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Abstract
We investigate an interacting particle system consisting of two types of particles located at a finite point-lattice. The particles randomly change their type and neighboring particles randomly interchange positions. The system seems to remain at equilibrium for a substantial amount of time until it suddenly, at a critical time T , leaves equilibrium along what seems to be a deterministic trajectory. The analysis reveals, however, that the trajectories are determined randomly, but only by the systems behavior at very early times, much prior to T. In the nonstandard model used, the system randomly 'chooses' the trajectory in an infinitesimal interval (0;), 0, but this choice only becomes visible in the interval (T ; T). The underlying reason for this behavior is revealed by a decomposition of the systems trajectories with respect to an eigenbasis (gk)k2K of the discrete Laplace operator 4. It shows that after an initial random period the system's dynamics behaves, coordinate-wise, like t 7! e ( + k)(t T) k(!), where is unlimited ('infinitely large'), kgk =4gk and k(!) denotes a random quantity. The hyperfinite result obtained is translated into a standard limit theorem.
Highlights
We briefly indicate how our simple interacting particle systems may relate to more complex systems in chemical reaction kinetics
We further introduce the discrete Laplace operator and its eigenbasis, which becomes in Section 5 the fundamental tool for the investigation of the extended dynamics
Considering particles of type A only and regarding particles of type B as holes, our dynamical system is described by hopping of particles to neighboring positions3 and the overall particle number is not conserved any more. It shares these properties with discrete-time zero-range processes with non-conservation of particle numbers in the sense of [10] or reaction-diffusion processes in the sense of [7, Section 13.2]
Summary
Interacting particle systems have been a prospering field of mathematical studies in a standard setting (Griffeath [11] and Liggett [15]) as well as a nonstandard one (Helms and Loeb [12], and Albeverio, Fenstad, Høegh-Krohn and Lindstrøm [1, Chapter 7]), the most prominent being the Ising model. The system shows a Gaussian behavior: Projections of the system’s random-state onto orthogonal eigenvectors of the Laplacian are approximately independent, approximately normally distributed random variables The variances of these variables increase geometrically with time. We are interested in the system’s behavior for large numbers of particles This is within standard mathematics expressed by limit theorems. It is further possible to obtain from results concerning the hyperfinite situation corresponding limit results in standard mathematical terms In this way Lindeberg type limit theorems have been proved in Weisshaupt [28]. Following this idea we characterize the system’s dynamics for hyperfinite particle-collections first (Theorem 6.5), and apply afterward transfer and the permanence principle to obtain a corresponding standard limit result (Theorem 7.9). That appendix B is still radically elementary in the sense that it only uses idealization and transfer to establish this connection
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