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Field Theoretic Methods

  • Reference work entry
Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

Correlation Functions and Field Theory

Discrete Stochastic Interacting Particle Systems

Stochastic Differential Equations

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Absorbing state:

State from which, once reached, an interacting many‐particle system cannot depart, not even through the aid of stochastic fluctuations.

Correlation function:

Quantitative measure of the correlation of random variables; usually set to vanish for statistically independent variables.

Critical dimension:

Borderline dimension d c above which mean-field theory yields reliable results, while for \( { d \leq d_{\mathrm{c}} } \) fluctuations crucially affect the system's large scale behavior.

External noise:

Stochastic forcing of a macroscopic system induced by random external perturbations, such as thermal noise from a coupling to a heat bath.

Field theory:

A representation of physical processes through continuous variables, typically governed by an exponential probability distribution.

Generating function:

Laplace transform of the probability distribution; all moments and correlation functions follow through appropriate partial derivatives.

Internal noise:

Random fluctuations in a stochastic macroscopic system originating from its internal kinetics.

Langevin equation:

Stochastic differential equation describing time evolution that is subject to fast random forcing.

Master equation:

Evolution equation for a configurational probability obtained by balancing gain and loss terms through transitions into and away from each state.

Mean-field approximation:

Approximative analytical approach to an interacting system with many degrees of freedom wherein spatial and temporal fluctuations as well as correlations between the constituents are neglected.

Order parameter:

A macroscopic density corresponding to an extensive variable that captures the symmetry and thereby characterizes the ordered state of a thermodynamic phase in thermal equilibrium. Nonequilibrium generalizations typically address appropriate stationary values in the long-time limit.

Perturbation expansion:

Systematic approximation scheme for an interacting and/or nonlinear system that involves a formal expansion about an exactly solvable simplication by means of a power series with respect to a small coupling.

Bibliography

  1. Lindenberg K, Oshanin G, Tachiya M (eds) (2007) J Phys: Condens Matter 19(6): Special issue containing articles on Chemical kinetics beyond the textbook: fluctuations, many‐particle effects and anomalous dynamics; see: http://www.iop.org/EJ/toc/0953-8984/19/6

  2. Alber M, Frey E, Goldstein R (eds) (2007) J Stat Phys 128(1/2): Special issue on Statistical physics in biology; see: http://springerlink.com/content/j4q1ln243968/

  3. Murray JD (2002) Mathematical biology, vols. I, II, 3rd edn. Springer, New York

    Google Scholar 

  4. Mobilia M, Georgiev IT, Täuber UC (2007) Phase transitions and spatio‐temporal fluctuations in stochastic lattice Lotka–Volterra models. J Stat Phys 128:447–483. several movies with Monte Carlo simulation animations can be accessed at http://www.phys.vt.edu/~tauber/PredatorPrey/movies/

    Google Scholar 

  5. Washenberger MJ, Mobilia M, Täuber UC (2007) Influence of local carrying capacity restrictions on stochastic predator‐prey models. J Phys: Condens Matter 19:065139, 1–14

    Google Scholar 

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

    Google Scholar 

  7. Amit DJ (1984) Field theory, the renormalization group, and critical phenomena. World Scientific, Singapore

    Google Scholar 

  8. Negele JW, Orland H (1988) Quantum many‐particle systems. Addison-Wesley, Redwood City

    MATH  Google Scholar 

  9. Parisi G (1988) Statistical field theory. Addison-Wesley, Redwood City

    MATH  Google Scholar 

  10. Itzykson C, Drouffe JM (1989) Statistical field theory. Cambridge University Press, Cambridge

    Book  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  14. Janssen HK (1976) On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties. Z Phys B 23:377–380

    Article  Google Scholar 

  15. De Dominicis C (1976) Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques. J Physique (France) Colloq 37:C247–C253

    Google Scholar 

  16. Janssen HK (1979) Field-theoretic methods applied to critical dynamics. In: Enz CP (ed) Dynamical critical phenomena and related topics. Lecture Notes in Physics, vol 104. Springer, Heidelberg, pp 26–47

    Google Scholar 

  17. Doi M (1976) Second quantization representation for classical many‐particle systems. J Phys A: Math Gen 9:1465–1477

    Article  Google Scholar 

  18. Doi M (1976) Stochastic theory of diffusion‐controlled reactions. J Phys A: Math Gen 9:1479–1495

    Article  Google Scholar 

  19. Grassberger P, Scheunert M (1980) Fock-space methods for identical classical objects. Fortschr Phys 28:547–578

    Article  MathSciNet  Google Scholar 

  20. Peliti L (1985) Path integral approach to birth-death processes on a lattice. J Phys (Paris) 46:1469–1482

    Article  Google Scholar 

  21. Peliti L (1986) Renormalisation of fluctuation effects in the \( { A + A \to A } \) reaction. J Phys A: Math Gen 19:L365–L367

    Article  MathSciNet  Google Scholar 

  22. Lee BP (1994) Renormalization group calculation for the reaction \( { kA \to \emptyset } \). J Phys A: Math Gen 27:2633–2652

    Article  Google Scholar 

  23. Lee BP, Cardy J (1995) Renormalization group study of the \( { A + B \to \emptyset } \) diffusion‐limited reaction. J Stat Phys 80:971–1007

