Decay of Correlations. IV. Necessary and Sufficient Conditions for a Rapid Decay of Correlations
Abstract
The authors consider N particle systems whose probability distributions obey the master equation. For these systems, the authors derive the necessary and sufficient conditions under which the reduced n particle (n < N) probabilities also obey master equations and under which the Ursell functions decay to their equili-brium values faster than the probability distributions. These conditions impose restrictions on the form of the transition rate matrix and thus on the form of its eigenfunctions. The authors first consider systems in which the eigenfunctions of the N particle transition rate matrix are completely factorized and demonstrate that for such systems the reduced probabilities obey master equations and the Ursell functions decay rapidly if certain additional conditions are imposed. As an example of such a system the authors discuss a random walk of N pairwise and interacting walkers. The authors then demonstrate that for systems whose N particle transition matrix can be written as a sum of one particle, two particle, etc. contributions, and for which the reduced probabilities obey master equations, the reduced master equtions become in the thermodynamic limit those for independent particles, which have been discussed previously. As an example of such N particle systems the authors discuss the relaxation of a gas of interacting harmonic oscillators.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1971
- Accession Number
- AD0726762
Entities
People
- Dick Bedeaux
- Irwin Oppenheim
- Kurt E. Shuler
Organizations
- University of California, San Diego