Passive advection in a percolation process: two-loop approximation. (English. Russian original) Zbl 1436.82022
Theor. Math. Phys. 200, No. 3, 1335-1347 (2019); translation from Teor. Mat. Fiz. 200, No. 3, 478-493 (2019).
Summary: We study an instructive model of the directed percolation process near its second-order phase transition between absorbing and active states. We first express the model as a Langevin equation and then rewrite it in a field theory formulation. Using the Feynman diagram technique and the perturbative renormalization group method, we then analyze the resulting response functional. The percolation process is assumed to occur in an external velocity field, which has an additional effect on the properties of spreading. We use the Kraichnan rapid-change ensemble to generate velocity fluctuations. We obtain the structure of the set of fixed points in the two-loop approximation.
MSC:
| 82C43 | Time-dependent percolation in statistical mechanics |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
| 82C28 | Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 92B10 | Taxonomy, cladistics, statistics in mathematical biology |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
References:
| [1] | B. Schmittmann and R. K. P. Zia, eds., Statistical Mechanics of Driven Diffusive Systems (Phase Trans. Crit. Phen., Vol. 17), Acad. Press, New York (1995). |
| [2] | Henkel, M.; Hinrichsen, H.; Lübeck, S., Non-Equilibrium Phase Transitions (2008), Dordrecht · Zbl 1165.82002 |
| [3] | U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior, Cambridge Univ. Press, Cambridge (2014). |
| [4] | P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics, Cambridge Univ. Press, Cambridge (2010). · Zbl 1235.82040 · doi:10.1017/CBO9780511780516 |
| [5] | J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press, Oxford (2002). · doi:10.1093/acprof:oso/9780198509233.001.0001 |
| [6] | D. J. Amit and V. Martín-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena, World Scientific, Singapore (2005). · Zbl 1088.81001 · doi:10.1142/5715 |
| [7] | D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor and Francis, London (1992). · Zbl 0862.60092 |
| [8] | J. L. Cardy and R. L. Sugar, “Directed percolation and Reggeon field theory,” J. Phys. A, 13, L423-L427 (1980). · doi:10.1088/0305-4470/13/12/002 |
| [9] | G. Ódor, “Universality classes in nonequilibrium lattice systems,” Rev. Modern Phys., 76, 663-724 (2004); arXiv:cond-mat/0205644v7 (2002). · Zbl 1205.82102 · doi:10.1103/RevModPhys.76.663 |
| [10] | H.-K. Janssen and U. C. Täuber, “The field theory approach to percolation processes,” Ann. Phys., 315, 147-192 (2004); arXiv:cond-mat/0409670v1 (2004). · Zbl 1083.82030 · doi:10.1016/j.aop.2004.09.011 |
| [11] | H. K. Janssen, “On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state,” Z. Phys. B, 42, 151-154 (1981). · doi:10.1007/BF01319549 |
| [12] | P. Grassberger, “On phase transitions in Schlögl’s second model,” Z. Phys. B, 47, 365-374 (1982). · doi:10.1007/BF01313803 |
| [13] | A. N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics [in Russian], Petersburg Inst. Nucl. Phys. Press, St. Petersburg (1998); English transl., CRC, Boca Raton, Fla. (2004). |
| [14] | H. Hinrichsen, “Non-equilibrium critical phenomena and phase transitions into absorbing states,” Adv. Phys., 49, 815-958 (2000); arXiv:cond-mat/0001070v2 (2000). · doi:10.1080/00018730050198152 |
| [15] | L. Ts. Adzhemyan, M. Hnatič, M. V. Kompaniets, T. Lučivjanský, and L. Mižišin, “Numerical calculation of scaling exponents of percolation process in the framework of renormalization group approach,” EPJ Web Conf., 108, 02005 (2016). · doi:10.1051/epjconf/201610802005 |
