Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Infinite particle systems
HTML articles powered by AMS MathViewer

by Sidney C. Port and Charles J. Stone PDF
Trans. Amer. Math. Soc. 178 (1973), 307-340 Request permission

Abstract:

We consider a system of denumerably many particles that are distributed at random according to a stationary distribution P on some closed subgroup X of Euclidean space. We assume that the expected number of particles in any compact set is finite. We investigate the relationship between P and the distribution Q of particles as viewed from a particle selected “at random” from some set. The distribution Q is called the tagged particle distribution. We give formulas for computing P in terms of Q and Q in terms of P and show that, with the appropriate notion of convergences, ${P_n} \to P$ implies ${Q_n} \to Q$ and vice versa. The particles are allowed to move in an appropriate translation invariant manner and we show that the tagged particle distribution Q’ at a later time 1 is the same as the distribution of particles at time 1 as viewed from a particle selected “at random” from those initially in some set. We also show that Q’ is the same as the distribution of particles at time 1 as viewed from a particle selected at random from those at the origin, when initially the particles are distributed according to Q. The one-dimensional case is treated in more detail. With appropriate topologies, we show that in this case there is a homeomorphism between the collection of stationary distributions P and tagged particle distributions Q. A stationary spacings distribution ${Q_0}$ related to Q is introduced, and we show that with the appropriate topology the map taking Q to ${Q_0}$ is a homeomorphism. Explicit expressions are found for all these maps and their inverses. The paper concludes by using the one-dimensional results to find stationary distributions for a class of motions of denumerably many unit intervals and to establish criteria for convergence to one of these distributions.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 60K35, 82.60
  • Retrieve articles in all journals with MSC: 60K35, 82.60
Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 178 (1973), 307-340
  • MSC: Primary 60K35; Secondary 82.60
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0326868-3
  • MathSciNet review: 0326868