Skip to main content
SpringerLink
Log in

Field-Theory Approaches to Nonequilibrium Dynamics

  • Chapter
Ageing and the Glass Transition

Part of the book series: Lecture Notes in Physics ((LNP,volume 716))

Abstract

It is explained how field-theoretic methods and the dynamic renormalisation group (RG) can be applied to study the universal scaling properties of systems that either undergo a continuous phase transition or display generic scale invariance, both near and far from thermal equilibrium. Part 1 introduces the response functional field theory representation of (nonlinear) Langevin equations. The RG is employed to compute the scaling exponents for several universality classes governing the critical dynamics near second-order phase transitions in equilibrium. The effects of reversible mode-coupling terms, quenching from random initial conditions to the critical point, and violating the detailed balance constraints are briefly discussed. It is shown how the same formalism can be applied to nonequilibrium systems such as driven diffusive lattice gases. Part 2 describes how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then used to analyse simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which are characterised by the power laws of (critical) directed percolation. Certain other important universality classes are mentioned, and some open issues are listed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Ramond: Field theory – a modern primer, (Benjamin/Cummings, Reading 1981)

    Google Scholar 

  2. D.J. Amit: Field theory, the renormalization group and critical phenomena (World Scientific, Singapore 1984)

    Google Scholar 

  3. C. Itzykson and J.M. Drouffe: Statistical field theory (Cambridge University Press, Cambridge 1989)

    Google Scholar 

  4. M. Le Bellac: Quantum and statistical field theory, (Oxford University Press, Oxford 1991)

    Google Scholar 

  5. J. Zinn-Justin: Quantum field theory and critical phenomena (Clarendon Press, Oxford 1993)

    Google Scholar 

  6. J. Cardy: Scaling and renormalization in statistical physics (Cambridge University Press, Cambridge 1996)

    Google Scholar 

  7. P.C. Hohenberg and B.I. Halperin: Rev. Mod. Phys. 49, 435 (1977)

    Article  ADS  Google Scholar 

  8. H.K. Janssen: Field-theoretic methods applied to critical dynamics. In: Dynamical critical phenomena and related topics, Lecture Notes in Physics, vol. 104, ed by C.P. Enz (Springer, Heidelberg 1979), pp. 26–47

    Google Scholar 

  9. R. Bausch, H.K. Janssen and H. Wagner: Z. Phys. B24, 113 (1976)

    ADS  Google Scholar 

  10. J.L. Cardy: Renormalisation group approach to reaction-diffusion problems. In: Proceedings of Mathematical Beauty of Physics, ed by J.-B. Zuber, Adv. Ser. in Math. Phys. 24, 113 (1997)

    MathSciNet  Google Scholar 

  11. D.C. Mattis and M.L. Glasser: Rev. Mod. Phys. 70, 979 (1998)

    Article  ADS  Google Scholar 

  12. U.C. Täuber, M.J. Howard and B.P. Vollmayr-Lee: J. Phys. A: Math. Gen. 38, R79 (2005)

    Article  MATH  Google Scholar 

  13. U.C. Täuber: Critical dynamics: a field theory approach to equilibrium and non-equilibrium scaling behavior, in preparation (to be published at Cambridge University Press, Cambridge); for completed chapters, see: http://www.phys.vt.edu/ 3 tauber/utaeuber.html

    Google Scholar 

  14. R.A. Ferrell, N. Menyhàrd, H. Schmidt, F. Schwabl and P. Szépfalusy: Phys. Rev. Lett. 18, 891 (1967); Ann. of Phys. 47, 565 (1968)

