Radically elementary analysis of an interacting particle system at an unstable equilibrium

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, ε], ε ≈ 0, but this choice only becomes visible in the interval [T − ε,T]. The underlying reason for this behavior is revealed by a decomposition of the systems trajectories with respect to an eigenbasis (gk)k∈K of the discrete Laplace operator 4 . It shows that after an initial random period the system’s dynamics behaves, coordinate-wise, like t 7→ ekυk(ω), where λ is unlimited (‘infinitely large’), μkgk = 4gk and υk(ω) denotes a random quantity. The hyperfinite result obtained is translated into a standard limit theorem. 2000 Mathematics Subject Classification 26E35 (primary); 82C22, 60J60, 60F05, 82C20, 60J10 (secondary)