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From dynamical scaling to local scale-invariance: a tutorial

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Abstract

Dynamical scaling arises naturally in various many-body systems far from equilibrium. After a short historical overview, the elements of possible extensions of dynamical scaling to a local scale-invariance will be introduced. Schrödinger-invariance, the most simple example of local scale-invariance, will be introduced as a dynamical symmetry in the Edwards-Wilkinson universality class of interface growth. The Lie algebra construction, its representations and the Bargman superselection rules will be combined with non-equilibrium Janssen-de Dominicis field-theory to produce explicit predictions for responses and correlators, which can be compared to the results of explicit model studies. At the next level, the study of non-stationary states requires to go over, from Schrödinger-invariance, to ageing-invariance. The ageing algebra admits new representations, which acts as dynamical symmetries on more general equations, and imply that each non-equilibrium scaling operator is characterised by two distinct, independent scaling dimensions. Tests of ageing-invariance are described, in the Glauber-Ising and spherical models of a phase-ordering ferromagnet and the Arcetri model of interface growth.

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References

  1. H. Weyl, Symmetry (Princeton University Press, Princeton NJ, 1952)

  2. H. Weyl, Symmetrie (Birkhäuser, Basel, 1955)

  3. A. Einstein, Ann. Physik 17, 891 (1905)

    Article  ADS  Google Scholar 

  4. E. Cunningham, Proc. London Math. Soc. 8, 77 (1909)

    Google Scholar 

  5. H. Bateman, Proc. London Math. Soc. 8, 223 (1910)

    Article  Google Scholar 

  6. S. El-Showk, Yu. Nakayama, S. Rychkov, Nucl. Phys. B 848, 578 (2011)

    Article  ADS  Google Scholar 

  7. R. Jackiw, S.-Y. Pi, J. Phys. A 44, 223001 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  8. E.C.G. Stueckelberg, A. Petermann, Helv. Phys. Acta 26, 499 (1952)

    Google Scholar 

  9. B. Widom, J. Chem. Phys. 43, 3898 (1965)

    Article  ADS  Google Scholar 

  10. L. Kadanoff, Physics 2, 263 (1966)

    Google Scholar 

  11. K.G. Wilson, Phys. Rev. B 4, 3174 (1971)

    Article  ADS  Google Scholar 

  12. C. Domb, The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena (Taylor & Francis, London, 1996)

  13. M.E. Fisher, Rev. Mod. Phys. 70, 653 (1998)

    Article  ADS  Google Scholar 

  14. B. Berche, M. Henkel, R. Kenna, Rev. Bras. Ens. Fís. 31, 2602 (2009)

    Google Scholar 

  15. B. Berche, M. Henkel, R. Kenna, J. Phys. Stud. 13, 3201 (2009)

    Google Scholar 

  16. A.M. Polyakov, Sov. Phys. JETP Lett. 12, 381 (1970)

    ADS  Google Scholar 

  17. A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984)

    Article  ADS  Google Scholar 

  18. B. Delamotte, N. Tissier, N. Wschebor, Phys. Rev. E 93, 012144 (2016)

    Article  ADS  Google Scholar 

  19. J. Polchinsky, String Theory (Cambridge University Press, Cambridge, 1998), Vol 2

  20. B. Zwiebach, A First Course in String Theory (Cambridge University Press, Cambridge, 2005)

  21. A. Einstein, Ann. Physik 17, 549 (1905)

    Article  ADS  Google Scholar 

  22. P. Langevin, C.R. Acad. Sci. (Paris) 146, 530 (1908)

    Google Scholar 

  23. S. Lie, Arch. Math. Vid. (Kristiania) 6, 328 (1882)

    Google Scholar 

  24. S. Lie, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (Teubner, Leipzig, 1891)

  25. C.G. Jacobi, Vorlesungen über Dynamik, 4. Vorlesung (Königsberg 1842/43), in A. Clebsch, A. Lottner (Eds.), Gesammelte Werke von C.G. Jacobi (Akademie der Wissenschaften, Berlin 1866/1884)

  26. U. Niederer, Helv. Phys. Acta 45, 802 (1972)

    MathSciNet  Google Scholar 

  27. C.R. Hagen, Phys. Rev. D 5, 377 (1972)

    Article  ADS  Google Scholar 

  28. G. Burdet, M. Perrin, Lett. Nuov. Cim. 4, 651 (1972)

    Article  Google Scholar 

  29. R. Jackiw, Phys. Today 25, 23 (1972)

    Article  ADS  Google Scholar 

  30. A.J. Bray, Adv. Phys. 43, 357 (1994)

    Article  ADS  Google Scholar 

  31. A.-L. Barabási, H.E. Stanley, Fractals Concepts Insurface Growth (Cambridge University Press, Cambridge, 1995)

  32. T. Halpin-Healy, Y.-C. Zhang, Phys. Rep. 254, 215 (1995)

    Article  ADS  Google Scholar 

  33. J. Krug, Adv. Phys. 46, 139 (1997)

    Article  ADS  Google Scholar 

  34. L.F. Cugliandolo, in J.-L. Barrat et al. (Eds.), Slow Relaxation and Non-Equilibrium Dynamics in Condensed Matter (Springer, Heidelberg, 2002)