    Article  ADS  Google Scholar 

  15. B.I. Halperin and P.C. Hohenberg: Phys. Rev. 177, 952 (1969)

    Article  ADS  Google Scholar 

  16. B.I. Halperin, P.C. Hohenberg and S.-k. Ma: Phys. Rev. Lett. 29, 1548 (1972)

    Article  ADS  Google Scholar 

  17. C. De Dominicis, E. Brézin and J. Zinn-Justin: Phys. Rev. B12, 4945 (1975)

    ADS  Google Scholar 

  18. H.K. Janssen: Z. Phys. B23, 377 (1976)

    MathSciNet  ADS  Google Scholar 

  19. C. De Dominicis: J. Physique Colloque 37, C2247 (1976)

    Google Scholar 

  20. P.C. Martin, E.D. Siggia and H.A. Rose: Phys. Rev. A8, 423 (1973)

    ADS  Google Scholar 

  21. H. Wagner: Z. Phys. 195, 273 (1966)

    Article  ADS  Google Scholar 

  22. N.D. Mermin and H. Wagner: Phys. Rev. Lett. 17, 1133 (1966)

    Article  ADS  Google Scholar 

  23. P.C. Hohenberg: Phys. Rev. 158, 383 (1967)

    Article  ADS  Google Scholar 

  24. P.M. Chaikin and T.C. Lubensky: Principles of condensed matter physics, (Cambridge University Press, Cambridge 1995)

    Google Scholar 

  25. S.-k. Ma and G.F. Mazenko: Phys. Rev. Lett. 33, 1383 (1974); Phys. Rev. B11, 4077 (1975)

    Article  ADS  Google Scholar 

  26. E. Frey and F. Schwabl: Adv. Phys. 43, 577 (1994)

    Article  ADS  Google Scholar 

  27. H.K. Janssen, B. Schaub and B. Schmittmann: Z. Phys. B73, 539 (1989)

    Article  ADS  Google Scholar 

  28. H.K. Janssen: On the renormalized field theory of nonlinear critical relaxation. In: From phase transitions to chaos, ed by G. Györgyi, I. Kondor, L. Sasvári and T. Tél (World Scientific, Singapore 1992), pp. 68–91.

    Google Scholar 

  29. P. Calabrese and A. Gambassi: Phys. Rev. E66, 066101 (2002); J. Phys. A: Math. Gen. 38, R133 (2005)

    ADS  Google Scholar 

  30. A. Gambassi: In: Proceedings of the International Summer School “Ageing and the Glass Transition”, to appear in J. Phys. Conf. Proc. (July 2006)

    Google Scholar 

  31. H.W. Diehl: In: Phase Transitions and Critical Phenomena, vol. 10, ed by C. Domb and J.L. Lebowitz (Academic Press, London 1986)

    Google Scholar 

  32. K. Oerding and H.K. Janssen: J. Phys. A: Math. Gen. 26, 5295 (1993)

    Article  ADS  Google Scholar 

  33. U.C. Täuber, V.K. Akkineni and J.E. Santos: Phys. Rev. Lett. 88, 045702 (2002)

    Article  ADS  Google Scholar 

  34. F. Haake, M. Lewenstein and M. Wilkens: Z. Phys. B55, 211 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  35. G. Grinstein, C. Jayaprakash and Y. He: Phys. Rev. Lett. 55, 2527 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  36. K.E. Bassler and B. Schmittmann: Phys. Rev. Lett. 73, 3343 (1994)

    Article  ADS  Google Scholar 

  37. U.C. Täuber and Z. Rácz: Phys. Rev. E55, 4120 (1997)

    Google Scholar 

  38. U.C. Täuber, J.E. Santos and Z. Rácz: Eur. Phys. J. B7, 309 (1999)

    ADS  Google Scholar 

  39. B. Schmittmann and R.K.P. Zia: Phys. Rev. Lett. 66, 357 (1991)

    Article  ADS  Google Scholar 

  40. B. Schmittmann: Europhys. Lett. 24, 109 (1993)

    Article  ADS  Google Scholar 

  41. K.E. Bassler and Z. Rácz: Phys. Rev. Lett. 73, 1320 (1994); Phys. Rev. E52, R9 (1995)

    Article  ADS  Google Scholar 

  42. B. Schmittmann and R.K.P. Zia: Statistical mechanics of driven diffusive systems. In: Phase Transitions and Critical Phenomena, vol. 17, ed by C. Domb and J.L. Lebowitz (Academic Press, London 1995)

