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Probabilistic Methods in Physics

  • N. G. van Kampen
Part of the Ettore Majorana International Science Series book series (EMISS, volume 46)

Abstract

The fundamental concepts and methods of probability theory are reviewed. It is emphasized that probability calculus merely transforms one probability distribution into another. Hence for every application some a priori distribution must be defined on the basis of the underlying physics. The mere use of the word ‘random’ is not enough, particularly since that word is often used in the sense of ‘irregular’ or ‘disordered’. The transition of a disordered system to a probabilistic (or stochastic) description is a fundamental problem, which cannot be solved by playing with words.

A stochastic process is an ensemble of functions of time, distinguished by a parameter for which a probability distribution is given. Alternatively it can be specified by a hierarchy of joint distributions for the values it takes at any set of time points tl, t2,...tr. An important role is played in physics by the Markov processes, roughly described as processes having no memory (in analogy with the differential equations of deterministic physics). They are fully specified by the distribution of values at any one time, and the transition probability between two times. This transition probability obeys the master equation, in which the coefficients have a direct physical meaning.

The natural way of solving the master equation is provided by the expansion in a parameter (often the size of the system), which separates the macroscopic scale from the scale in which the fluctuations take place. The lowest term in the expansion reproduces the deterministic macroscopic rate equation. The next term describes the fluctuations in terms of the Fokker-Planck equation, whose solution is a time dependent Gaussian distribution. Higher terms provide corrections to this Gaussian behavior of the fluctuations. In this way the dilemma that arises from applying the Langevin approach to nonlinear systems is resolved.

A condition for the validity of this expansion is that the macroscopic equation is dissipative and tends to a stable equilibrium. If the macroscopic equation vanishes, the expansion tends to a Fokker-Planck equation with nonlinear coefficients. As a special example, the hopping of a charge carrier in an inhomogeneous medium with an external field is worked out. It leads to a diffusion equation in a rather unexpected form.

Keywords

Markov Process Joint Distribution Master Equation Macroscopic Equation Probability Calculus 
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. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam 1981 ).MATHGoogle Scholar
  2. C.W. Gardiner, Handbook of Stochastic Methods ( Springer, Berlin 1983 ).MATHGoogle Scholar
  3. W. Feller, An Introduction to Probability Theory and its Applications, I and II ( Wiley, New York 1957, 1966 ).Google Scholar
  4. N. Wax ed., Selected Papers on Noise and Stochastic Processes ( Dover, New York 1954 ).MATHGoogle Scholar
  5. R. L. Stratonovich, Topics in the Theory of Random Noise. I and II ( Gordon and Breach, New York 1963, 1967 ).Google Scholar
  6. 1.
    E. W. Montroll and B. J. West, in Fluctuation Phenomena ( E. W. Montroll and J. L. Lebowitz eds., North-Holland, Amsterdam 1979 ).Google Scholar
  7. 2.
    M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels (Acad. Press, New York 1967);MATHGoogle Scholar
  8. R. J. Elliott, J. A. Krumhansl, and P. L. Leath, R.v. Mod. Phys. 46, 465 (1974).MathSciNetADSCrossRefGoogle Scholar
  9. 3.
    S. Alexander, J. Bernasconi, W. R. Schneider, and R. Orbach, Rev. Mod. Phys. 53, 175 (1981).MathSciNetADSCrossRefGoogle Scholar
  10. 4.
    M. B. Weissman, in Proc. 6th Intern. Conf. on Noise in Physical Systems (P. H. E. Meijer et al. eds., NBS Washington, D.C. 1981);Google Scholar
  11. F. N. Hooge, T. G. M. Kleinpenning, and L. K. J. Vandamme, Repts. Prog. Phys. 44, 479 (1981).ADSCrossRefGoogle Scholar
  12. 5.
    G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930).ADSMATHCrossRefGoogle Scholar
  13. 6.
    G. Czycholl, Phys. Repts. 143, 277 (1986).ADSCrossRefGoogle Scholar
  14. 7.
    J. L. Doob, Annals Math. 43, 351 (1942).MathSciNetADSMATHCrossRefGoogle Scholar
  15. 8.
    A. Ramakrishnan, in Encyclopedia of Physics 3/2 (S. Flügge ed., Springer. Berlin 1959 ), Sec. 33; Stratonovich, I. Ch. 6; Van Kampen, Ch. 2.Google Scholar
  16. M. Planck, Sitzber. Preuss. Akad. Wissens. (1917), p. 324.Google Scholar
  17. 10.
    A. Einstein, Ann. Phys. [4] 17, 549 (1905); 19, 371 (1906).Google Scholar
  18. 11.
    Lord Rayleigh, Phil. Mag. 32, 424 (1891).MATHCrossRefGoogle Scholar
  19. 12.
    H. A. Kramers, Physica 7, 284 (1940);MathSciNetADSMATHCrossRefGoogle Scholar
  20. U. M. Titulaer, Physica 91A, 321 (1978); 100A, 234, 251 (1980).Google Scholar
  21. 13.
    A. Kolmogorov, Mathem. Annalen 104, 415 (1931).MATHCrossRefGoogle Scholar
  22. 14.
    R. F. Pawula, Phys. Rev. 162, 186 (1967).ADSCrossRefGoogle Scholar
  23. 15.
    P. Langevin, Comptes Rendus (Paris) 146, 530 (1908).MATHGoogle Scholar
  24. 16.
    L. Brillouin, Phys. Rep. 78, 627 (1950).MATHGoogle Scholar
  25. 17.
    R. E. Mortensen, J. Stat. Phys. 1, 271 (1969);ADSCrossRefGoogle Scholar
  26. 17.
    N. G. van Kampen, J. Stat. Phys. 24, 175 (1981).ADSMATHCrossRefGoogle Scholar
  27. 18.
    S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).MathSciNetADSMATHCrossRefGoogle Scholar
  28. 19.
    R. Kubo, K. Matsuo, and K. Kitahara, J. Stat. Phys. 9, 51 (1973).ADSCrossRefGoogle Scholar
  29. 20.
    R. Landauer, Phys. Rev. Al2, 636 (1975);ADSGoogle Scholar
  30. R. Landauer, Heiv. Phys. Acta 56, 847 (1983);Google Scholar
  31. N. G. van Kampen, IBM J. Res. Dey. 32, 107 (1988).CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • N. G. van Kampen
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of UtrechtUtrechtNetherlands

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