Advertisement

Scaling behavior in interacting systems: joint effect of anisotropy and compressibility

  • Michal Hnatič
  • Georgii Kalagov
  • Tomáš Lučivjanský
Regular Article

Abstract

Motivated by the ubiquity of turbulent flows in realistic conditions, effects of turbulent advection on two models of classical non-linear systems are investigated. In particular, we analyze model A (according to the Hohenberg–Halperin classification [P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977)]) of a non-conserved order parameter and a model of the direct bond percolation process. Having two paradigmatic representatives of distinct stochastic dynamics, our aim is to elucidate to what extent velocity fluctuations affect their scaling behavior. The main emphasis is put on an interplay between anisotropy and compressibility of the velocity flow on their respective scaling regimes. Velocity fluctuations are generated by means of the Kraichnan rapid-change model, in which the anisotropy is due to a distinguished spatial direction n and a correlator of the velocity field obeys the Gaussian distribution law with prescribed statistical properties. As the main theoretical tool, the field-theoretic perturbative renormalization group is adopted. Actual calculations are performed in the leading (one-loop) approximation. Having obtained infrared stable asymptotic regimes, we have found four possible candidates for macroscopically observable behavior for each model. In contrast to the isotropic case, anisotropy brings about enhancement of non-linearities and non-trivial regimes are proved to be more stable.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    D. Yu. Ivanov, Critical Behaviour of Non-Ideal Systems (Wiley-VCH, Weinhein, 2008) Google Scholar
  2. 2.
    P.K. Khabibullaev, A.A. Saidov, Phase Separation in Soft Matter Physics: Micellar Solutions, Microemulsions, Critical Phenomena (Springer Series in Solid-State Sciences, Weinhein, 2003) Google Scholar
  3. 3.
    M.A. Anisimov, Critical Phenomena in Liquids and Liquid Crystals (Gordon and Breach, Amsterdam, 1991) Google Scholar
  4. 4.
    M. Barmatz, H. Inseob, J.A. Lipa, R.V. Duncan, Rev. Mod. Phys. 79, 1 (2007) ADSCrossRefGoogle Scholar
  5. 5.
    R.F. Berg, M.R. Moldover, J. Chem. Phys. 93, 1926 (1990) ADSCrossRefGoogle Scholar
  6. 6.
    H.K. Janssen, Phys. Rev. E 55, 6253 (1997) ADSCrossRefGoogle Scholar
  7. 7.
    P.A. Davidson, Turbulence: An Introduction for Scientists and Engineers, 2nd edn. (Oxford University Press, Oxford, 2015) Google Scholar
  8. 8.
    U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, 1995) Google Scholar
  9. 9.
    A.S. Monin, A.M. Yaglom, in Statistical Fluid Mechanics, (MIT Press, Cambridge, MA, 1975), Vol. 2 Google Scholar
  10. 10.
    G. Falkovich, K. Gawedzki, M. Vergassola, Rev. Mod. Phys. 73, 913 (2001) ADSCrossRefGoogle Scholar
  11. 11.
    L.Ts. Adzhemyan, N.V. Antonov, A.N. Vasil’ev, The Field Theoretic Renormalization Group in Fully Developed Turbulence (Gordon and Breach, Amsterdam,] 1999) Google Scholar
  12. 12.
    M. Hnatič, J. Honkonen, T. Lučivjanský, Acta Phys. Slovaca 66, 69 (2016) Google Scholar
  13. 13.
    R.H. Kraichnan, Phys. Rev. Lett. 72, 1016 (1994) ADSCrossRefGoogle Scholar
  14. 14.
    M. Chertkov, I. Kolokolov, M. Vergassola, Phys. Rev. E 56, 5483 (1997) ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    L.Ts. Adzhemyan, N.V. Antonov, Phys. Rev. E 58, 7381 (1998) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    N.V. Antonov, Phys. Rev. E 60, 6691 (1999) ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    A.N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Chapman and Hall, Boca Raton, 2004) Google Scholar
  18. 18.
