Probabilistic Methods in Physics
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.
KeywordsMarkov Process Joint Distribution Master Equation Macroscopic Equation Probability Calculus
Unable to display preview. Download preview PDF.
- W. Feller, An Introduction to Probability Theory and its Applications, I and II ( Wiley, New York 1957, 1966 ).Google Scholar
- R. L. Stratonovich, Topics in the Theory of Random Noise. I and II ( Gordon and Breach, New York 1963, 1967 ).Google Scholar
- 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
- 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
- 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
- M. Planck, Sitzber. Preuss. Akad. Wissens. (1917), p. 324.Google Scholar
- 10.A. Einstein, Ann. Phys.  17, 549 (1905); 19, 371 (1906).Google Scholar
- U. M. Titulaer, Physica 91A, 321 (1978); 100A, 234, 251 (1980).Google Scholar
- R. Landauer, Heiv. Phys. Acta 56, 847 (1983);Google Scholar