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How do stochastic processes enter into physics?

Part of the Lecture Notes in Mathematics book series (LNM,volume 1250)

Abstract

Fluctuations in non-equilibrium systems do not arise from a probability distribution of the initial state, but are continually generated by the equations of motion. In order to derive them from statistical mechanics a drastic repeated randomness assumption is indispensable. One is then led to a master equation, from which both the deterministic macroscopic equation and the fluctuations are obtained by a limiting process. The approximate nature of the whole procedure makes the use of strictly mathematical delta-correlations and Itô calculus illusory.

Keywords

  • Master Equation
  • Stochastic Differential Equation
  • Langevin Equation
  • Brownian Particle
  • Random Force

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. A. Einstein, Ann. Physik (4) 17, 549 (1905); 19, 371 (1906); M. v. Smoluchowski, Ann. Physik (4) 21, 756 (1906).

    CrossRef  Google Scholar 

  2. H. Haken, in Encyclopaedia of Physics 25/2c (Springer, Berlin 1970); M. Sargent, M.O. Scully, and W.E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass. 1974).

    Google Scholar 

  3. H. Haken, Synergetics (Springer, Berlin 1976, 1978); C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin 1983).

    MATH  Google Scholar 

  4. With the exception of a few soluble cases, viz., the linear harmonic chain, see e.g. G.W. Ford, M. Kac, and P. Mazur, J. Math. Phys. 6, 504 (1965); P. Ullersma, Physica 32, 27, 56, 74, 90 (1966).

    CrossRef  MathSciNet  Google Scholar 

  5. S. Nakajima, Prog. Theor. Phys. 20, 948 (1958); R. Zwanzig, J. Chem. Phys. 33, 1338 (1960); M. Mori, Prog. Theor. Phys. 33, 423 (1965).

    CrossRef  MathSciNet  Google Scholar 

  6. R. Kubo, J. Phys. Soc. Japan 12, 570 (1957).

    CrossRef  MathSciNet  Google Scholar 

  7. N.G. van Kampen, Physica Norvegica 5, 279 (1971).

    Google Scholar 

  8. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam 1981).

    MATH  Google Scholar 

  9. The vital role of such margins was forcefully argued by P. and T. Ehrenfest, in: Enzyklopädie der mathematischen Wissenschaften 4, Nr. 32 (Teubner, Leipzig 1912); translated by M.J. Moravcsik with the title Conceptual Foundations of the Statistical Approach in Mechanics (Cornell Univ. Press, Ithaca 1959).

    Google Scholar 

  10. The founding fathers were of course fully aware of this: G.E. Uhlenbeck and L.S. Ornstein, Phys. Rev. 36, 823 (1930).

    CrossRef  Google Scholar 

  11. T.S. van Albada, Bull. Astr. Inst. Neth. 19, 479 (1968); W. Thirring, Z. Physik 235, 339 (1970); R. Miller, in: Advances in Chemical Physics 26 (Wiley, New York 1970).

    Google Scholar 

  12. N.N. Bogolubov, Problems of Dynamical Theory in Statistical Physics, in: Studies in Statistical Mechanics I (G.E. Uhlenbeck and J. de Boer eds., North-Holland, Amsterdam 1962); G.E. Uhlenbeck, in: Probability and Related Topics in Physical Sciences I (Proceedings of the Summer Seminar in Boulder, Colorado in 1957; Interscience, London and New York 1959) p. 195 ff.; E.G.D. Cohen, in: Fundamental Problems in Statistical Mechanics II (E.G.D. Cohen ed., North-Holland, Amsterdam 1968).

    Google Scholar 

  13. R.E. Nettleton, J. Chem Phys. 40, 112 (1964); I. Müller, Z. Physik 198, 329 (1967); L.S. García-Colin, M. López de Haro, R.F. Rodriguez, and D. Jou, J. Stat. Phys. 37, 465 (1984).

    CrossRef  MathSciNet  Google Scholar 

  14. N.G. van Kampen, Can. J. Phys. 39, 551 (1961) and in: Advances in Chemical Physics 34 (Wiley, New York 1976); R. Kubo, K. Matsuo, and K. Kitahara, J. Statis. Phys. 9, 51 (1973).

    CrossRef  Google Scholar 

  15. N.G. van Kampen, Phys. Letters 62A, 383 (1977).

    CrossRef  Google Scholar 

  16. H. Grabert and M.S. Green, Phys. Rev. A19, 1747 (1979); H. Grabert, R. Graham, and M.S. Green, Phys. Rev. A21, 2136 (1980).

    CrossRef  MathSciNet  Google Scholar 

  17. C.P. Slichter, Principles of Magnetic Resonance (Harper and Row, New York 1963; Springer, Berlin 1978).

    Google Scholar 

  18. V.I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York 1961); U. Frisch, in: Probabilistic Methods in Applied Mathematics 1 (A.T. Bharucha-Reid ed., Acad. Press, New York 1968); V.I. Klyatskin and V.I. Tatarski, Sov. Phys. Usp. 16, 494 (1974).

    MATH  Google Scholar 

  19. P. Mazur and I. Oppenheim, Physica 50, 241 (1970), and literature quoted there.

    CrossRef  Google Scholar 

  20. T.G. Kurtz, J. Appl. Prob. 7, 49 (1970); 8, 344 (1977); J. Chem. Phys. 57, 2976 (1972); Z.A. Akcasu, J. Statis. Phys. 16, 33 (1977); N.G. van Kampen, J. Statis. Phys. 25, 431 (1981).

    CrossRef  MathSciNet  Google Scholar 

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© 1987 Springer-Verlag

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van Kampen, N.G. (1987). How do stochastic processes enter into physics?. In: Albeverio, S., Blanchard, P., Streit, L. (eds) Stochastic Processes — Mathematics and Physics II. Lecture Notes in Mathematics, vol 1250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077353

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  • DOI: https://doi.org/10.1007/BFb0077353

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