## Infinite particle systems

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- 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.

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## 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