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

Abstract

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.

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References

  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).

    Google 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.

    ADS  MathSciNet  Article  MATH  Google 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).

    ADS  Article  Google Scholar 

  5. 5.

    H. K. Janssen, “Renormalized field theory of the Gribov process with quenched disorder,” Phys. Rev. E 55, 6253–6256 (1997).

    ADS  Article  Google Scholar 

  6. 6.

    H. Hinrichsen, “Non-equilibrium phase transitions with long-range interactions,” J. Stat. Mech.: Theor. Exp., 07066 (2007); arXiv:cond-mat/0702169.

  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].

    ADS  Article  Google 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.

    ADS  MathSciNet  Article  MATH  Google 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.

    ADS  MathSciNet  Article  MATH  Google 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].

    ADS  MathSciNet  Article  MATH  Google 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].

    MathSciNet  Article  MATH  Google 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].

    MathSciNet  Article  MATH  Google 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].

    MathSciNet  Article  MATH  Google 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].

    Article  MATH  Google 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].

    ADS  Article  Google 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).

    ADS  Article  Google Scholar 

  17. 17.

    D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group and Critical Phenomena (World Scientific, Singapore, 2005).

    Google Scholar 

  18. 18.

    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford Univ. Press, Oxford, 1996).

    Google 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).

    Google 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).

    Google 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.

    ADS  MathSciNet  Article  Google 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).

    Article  MATH  Google 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.

    ADS  Article  Google 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.

    ADS  Article  Google Scholar 

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Antonov, N.V., Hnatich, M., Kapustin, A.S. et al. Active-to-absorbing phase transition subjected to the velocity fluctuations in the frozen limit case. Phys. Part. Nuclei Lett. 14, 944–952 (2017). https://doi.org/10.1134/S154747711706005X

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