Infinite particle systems

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 → P {P_n} \to P implies Q n → Q {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 {Q_0} related to Q is introduced, and we show that with the appropriate topology the map taking Q to Q 0 {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.