    Google Scholar 

  43. H.K. Janssen and B. Schmittmann: Z. Phys. B63, 517 (1986)

    Article  ADS  Google Scholar 

  44. K.-t. Leung and J.L. Cardy: J. Stat. Phys. 44, 567 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  45. D. Forster, D.R. Nelson and M.J. Stephen: Phys. Rev. A16, 732 (1977)

    ADS  MathSciNet  Google Scholar 

  46. J.D. Murray: Mathematical Biology, vols. I/II (Springer, New York, 3rd ed 2002)

    Google Scholar 

  47. M. Mobilia, I.T. Georgiev and U.C. Täuber: e-print q-bio.PE/0508043 (2005)

    Google Scholar 

  48. M. Doi: J. Phys. A: Math. Gen. 9, 1465 & 1479 (1976)

    Google Scholar 

  49. P. Grassberger and M. Scheunert: Fortschr. Physik 28, 547 (1980)

    Article  MathSciNet  Google Scholar 

  50. L. Peliti: J. Physique 46, 1469 (1985)

    Article  MathSciNet  Google Scholar 

  51. F.C. Alcaraz, M. Droz, M. Henkel and V. Rittenberg: Ann. of Phys. 230, 250 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. M. Henkel, E. Orlandini and J. Santos: Ann. of Phys. 259, 163 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. G.M. Schütz: In: Phase Transitions and Critical Phenomena, vol. 19, ed by C. Domb and J.L. Lebowitz (Academic Press, London 2001)

    Google Scholar 

  54. R. Stinchcombe: Adv. Phys. 50, 431 (2001)

    Article  ADS  Google Scholar 

  55. F. van Wijland: Phys. Rev. E63, 022101 (2001)

    ADS  Google Scholar 

  56. J.W. Negele and H. Orland: Quantum many-particle systems (Addison-Wesley, Redwood City 1988)

    MATH  Google Scholar 

  57. L. Peliti: J. Phys. A: Math. Gen. 19, L365 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  58. H.K. Janssen: J. Stat. Phys. 103, 801 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  59. B.P. Lee: J. Phys. A: Math. Gen. 27, 2633 (1994)

    Article  ADS  Google Scholar 

  60. B.P. Lee and J. Cardy: Phys. Rev. E50, R3287 (1994)

    ADS  Google Scholar 

  61. D. Toussaint and F. Wilczek: J. Chem. Phys. 78, 2642 (1983)

    Article  ADS  Google Scholar 

  62. B.P. Lee and J. Cardy: J. Stat. Phys. 80, 971 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  63. O. Deloubrière, H.J. Hilhorst and U.C. Täuber: Phys. Rev. Lett. 89, 250601 (2002); H.J. Hilhorst, O. Deloubrière, M.J. Washenberger and U.C. Täuber: J. Phys. A: Math. Gen. 37, 7063 (2004)

    Article  ADS  Google Scholar 

  64. H.J. Hilhorst, M.J. Washenberger and U.C. Täuber: J. Stat. Mech. P10002 (2004)

    Google Scholar 

  65. M. Moshe: Phys. Rep. 37C, 255 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  66. P. Grassberger and K. Sundermeyer: Phys. Lett. B77, 220 (1978)

    ADS  Google Scholar 

  67. P. Grassberger and A. De La Torre: Ann. of Phys. 122, 373 (1979)

    Article  ADS  Google Scholar 

  68. H.K. Janssen and U.C. Täuber: Ann. of Phys. 315, 147 (2005)

    Article  MATH  ADS  Google Scholar 

  69. S.P. Obukhov: Physica A101, 145 (1980)

    ADS  MathSciNet  Google Scholar 

  70. J.L. Cardy and R.L. Sugar: J. Phys. A: Math. Gen. 13, L423 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  71. H.K. Janssen: Z. Phys. B42, 151 (1981)