    E. Jurčišinová, M. Jurčišin, R. Remecký, Phys. Rev. E 80, 046302 (2009) ADSCrossRefGoogle Scholar
  19. 19.
    A.Z. Patashinskii, V.L. Pokrokovskii, Fluctuation Theory of Phase Transitions (Pergamon Press, Oxford, 1979) Google Scholar
  20. 20.
    R.H. Kraichnan, Phys. Fluids 11, 945 (1968) ADSCrossRefGoogle Scholar
  21. 21.
    N.V. Antonov, J. Phys. A: Math. Gen. 39, 7825 (2006) ADSCrossRefGoogle Scholar
  22. 22.
    S.N. Shore, Astrophysical Hydrodynamics: An Introduction (Wiley-VCH Verlag GmbH KGaA, Weinheim, 2007) Google Scholar
  23. 23.
    J. Kim, D. Ryu, Astrophys. J. 630, L45 (2005) ADSCrossRefGoogle Scholar
  24. 24.
    F. Sahraoui, M.L. Goldstein, P. Robert, Y.V. Khotyainstsev, Phys. Rev. Lett. 102, 231102 (2009) ADSCrossRefGoogle Scholar
  25. 25.
    S. Galtier, S. Banerjee, Phys. Rev. Lett. 107, 134501 (2011) ADSCrossRefGoogle Scholar
  26. 26.
    S. Banerjee, S. Galtier, Phys. Rev. E 87, 013019 (2013) ADSCrossRefGoogle Scholar
  27. 27.
    S. Banerjee, L.Z. Hadid, F. Sahraoui, S. Galtier, Astrophys. J. Lett. 829, L27 (2016) ADSCrossRefGoogle Scholar
  28. 28.
    L.Z. Hadid, F. Sahraoui, S. Galtier, Astrophys. J. 838, 9 (2017) ADSCrossRefGoogle Scholar
  29. 29.
    A. Celani, A. Lanotte, A. Mazzino, M. Vergassola, Phys. Rev. Lett. 84, 2385 (2000) ADSCrossRefGoogle Scholar
  30. 30.
    S.G. Saddoughi, S.V. Veeravalli, J. Fluid. Mech. 268, 333 (1994) ADSCrossRefGoogle Scholar
  31. 31.
    I. Arad, B. Dhruva, S. Kurien, V.S. L’vov, I. Procaccia, K.R. Sreenivasan, Phys. Rev. Lett. 81, 5330 (1998) ADSCrossRefGoogle Scholar
  32. 32.
    R. Rubinstein, J.M. Barton, Phys. Fluids 30, 2987 (1987) ADSCrossRefGoogle Scholar
  33. 33.
    L.Ts. Adzhemyan, N.V. Antonov, M. Hnatič, S.V. Novikov, Phys. Rev. E 63, 016309 (2000) ADSCrossRefGoogle Scholar
  34. 34.
    D. Carati, L. Brenig, Phys. Rev. A 40, 5193 (1989) ADSCrossRefGoogle Scholar
  35. 35.
    T.L. Kim, A.V. Serdyukov, Theor. Math. Phys. 105, 1525 (2005) CrossRefGoogle Scholar
  36. 36.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th edn. (Oxford University Press, Oxford, 2002) Google Scholar
  37. 37.
    D.J. Amit, V. Martín-Mayor, Field Theory, the Renormalization Group and Critical Phenomena (World Scientific, Singapore, 2005) Google Scholar
  38. 38.
    K.G. Wilson, J. Kogut, Phys. Rep. 12, 75 (1974) ADSCrossRefGoogle Scholar
  39. 39.
    J. Zinn-Justin, Phase Transitions and Renormalization Group (Clarendon, Oxford, 2007) Google Scholar
  40. 40.
    F. Wegner, in Phase Transitions and Critical Phenomena, edited by C. Domb, M.S. Green (Academic Press, London and New York, 1976), Vol. 6 Google Scholar
  41. 41.
    G. Parisi, J. Stat. Phys. 23, 49 (1980) ADSCrossRefGoogle Scholar
  42. 42.
    J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980) ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Pelissetto, E. Vicari, Phys. Rep. 368, 549 (2002) ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    H. Kleinert, V. Schulte-Frohlinde, Critical Properties of ϕ4 Theories (World Scientific, Singapore, 2001) Google Scholar
  45. 45.
    M.V. Kompaniets, E. Panzer, Phys. Rev. D 96, 036016 (2017) ADSCrossRefGoogle Scholar
  46. 46.
    L.Ts. Adzhemyan, N.V. Antonov, M.V. Kompaniets, A.N. Vasil’ev, Int. J. Mod. Phys. B 17, 2137 (2003) ADSCrossRefGoogle Scholar
  47. 47.
    N.V. Antonov, A.N. Vasil’ev, Theor. Math. Phys. 60, 671 (1984) CrossRefGoogle Scholar
  48. 48.
    L.Ts. Adzhemyan, S.V. Novikov, L. Sladkoff, Vestnik St. Petersburg Univ. 4, 109 (2008) Google Scholar
  49. 49.
    L.Ts. Adzhemyan, N.V. Antonov, V.A. Barinov, Yu.S. Kabrtits, A.N. Vasil’ev, Phys. Rev. E 64, 056306 (2001) ADSCrossRefGoogle Scholar
  50. 50.