    Article  MathSciNet  MATH  Google Scholar 

  24. Mattis DC, Glasser ML (1998) The uses of quantum field theory in diffusion‐limited reactions. Rev Mod Phys 70:979–1002

    Article  Google Scholar 

  25. Täuber UC, Howard MJ, Vollmayr-Lee BP (2005) Applications of field‐theoretic renormalization group methods to reaction‐diffusion problems. J Phys A: Math Gen 38:R79–R131

    Google Scholar 

  26. Täuber UC (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, pp 295–348

    Google Scholar 

  27. Täuber UC, Critical dynamics: a field theory approach to equilibrium and nonequilibrium scaling behavior. To be published at Cambridge University Press, Cambridge. for completed chapters, see: http://www.phys.vt.edu/~tauber/utaeuber.html

  28. Schütz GM (2000) Exactly solvable models for many-body systems far from equilibrium. In: Domb C, Lebowitz JL (eds) Phase transitions and critical phenomena, vol 19. Academic Press, London

    Google Scholar 

  29. Stinchcombe R (2001) Stochastic nonequilibrium systems. Adv Phys 50:431–496

    Article  Google Scholar 

  30. Van Wijland F (2001) Field theory for reaction‐diffusion processes with hard-core particles. Phys Rev E 63:022101, 1–4

    Article  Google Scholar 

  31. Chopard B, Droz M (1998) Cellular automaton modeling of physical systems. Cambridge University Press, Cambridge

    Book  Google Scholar 

  32. Marro L, Dickman R (1999) Nonequilibrium phase transitions in lattice models. Cambridge University Press, Cambridge

    Book  Google Scholar 

  33. Hinrichsen H (2000) Nonequilibrium critical phenomena and phase transitions into absorbing states. Adv Phys 49:815–958

    Article  Google Scholar 

  34. Ódor G (2004) Phase transition universality classes of classical, nonequilibrium systems. Rev Mod Phys 76:663–724

    Google Scholar 

  35. Moshe M (1978) Recent developments in Reggeon field theory. Phys Rep 37:255–345

    Article  MathSciNet  Google Scholar 

  36. Obukhov SP (1980) The problem of directed percolation. Physica A 101:145–155

    Article  MathSciNet  Google Scholar 

  37. Cardy JL, Sugar RL (1980) Directed percolation and Reggeon field theory. J Phys A: Math Gen 13:L423–L427

    Article  MathSciNet  Google Scholar 

  38. Janssen HK (1981) On the nonequilibrium phase transition in reaction‐diffusion systems with an absorbing stationary state. Z Phys B 42:151–154

    Article  Google Scholar 

  39. Janssen HK, Täuber UC (2005) The field theory approach to percolation processes. Ann Phys (NY) 315:147–192

    Google Scholar 

  40. Grassberger P (1982) On phase transitions in Schlögl's second model. Z Phys B 47:365–374

    Article  MathSciNet  Google Scholar 

  41. Janssen HK (2001) Directed percolation with colors and flavors. J Stat Phys 103:801–839

    Article  MATH  Google Scholar 

  42. Martin PC, Siggia ED, Rose HA (1973) Statistical dynamics of classical systems. Phys Rev A 8:423–437

    Article  Google Scholar 

  43. Bausch R, Janssen HK, Wagner H (1976) Renormalized field theory of critical dynamics. Z Phys B 24:113–127

    Article  Google Scholar 

  44. Chaikin PM, Lubensky TC (1995) Principles of condensed matter physics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  45. Hohenberg PC, Halperin BI (1977) Theory of dynamic critical phenomena. Rev Mod Phys 49:435–479

    Article  Google Scholar 

  46. Schmittmann B, Zia RKP (1995) Statistical mechanics of driven diffusive systems. In: Domb C, Lebowitz JL (eds) Phase transitions and critical phenomena, vol 17. Academic Press, London

    Google Scholar 

  47. Janssen HK, Schmittmann B (1986) Field theory of long time behaviour in driven diffusive systems. Z Phys B 63:517–520

    Article  Google Scholar 

  48. Leung KT, Cardy JL (1986) Field theory of critical behavior in a driven diffusive system. J Stat Phys 44:567–588

    Article  MathSciNet  Google Scholar 

  49. Forster D, Nelson DR, Stephen MJ (1977) Large‐distance and long-time properties of a randomly stirred fluid. Phys Rev A 16:732–749

    Article  MathSciNet  Google Scholar 

  50. Kardar M, Parisi G, Zhang YC (1986) Dynamic scaling of growing interfaces. Phys Rev Lett 56:889–892

    Article  MATH  Google Scholar 

  51. Barabási AL, Stanley HE (1995) Fractal concepts in surface growth. Cambridge University Press, Cambridge

    Google Scholar 

  52. Halpin-Healy T, Zhang YC (1995) Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Phys Rep 254:215–414

    Article  Google Scholar 

  53. Krug J (1997) Origins of scale invariance in growth processes. Adv Phys 46:139–282

    Article  Google Scholar 

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Acknowledgments

The author would like to acknowledge financial support through the US National Science Foundation grant NSF DMR-0308548. This article is dedicated to the victims of the terrible events at Virginia Tech on April 16, 2007.

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Authors and Affiliations

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

    Uwe Claus Täuber

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  1. Uwe Claus Täuber
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Täuber, U.C. (2012). Field Theoretic Methods. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_69

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