Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena
- 1 August 1952
- journal article
- Published by AIP Publishing in The Journal of Chemical Physics
- Vol. 20 (8), 1281-1295
- https://doi.org/10.1063/1.1700722
Abstract
Two principles of a statistical mechanics of time‐dependent phenomena are proposed and argued for. The first states that the proper mathematical object to describe the physical situation is the stationary random process specified by the ensemble of time series ai(Xt)i=1···s and the distribution ρ(X). The set phase functions ai(X)i=1···s represent the set of grossly observable features of the system. Xt is the image of the phase X after time t. ρ(X) is a stationary distribution. The second principle is concerned with the very common case in which the phenomenological equations are of the first order in time and states that in this case the random process in question is a Markoff process. A Fokker‐Planck equation is derived for the process, and an entropy is defined and is shown always to increase. Phenomenological equations are derived as a first approximation to the Markoff process. These involve a certain matrix ξij which is shown to satisfy symmetry relations which are a generalization of Onsager's. The theory is applied to the uniformization of the direction of momentum of a particle on a two‐dimensional torus subject to a small perturbing potential.Keywords
This publication has 12 references indexed in Scilit:
- A general kinetic theory of liquids III. Dynamical propertiesProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1947
- The Statistical Mechanical Theory of Transport Processes I. General TheoryThe Journal of Chemical Physics, 1946
- On Onsager's Principle of Microscopic ReversibilityReviews of Modern Physics, 1945
- On the Theory of the Brownian Motion IIReviews of Modern Physics, 1945
- Stochastic Problems in Physics and AstronomyReviews of Modern Physics, 1943
- Proof of the Quasi-Ergodic HypothesisProceedings of the National Academy of Sciences, 1932
- ber die analytischen Methoden in der WahrscheinlichkeitsrechnungMathematische Annalen, 1931
- Hamiltonian Systems and Transformation in Hilbert SpaceProceedings of the National Academy of Sciences, 1931
- Reciprocal Relations in Irreversible Processes. I.Physical Review B, 1931
- Generalized harmonic analysisActa Mathematica, 1930