    B. Delamotte, An introduction to the nonperturbative renormalization group, in Renormalization Group and Effective Field Theory Approaches to Many-Body Systems, edited by A. Schwenk, J. Polonyi, Lecture Notes in Physics (Springer, Berlin, Heidelberg, 2012), Vol. 852 Google Scholar
  51. 51.
    J. Polonyi, Central Eur. J. Phys. 1, 1 (2003) ADSGoogle Scholar
  52. 52.
    L. Canet, H. Chate, J. Phys. A: Math. Gen. 40, 1937 (2007) ADSCrossRefGoogle Scholar
  53. 53.
    L. Canet, B. Delamotte, N. Wschebor, Phys. Rev. E 93, 063101 (2016) ADSCrossRefGoogle Scholar
  54. 54.
    C. Pagani, Phys. Rev. E 92, 033016 (2015) ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    U.C. Täuber, Critical dynamics, in A Field Theory Approach to Equilibrium and Non-Equilibrium Scailing Behavior (Cambridge University Press, Cambridge, 2014) Google Scholar
  56. 56.
    P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977) ADSCrossRefGoogle Scholar
  57. 57.
    R. Kubo, Rep. Prog. Phys. 29, 255 (1966) ADSCrossRefGoogle Scholar
  58. 58.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 2007) Google Scholar
  59. 59.
    G. Ódor, Phys. Rev. E 70, 26119 (2004) CrossRefGoogle Scholar
  60. 60.
    U.C. Täuber, M. Howard, B.P. Vollmayr-Lee, J. Phys. A: Math. Gen. 38, R79–R129 (2005) CrossRefGoogle Scholar
  61. 61.
    B. Schmittmann, R.K.P. Zia, Statistical mechanics of driven diffusive systems, Vol. 17 of Phase Transitions and Critical Phenomena (Academic Press, London, 1995) Google Scholar
  62. 62.
    D.C. Mattis, M.L. Glasser, Rev. Mod. Phys. 70, 979 (1998) ADSCrossRefGoogle Scholar
  63. 63.
    H. Hinrichsen, Adv. Phys. 49, 815 (2000) ADSCrossRefGoogle Scholar
  64. 64.
    H.K. Janssen, U.C. Täuber, Ann. Phys. 315, 147 (2004) ADSCrossRefGoogle Scholar
  65. 65.
    M. Henkel, H. Hinrichsen, S. Lübeck, Non-Equilibrium Phase Transitions: Volume 1-Absorbing Phase Transitions (Springer, Dordrecht, 2008) Google Scholar
  66. 66.
    S.R. Broadbent, I.M. Hamersley, Proc. Camb. Philos. Soc. 53, 629 (1957) ADSCrossRefGoogle Scholar
  67. 67.
    U.C. Täuber, Adv. Solid State Phys. 43, 629 (2003) Google Scholar
  68. 68.
    N.V. Antonov, M. Hnatič, A.S. Kapustin, T. Lučivjanský, L. Mižišin, Phys. Rev. E 93, 012151 (2016) ADSCrossRefGoogle Scholar
  69. 69.
    M. Dančo, M. Hnatič, T. Lučivjanský, L. Mižišin, Theor. Math. Phys. 176, 79 (2013) Google Scholar
  70. 70.
    N. Antonov, A. Ignatieva, A. Malyshev, Phys. Part. Nuclei 41, 998 (2010) ADSCrossRefGoogle Scholar
  71. 71.
    H.K. Janssen, K. Oerding, F. van Wijland, H.J. Hilhorst, Eur. Phys. J. B 7, 137 (1999) ADSCrossRefGoogle Scholar
  72. 72.
    H.K. Janssen, O. Stenull, Phys. Rev. E 78, 061117 (2008) ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    C.W. Gardiner, Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences (Springer, Berlin, 2009) Google Scholar
  74. 74.
    N.V. Antonov, A.S. Kapustin, A.V. Malyshev, Theor. Math. Phys. 169, 1470 (2011) CrossRefGoogle Scholar
  75. 75.
    R. Folk, G. Moser, J. Phys. A: Math. Gen. 39, 207 (2006) ADSCrossRefGoogle Scholar
  76. 76.
    N.V. Antonov, Physica D 144, 370 (2000) ADSMathSciNetCrossRefGoogle Scholar
  77. 77.
    N.V. Antonov, A.S. Kapustin, J. Phys. A: Math. Gen. 43, 405001 (2010) CrossRefGoogle Scholar
  78. 78.
    Maplesoft. Maple (Waterloo Maple Inc., Waterloo, Ontario, Canada, 2012) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Michal Hnatič
    • 1
    • 2
  • Georgii Kalagov
    • 1
    • 3
  • Tomáš Lučivjanský
    • 1
  1. 1.Faculty of Science, Šafárik UniversityKošiceSlovakia
  2. 2.Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear ResearchDubnaRussia
  3. 3.Department of Theoretical PhysicsSt. Petersburg State UniversitySt. Petersburg, PetrodvoretsRussia

Personalised recommendations