| [16] | L. Ts. Adzhemyan, M. Hnatič, M. V. Kompaniets, T. Lučivjanský, and L. Miž iš in, “Directed percolation: Calculation of Feynman diagrams in the three-loop approximation,” EPJ Web Conf., 173, 02001 (2018). · doi:10.1051/epjconf/201817302001 |
| [17] | P. Rupp, R. Richter, and I. Rehberg, “Critical exponents of directed percolation measured in spatiotemporal intermittency,” Phys. Rev. E, 67, 036209 (2003); arXiv:cond-mat/0211137v1 (2002). · doi:10.1103/PhysRevE.67.036209 |
| [18] | K. A. Takeuchi, M. Kuroda, H. Chaté, and M. Sano, “Directed percolation criticality in turbulent liquid crystals,” Phys. Rev. Lett., 99, 234503 (2007); arXiv:0706.4151v1 [cond-mat.stat-mech] (2007). · doi:10.1103/PhysRevLett.99.234503 |
| [19] | M. Sano and K. Tamai, “A universal transition to turbulence in channel flow,” Nature Phys., 12, 249-253 (2016); arXiv:1510.07868v1 [cond-mat.stat-mech] (2015). · doi:10.1038/nphys3659 |
| [20] | G. Lemoult, L. Shi, K. Avila, S. V Jalikop, M. Avila, and B. Hof, “Directed percolation phase transition to sustained turbulence in Couette flow,” Nature Phys., 12, 254-258 (2016). · doi:10.1038/nphys3675 |
| [21] | H. Hinrichsen, “Non-equilibrium phase transitions,” J. Phys. A, 369, 1-28 (2006). |
| [22] | H. K. Janssen, K. Oerding, F. van Wijland, and H. Hilhorst, “Levy-flight spreading of epidemic processes leading to percolating clusters,” Eur. Phys. J. B, 7, 137-145 (1999). · doi:10.1007/s100510050596 |
| [23] | H. Hinrichsen, “Non-equilibrium phase transitions with long-range interactions,” J. Stat. Mech., 2007, P07006 (2007). · Zbl 1456.82736 · doi:10.1088/1742-5468/2007/07/P07006 |
| [24] | H. K. Janssen, “Renormalized field theory of the Gribov process with quenched disorder,” Phys. Rev. E, 55, 6253-6256 (1997). · doi:10.1103/PhysRevE.55.6253 |
| [25] | N. V. Antonov, V. I. Iglovikov, and A. S. Kapustin, “Effects of turbulent mixing on the nonequilibrium critical behaviour,” J. Phys. A, 42, 135001 (2008); arXiv:0808.0076v1 [cond-mat.stat-mech] (2008). · Zbl 1159.82011 · doi:10.1088/1751-8113/42/13/135001 |
| [26] | R. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids, 11, 945-953 (1968). · Zbl 0164.28904 · doi:10.1063/1.1692063 |
| [27] | N. V. Antonov, “Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field,” Phys. Rev. E, 60, 6691-6707 (1999). · Zbl 1062.76512 · doi:10.1103/PhysRevE.60.6691 |
| [28] | G. Falkovich, K. Gawȩdzki, and M. Vergassola, “Particles and fields in fluid turbulence,” Rev. Modern Phys., 73, 913-975 (2001). · Zbl 1205.76133 · doi:10.1103/RevModPhys.73.913 |
| [29] | H. K. Janssen, “On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties,” Z. Phys. B, 23, 377-380 (1976). · doi:10.1007/BF01316547 |
| [30] | C. De Dominicis, “FrTechniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques,” J. Phys. Colloques France, 37, C1-247-C1-254 (1976). · doi:10.1051/jphys:019760037010100 |
| [31] | H. K. Janssen, Dynamical Critical Phenomena and Related Topics (Lect. Notes Phys., Vol. 104), Springer, Berlin (1979). |
| [32] | N. V. Antonov, M. Hnatich, and J. Honkonen, “Effects of mixing and stirring on the critical behaviour,” J. Phys. A: Math. Gen., 39, 7867-7887 (2006); arXiv:cond-mat/0604434v1 (2006). · Zbl 1122.82032 · doi:10.1088/0305-4470/39/25/S05 |
| [33] | Landau, L. D.; Lifshitz, E. M., Course of Theoretical Physics (1959), Oxford |
| [34] | M. Hnatič, J. Honkonen, and T. Lučivjanský, “Advanced field-theoretical methods in stochastic dynamics and theory of developed turbulence,” Acta Phys. Slovaca, 66, 69-264 (2016). |
| [35] | J. C. Collins, Renormalization, Cambridge Univ. Press, Cambridge (1984). · Zbl 1094.53505 · doi:10.1017/CBO9780511622656 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