  35. M. Henkel, M. Pleimling, Non-Equilibrium Phase Transitions, Vol. 2: Ageing and Dynamical Scaling Far From Equilibrium (Springer, Heidelberg, 2010)

  36. U.C. Täuber, Critical dynamics (Cambridge University Press, Cambridge, 2014)

  37. L.C.E. Struik, Physical Ageing in Amorphous Polymers and Other Materials (Elsevier, Amsterdam, 1978)

  38. S.F. Edwards, D.R. Wilkinson, Proc. Roy. Soc. A 381, 17 (1982)

    Article  ADS  Google Scholar 

  39. F. Family, T. Vicsek, J. Phys. A 18, L75 (1985)

    Article  ADS  Google Scholar 

  40. T. Enss, M. Henkel, A. Picone, U. Schollwöck, J. Phys. A 37, 10479 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  41. F. Baumann, A. Gambassi, J. Stat. Mech. P01002 (2007)

  42. V.S. L'vov, V.V. Lebedev, M. Paton, I. Procaccia, Nonlinearity 6, 25 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  43. E. Frey, U.C. Täuber, T. Hwa, Phys. Rev. E 53, 4424 (1996)

    Article  ADS  Google Scholar 

  44. M. Henkel, M. Paeßens, M. Pleimling, Europhys. Lett. 62, 644 (2003)

    Article  ADS  Google Scholar 

  45. M. Henkel, M. Paeßens, M. Pleimling, Phys. Rev. E 69, 056109 (2004)

    Article  ADS  Google Scholar 

  46. M. Henkel, J.D. Noh, M. Pleimling, Phys. Rev. E 85, 030102(R) (2012)

    Article  ADS  Google Scholar 

  47. V. Bargman, Ann. Math. 56, 1 (1954)

    Article  Google Scholar 

  48. A. Picone, M. Henkel, Nucl. Phys. B 688, 217 (2004)

    Article  ADS  Google Scholar 

  49. M. Henkel, in H.J. Herrmann, W.J. Janke, F. Karsch (Eds.), textitDynamics of First-Order Transitions (World Scientific, Singapour, 1992), p. 239

  50. M. Henkel, Int. J. Mod. Phys. C 3, 1011 (1992)

    Article  ADS  Google Scholar 

  51. M. Henkel, J. Stat. Phys. 75, 1023 (1994)

    Article  ADS  Google Scholar 

  52. M. Henkel, J. Unterberger, Nucl. Phys. B 660, 407 (2003)

    Article  ADS  Google Scholar 

  53. J. Unterberger, C. Roger, The Schrödinger-Virasoro Algebra (Springer, Heidelberg, 2012)

  54. C. Roger, J. Unterberger, Ann. Henri Poincaré 7, 1477 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  55. A. Röthlein, F. Baumann, M. Pleimling, Phys. Rev. E 74, 061604 (2006)

    Article  ADS  Google Scholar 

  56. A. Röthlein, F. Baumann, M. Pleimling, Phys. Rev. E 76, 019901 (2007)

    Article  ADS  Google Scholar 

  57. S. Bustingorry, L.F. Cugliandolo, J.L. Iguain, J. Stat. Mech. P09008 (2007)

  58. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)

  59. F. Family, J. Phys. A 19, L441 (1986)

    Article  ADS  Google Scholar 

  60. D.D. Vvedensky, Phys. Rev. E 67, 025102(R) (2003)

    Article  ADS  Google Scholar 

  61. H.K. Janssen, B. Schaub, B. Schmittmann, Z. Phys. B 73, 539 (1989)

    Article  ADS  Google Scholar 

  62. T.J. Newman, A.J. Bray, J. Phys. A 23, 4491 (1990)

    Article  ADS  Google Scholar 

  63. M. Henkel, T. Enss, M. Pleimling, J. Phys. A 39, L589 (2006)

    Article  ADS  Google Scholar 

  64. U. Niederer, Helv. Phys. Acta 47, 167 (1974)

    MathSciNet  Google Scholar 

  65. S. Stoimenov, M. Henkel, J. Phys. A 46, 245004 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  66. S. Stoimenov, M. Henkel, Nucl. Phys. B 847, 612 (2011)

