Map lattices coupled by collisions: hitting time statistics and collisions per lattice unit

Abstract

We study map lattices coupled by collision and show how perturbations of transfer operators associated with the spatially periodic approximation of the model can be used to extract information about the complete system. More precisely, we first study a map on a finite box of L sites with periodic boundary conditions, coupled by collision. We derive, via a non-trivial first order approximation for the leading eigenvalue of the rare event transfer operator, a formula for the first collision rate and a corresponding first hitting time law. For the former we show that the formula scales at the order of L • ε 2 , where ε is the coupling strength, and for the latter we show that the error in the law is of order O Lε 2 • | ln Lε 2 | . Using this information, we then derive an explicit formula for the first collision rate per lattice unit in the infinite dimensional system.