Rev. E 55, 6253–6256 (1997).ADSCrossRefGoogle Scholar
  6. 6.
    H. Hinrichsen, “Non-equilibrium phase transitions with long-range interactions,” J. Stat. Mech.: Theor. Exp., 07066 (2007); arXiv:cond-mat/0702169.Google Scholar
  7. 7.
    K. A. Takeuchi, M. Kuroda, H. Chat, and M. Sano, “Directed percolation criticality in turbulent liquid crystals,” Phys. Rev. Lett. 99, 234503 (2007); arXiv:0706.4151 [cond-mat.stat-mech].ADSCrossRefGoogle Scholar
  8. 8.
    N. V. Antonov, “Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field,” Phys. Rev. E 60, 6691–6707 (1999); arXiv:chaodyn/ 9808011.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. V. Antonov, “Anomalous scaling of a passive scalar advected by the synthetic compressible flow,” Phys. D (Amsterdam) 144, 370–386 (2000); arXiv:chaodyn/ 9907018.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    N. V. Antonov, V. I. Iglovikov, and A. S. Kapustin, “Effects of turbulent mixing on the nonequilibrium critical behaviour,” J. Phys. A: Math. Theor. 42, 135001 (2008); arXiv:0808.0076 [cond-mat.stat-mech].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    N. V. Antonov and A. S. Kapustin, “Effects of turbulent mixing on critical behaviour in the presence of compressibility: renormalization group analysis of two models,” J. Phys. A: Math. Theor. 43, 405001 (2010); arXiv:1006.3133 [cond-mat.stat-mech].MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    N. V. Antonov, A. S. Kapustin, and A. V. Malyshev, “Effects of turbulent transfer on the critical behavior,” Theor. Math. Phys. 169, 1470–1480 (2011); arXiv:1012.4317 [cond-mat.stat-mech].MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Dančo, H. Hnatič, T. Lučivjanský, and L. Mižišin, “Critical behavior of percolation process influenced by a random velocity field: one-loop approximation,” Theor. Math. Phys. 176, 898–905 (2013); arXiv:1302.1006 [nlin.CD].MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    N. V. Antonov, H. Hnatič, A. S. Kapustin, T. Lučivjanský, and L. Mižišin, “Gribov process advected by the synthetic compressible velocity ensemble: renormalization group approach,” Theor. Math. Phys. 190, 323–334 (2017); arXiv:1603.00310 [cond-mat.statmech].CrossRefzbMATHGoogle Scholar
  15. 15.
    N. V. Antonov, H. Hnatič, A. S. Kapustin, T. Lučivjanský, and L. Mižišin, “Directed percolation process in the presence of velocity fluctuations: effect of compressibility and finite correlation time,” Phys. Rev. E 93, 012151 (2016); arXiv:1602.00642 [cond-mat.statmech].ADSCrossRefGoogle Scholar
  16. 16.
    J. Honkonen and E. Karjalainen, “Diffusion in a random medium with long-range correlations,” J. Phys. A: Math. Gen. 21, 4217 (1988).ADSCrossRefGoogle Scholar
  17. 17.
    D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group and Critical Phenomena (World Scientific, Singapore, 2005).CrossRefzbMATHGoogle Scholar
  18. 18.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford Univ. Press, Oxford, 1996).zbMATHGoogle Scholar
  19. 19.
    A. N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Chapman Hall, CRC, Boca Raton, 2004).CrossRefzbMATHGoogle Scholar
  20. 20.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Pergamon, Oxford, 1959).Google Scholar
  21. 21.
    U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, Cambridge, 1995).zbMATHGoogle Scholar
  22. 22.
    K. Gawedzki and M. Vergasssola, “Phase transition in the passive scalar advection,” Phys. D (Amsterdam) 138, 63–90 (1999); arXiv:cond-mat/9811399.ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    N. V. Antonov, M. Yu. Nalimov, and A. A. Udalov, “Renormalization group in the problem of fully developed turbulence of a compressible fluid,” Theor. Math. Phys. 110, 305–315 (1997).CrossRefzbMATHGoogle Scholar
  24. 24.
    N. V. Antonov and M. M. Kostenko, “Anomalous scaling of passive scalar fields advected by the navier-stokes velocity ensemble: effects of strong compressibility and large-scale anisotropy,” Phys. Rev. E 90, 063016 (2014); arXiv:1410.1262.ADSCrossRefGoogle Scholar
  25. 25.
    N. V. Antonov, N. M. Gulitskiy, M. M. Kostenko, and T. Lučivjanský, “Turbulent compressible fluid: renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields,” Phys. Rev. E 95, 033120 (2017); arXiv:1611.00327.ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • N. V. Antonov
    • 1
    Email author
  • M. Hnatich
    • 2
    • 3
  • A. S. Kapustin
    • 1
  • T. Lučivjanský
    • 2
    • 3
  • L. Mižišin
    • 2
    • 4
  1. 1.Department of Theoretical PhysicsSt. Petersburg UniversitySt. Petersburg, PetrodvoretsRussia
  2. 2.Faculty of SciencesP.J. Šafárik UniversityKošiceSlovakia
  3. 3.Peoples’ Friendship University of Russia (RUDN University)MoscowRussia
  4. 4.Bogoliubov Laboratory of Theoretical Physics, JINRDubna, Moscow oblastRussia

Personalised recommendations