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Functional integrals and the Fokker-Planck equation

Функциональные интегралы и уравнение Фоккера-Планка

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

We give a precise definition of the functional integrals involved by means of different discretization prescriptions. The value of the functional integral is prescription dependent and this leads to a variety of representations with different Lagrangians. We discuss general covariance and we derive in a unified way, all the lagrangians proposed in the literature together with their associated discretizations. We make also a critical examination of previous results.

Riassunto

Si dà una definizione precisa degli integrali funzionali in questione, per mezzo di diverse prescrizioni di discretizzazione. Il valore dell'integrale funzionale è dipendente dalla prescrizione e questo porta ad una varietà di rappresentazioni con lagrangiane diverse. Si discute la covarianza generale e si derivano in maniera unificata tutte le lagrangiane proposte nella letteratura assieme con le loro discretizzazioni associate. Si effettua anche un esame critico di precedenti risultati.

Резюме

Мы даем точное определение функциональных интегралов. Величина функционального интеграла зависит от способа описания. Это привпдит к множеству представлений с различными Лагранжианами. Мы обсуждаем общую ковариантность и выводим единым образом все Лагранжианы, предложенные в литературе. Мы также проводим критическое обсуждение предыдущих редультатов.

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References

  1. F. Langouche, D. Roekaerts andE. Tirapegui:Physica,95 A, 252 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. F. Langouche, D. Roekaerts andE. Tirapegui:Physica (to appear).

  3. M. E. Brachet andE. Tirapegui:Phys. Lett.,71 A, 174 (1979); and preprint UCL-ITP-79-09 (Louvain-la-Neuve).

    MathSciNet  ADS  MATH  Google Scholar 

  4. F. Langouche, D. Roekaerts andE. Tirapegui:Prog. Theor. Phys. (to appear).

  5. I. W. Mayes andJ. S. Dowker:Journ. Math. Phys.,14, 434 (1973).

    Article  MathSciNet  ADS  Google Scholar 

  6. I. W. Mayes andJ. S. Dowker:Proc. Roy. Soc.,327 A, 131 (1972).

    Article  ADS  Google Scholar 

  7. J. S. Dowker:Journ. Math. Phys.,17, 1873 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  8. F. A. Berezin:Theor. Math. Phys.,6, 194 (1971).

    Article  MathSciNet  Google Scholar 

  9. F. A. Berezin andM. S. Marinov:Ann. of Phys.,104, 336 (1977).

    Article  ADS  Google Scholar 

  10. H. Leschke andM. Schmutz:Zeits. Phys.,27 B, 85 (1977).

    MathSciNet  ADS  Google Scholar 

  11. R. L. Stratonovich:Sel. Transl. Math. Stat. Prob.,10, 273 (1971).

    Google Scholar 

  12. P. Graham:Zeits. Phys.,26 B, 281 (1977).

    ADS  Google Scholar 

  13. H. Dekker:On the Fourier series analysis of the functional integral for general diffusion processes, Physical Laboratory TNO Report IR 49 (1978) (Den Haag).

  14. C. P. Enz:Variational principles for Langevin equations, inProceedings of the XIII IUPAP Conference on Statistical Physics, Ann. Isr. Phys. Soc., Vol.2 (1978).

  15. C. P. Enz:Lecture Notes in Physics, Vol.54, sect. 2–3 (Berlin, 1976), p. 79.

    Article  ADS  Google Scholar 

  16. W. Horsthemke andA. Bach:Zeits. Phys.,22 B, 189 (1975).

    ADS  Google Scholar 

  17. L. D. Faddeev: inMethods in Field Theory (Les Houches, 1975), edited byR. Balian andJ. Zinn-Justin (Amsterdam, 1976).

  18. B. S. DeWitt:Rev. Mod. Phys.,29, 377 (1957).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. K. S. Cheng:Journ. Math. Phys.,13, 1723 (1972).

    Article  ADS  Google Scholar 

  20. J. L. Gervais andA. Jevicki:Nucl. Phys.,110 B, 93 (1976).

    Article  MathSciNet  ADS  Google Scholar 

  21. A. C. Hirshfeld:Phys. Lett.,67 A, 5 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  22. F. Langouche, D. Roekaerts andE. Tirapegui:Functional integrals in phase space, to be published inProceedings of Symposium of Theoretical Physics, Santiago, December 1978.

  23. R. Graham: private communication.

  24. B. S. Agarwal andE. Wolf:Phys. Rev. D,2, 2161 (1970).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. R. Graham:Proceedings of the XIII IUPAP Conference on Statistical Physics, Ann. Isr. Phys. Soc. (1978), p. 948.

  26. H. Dekker:Physica,87 A, 419 (1977).

    Article  MathSciNet  ADS  Google Scholar 

  27. R. Graham:Springer Tracts on Modern Physics, Vol.66 (Berlin, 1973).

  28. H. Dekker:Physica,85 A, 598 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. R. Graham:Zeits. Phys.,26 B, 397 (1977).

    ADS  Google Scholar 

  30. A. Lichnerowicz:Eléments de calcul tensoriel, Collection Armand Colin (Paris, 1962).

  31. U. Deininghaus andR. Graham:Zeits. Phys. (to appear).

  32. F. Langouche, D. Roekaerts andE. Tirapegui:Phys. Rev. D (to appear).

  33. F. Langouche, D. Roekaerts andE. Tirapegui:Phys. Rev. D (to appear).

  34. F. Langouche, D. Roekaerts andE. Tirapegui:Lett. Nuovo Cimento (to appear).

  35. F. Langouche, D. Roekaerts andE. Tirapegui:Phys. Lett., A (to appear).

  36. H. Leschke, A. C. Hirshfeld andT. Suzuki:Phys. Rev. D,18, 2834 (1978).

    Article  ADS  Google Scholar 

  37. D. Dürr andA. Bach:Comm. Math. Phys.,60, 725 (1978).

    Article  Google Scholar 

  38. H. Ito:Prog. Theor. Phys.,59, 725 (1978).

    Article  ADS  MATH  Google Scholar 

  39. F. Langouche, D. Roekaerts andE. Tirapegui:J. Phys. A,11, L263 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  40. F. Langouche, D. Roekaerts andE. Tirapegui:J. Phys. A (to appear).

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Author information

Author notes

  1. F. Langouche

    Present address: Onderzoeker IIKW, Belgium

  2. D. Roekaerts

    Present address: Aspirant NFWO, Belgium

Authors and Affiliations

  1. Instituut voor Theoretische Fysica, K. U. Leuven, B-3030, Leuven, Belgium

    F. Langouche, D. Roekaerts & E. Tirapegui

Authors
  1. F. Langouche
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  2. D. Roekaerts
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  3. E. Tirapegui
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Traduzione a cura della Redazione.

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Langouche, F., Roekaerts, D. & Tirapegui, E. Functional integrals and the Fokker-Planck equation. Nuov Cim B 53, 135–159 (1979). https://doi.org/10.1007/BF02739307

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  • DOI: https://doi.org/10.1007/BF02739307

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