    Article  ADS  Google Scholar 

  72. W. Kinzel: In: Percolation structures and processes, ed by G. Deutsch, R. Zallen and J. Adler (Hilger, Bristol 1983)

    Google Scholar 

  73. P. Grassberger, Z. Phys. B47, 365 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  74. H. Hinrichsen: Adv. Phys. 49, 815 (2001)

    Article  ADS  Google Scholar 

  75. G. Ódor: Rev. Mod. Phys. 76, 663 (2004)

    Article  ADS  Google Scholar 

  76. P. Grassberger: Math. Biosc. 63, 157 (1983)

    Article  MATH  Google Scholar 

  77. H.K. Janssen: Z. Phys. B58, 311 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  78. J.L. Cardy and P. Grassberger: J. Phys. A: Math. Gen. 18, L267 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  79. J. Benzoni and J.L. Cardy: J. Phys. A: Math. Gen. 17, 179 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  80. E. Frey, U.C. Täuber and F. Schwabl: Europhys. Lett. 26, 413 (1994); Phys. Rev. E49, 5058 (1994)

    Article  ADS  Google Scholar 

  81. H.K. Janssen and O. Stenull: Phys. Rev. E62, 3173 (2000)

    ADS  Google Scholar 

  82. U.C. Täuber, M.J. Howard and H. Hinrichsen: Phys. Rev. Lett. 80, 2165 (1998); Y.Y. Goldschmidt, H. Hinrichsen, M.J. Howard and U.C. Täuber: Phys. Rev. E59, 6381 (1999)

    Article  ADS  Google Scholar 

  83. R. Kree, B. Schaub and B. Schmittmann: Phys. Rev. A39, 2214 (1989)

    ADS  Google Scholar 

  84. F. van Wijland, K. Oerding and H. Hilhorst: Physica A251, 179 (1998); K. Oerding, F. van Wijland, J.P. Leroy and H. Hilhorst: J. Stat. Phys. 99, 1365 (2000)

    Google Scholar 

  85. J. Cardy and U.C. Täuber: Phys. Rev. Lett. 77, 4780 (1996); J. Stat. Phys. 90, 1 (1998)

    Article  ADS  Google Scholar 

  86. L. Canet, H. Chaté and B. Delamotte: Phys. Rev. Lett. 92, 255703 (2004); L. Canet, H. Chaté, B. Delamotte, I. Dornic and M.A. Muñoz: e-print cond-mat/0505170 (2005)

    Article  ADS  Google Scholar 

  87. M. Henkel and H. Hinrichsen: J. Phys. A: Math. Gen. 37, R117 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  88. H.K. Janssen, F. van Wijland, O. Deloubrière and U.C. Täuber: Phys. Rev. E70, 056114 (2004)

    ADS  Google Scholar 

  89. H.K. Janssen: Phys. Rev. E55, 6253 (1997)

    ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Center for Stochastic Processes in Science and Engineering, Virginia Polytechnic Institute and State University Blacksburg, 24061-0435, Virginia, USA

    U. C. Täuber

Authors
  1. U. C. Täuber
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire Physique des Matériaux, Université Henri Poincaré Nancy I, 239, F-54506, Vandœuvre lés, Nancy Cedex, France

    Malte Henkel

  2. Department of Physics Robertson Hall Virginia Polytechnic Institute, State University, 24061-0435, Blacksburg, VA, USA

    Michel Pleimling

  3. Laboratoire de Physique des Matériaux, Université de Luxembourg, 162a, avenue de la Faïencerie, L-1511, Luxembourg

    Roland Sanctuary

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer

About this chapter

Cite this chapter

Täuber, U. (2007). Field-Theory Approaches to Nonequilibrium Dynamics. In: Henkel, M., Pleimling, M., Sanctuary, R. (eds) Ageing and the Glass Transition. Lecture Notes in Physics, vol 716. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69684-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation