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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U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge Univ. Press, New York, 2014).
M. Henkel, H. Hinrichsen, and S. Lübeck, Non-Equilibrium Phase Transitions, Vol. 1: Absorbing Phase Transitions (Springer, Dordrecht, 2008).
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.
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).
H. K. Janssen, “Renormalized field theory of the Gribov process with quenched disorder,” Phys. Rev. E 55, 6253–6256 (1997).
H. Hinrichsen, “Non-equilibrium phase transitions with long-range interactions,” J. Stat. Mech.: Theor. Exp., 07066 (2007); arXiv:cond-mat/0702169.
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].
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.
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.
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].
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].
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].
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].
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].
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].
J. Honkonen and E. Karjalainen, “Diffusion in a random medium with long-range correlations,” J. Phys. A: Math. Gen. 21, 4217 (1988).
D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group and Critical Phenomena (World Scientific, Singapore, 2005).
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford Univ. Press, Oxford, 1996).
A. N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Chapman Hall, CRC, Boca Raton, 2004).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Pergamon, Oxford, 1959).
U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, Cambridge, 1995).
K. Gawedzki and M. Vergasssola, “Phase transition in the passive scalar advection,” Phys. D (Amsterdam) 138, 63–90 (1999); arXiv:cond-mat/9811399.
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).
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.
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.
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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