    Article  ADS  Google Scholar 

  67. M.H. Jacobs, Diffusion Processes (Springer, Heidelberg, 1967)

  68. P.C. Bressloff, J.M. Newby, Rev. Mod. Phys. 85, 135 (2013)

    Article  ADS  Google Scholar 

  69. J.M. Romero, A. Gaona, J. Phys. Conf. Ser. 512, 012028 (2014)

    Article  Google Scholar 

  70. J.M. Romero, O. González-Gaxiola, G. Chacón-Acosta, Int. J. Pure Appl. Math. 82, 41 (2013)

    Google Scholar 

  71. D. Minic, D. Vaman, C. Wu, Phys. Rev. Lett. 109, 131601 (2012)

    Article  ADS  Google Scholar 

  72. M. Henkel, Symmetry 7, 2108 (2015)

    Article  MathSciNet  Google Scholar 

  73. T.H. Berlin, M. Kac, Phys. Rev. 86, 821 (1952)

    Article  ADS  Google Scholar 

  74. H.W. Lewis, G.H. Wannier, Phys. Rev. 88, 682 (1952)

    Article  ADS  Google Scholar 

  75. H.W. Lewis, G.H. Wannier, Phys. Rev. 90, 1131 (1953)

    Article  ADS  Google Scholar 

  76. G. Ronca, J. Chem. Phys. 68, 3737 (1978)

    Article  ADS  Google Scholar 

  77. C. Godrèche, J.-M. Luck, J. Phys. A 33, 9141 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  78. M. Henkel, X. Durang, J. Stat. Mech. P05022 (2015)

  79. J.M. Kim, J.M. Kosterlitz, Phys. Rev. Lett. 62, 2289 (1989)

    Article  ADS  Google Scholar 

  80. P. Calabrese, A. Gambassi, J. Phys. A 38, R133 (2005)

    Article  ADS  Google Scholar 

  81. R.J. Glauber, J. Math. Phys. 4, 294 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  82. C. Godrèche, J.-M. Luck, J. Phys. A 33, 1151 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  83. G.F. Mazenko, Phys. Rev. E 69, 016114 (2004)

    Article  ADS  Google Scholar 

  84. M. Henkel, M. Pleimling, Phys. Rev. E 68, 065101(R) (2003)

    Article  ADS  Google Scholar 

  85. M. Henkel, A. Picone, M. Pleimling, Europhys. Lett. 68, 191 (2004)

    Article  ADS  Google Scholar 

  86. E. Lorenz, W. Janke, Europhys. Lett. 77, 10003 (2007)

    Article  ADS  Google Scholar 

  87. D.S. Fisher, D.A. Huse, Phys. Rev. B 38, 373 (1988)

    Article  ADS  Google Scholar 

  88. S. Mujumder, W. Janke, Phys. Rev. E 93, 032306 (2016)

    ADS  Google Scholar 

  89. S. Mujumder, W. Janke, in D. Landau et al. (Eds.) Computer Simulation Studies in Condensed-Matter Physics XXVIII, J. Phys. Conf. Series 750, 012020 (2016)

  90. M. Pleimling, A. Gambassi, Phys. Rev. B 71, 180401(R) (2005)

    Article  ADS  Google Scholar 

  91. M. Kardar, G. Parisi, Y.-C. Zhang, Phys. Rev. Lett. 56, 889 (1986)

    Article  ADS  Google Scholar 

  92. M. Henkel, Nucl. Phys. B 869, 282 (2013)

    Article  ADS  Google Scholar 

  93. J. Kelling, G. Ódor, S. Gemming, Phys. Rev. E 94, 022107 (2016)

    Article  ADS  Google Scholar 

  94. J. Kelling, G. Ódor, S. Gemming, to be published in J. Phys. A [arXiv: 1609.05795]

  95. F. Sastre, private communication (2016)

  96. G. Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, 032146 (2014)

    Article  ADS  Google Scholar 

  97. T. Halpin-Healy, G. Palansantzas, Europhys. Lett. 105, 50001 (2014)

    Article  ADS  Google Scholar 

Download references

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  1. Rechnergestützte Physik der Werkstoffe, Institut für Baustoffe (IfB), ETH Zürich, Stefano-Franscini-Platz 3, 8093, Zürich, Switzerland

    Malte Henkel

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Henkel, M. From dynamical scaling to local scale-invariance: a tutorial. Eur. Phys. J. Spec. Top. 226, 605–625 (2017). https://doi.org/10.1140/epjst/e2016-60336-5

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