Simple, But Spatially Extended Dynamical Systems

Prof. Igor Mezic, Department of Mechanical and Environmental Engineering, University of California, Santa Barbara

Coupled Map Lattice and Probabilistic Cellular Automata seem to be gaining popularity as tools for modelling complex phenomena in physics, engineering, biology, chemistry, social sciences, economics, etc. (ordering of subjects does not reflect the importance - or anything else, for that matter). These sytems typically exhibit spatially and/or temporally chaotic behavior. Thus, probabilistically-statistically swayed studies are prevalent. So, in the first part of the talk we will address questions like: what are the fluctuations around the state of equilibrium for the system? For systems that exhibit monotonicity condition (analogous to that for PDE's), sufficiently fast decay of correlations, we show that the fluctuations around the equilibrium are Gaussian, and some other nice statistical properties folllow. This is achieved through the use of FKG (Fortuin-Kasteleyn-Ginibre) inequalities, well-known in statistical mechanics studies of spin systems.

In the second part of the talk, we discuss the problems of natural observation scales in Coupled Map Lattices and Probabilistic Cellular Automata. Time-signal at a particular site of a spatially extended model can be stochastic in nature. Thus, we are tempted to average away, and uncover the {\it macroscale} dynamics by averaging over measurements at all sites. But, this typically results in an equilibrium value (i.e. no time-dependence) for the average state of the system. Is there an averaging scale at which we keep some of the nontrivial time-dependent behavior of the system? We present a method to do find such a scale, backed up by some rigorous analysis using the above-mentioned FKG inequalities and the theory of hydrodynamic limits. At the scale that we find, the system is governed by a stochastic partial differential equation that is amenable to analysis in some simplified situations. The dynamics of the model is composed of a deterministic component and a small stochastic component.

We present a number of examples from biology, chemistry and engineering physics, the cruelest of which is an ecological game in which hawks eat doves, but hydrodynamic limits work.