Some Problems on Spatial Patterns in Nonequilibrium Systems.

In this thesis, we study the evolution of spatial patterns in two nonequilibrium systems. In Chapter 1, we study the steady state of a 1 -d cellular automata (CA) model of chemical turbulence. Empirically there are two interesting types of space-time patterns (depending on model parameters): a S phase which seems to contain solitons and a T phase which seems to be turbulent. We show that the macroscopic phases can be predicted from the microscopic dynamics. We define the thermodynamic limit of the steady state of CAs and show that the steady state of the S phase is trivial and the T phase exhibits a Gibbs state. We explicitly calculate the T phase steady state and find an approximate form for the energy functional which generates the Gibbs state. We show that there is no adequate characterization of turbulent behavior in CAs and introduce a quantity the "P-entropy" which is positive if the CA patterns are turbulent and zero otherwise. We show the P-entropy for the T phase is positive. In Chapter 2, we consider the consequences of the dynamical scaling hypothesis in phase ordering dynamics. We assume that the dynamics are governed by the Cahn-Hilliard -Cook (CHC) and time-dependent Ginzburg-Landau equations and show that the scaling hypothesis restricts the asymptotic growth rate of the length-scale of the patterns and the small wavevector behavior of the form factor. Specifically, if the form factor S_{k}( t) grows as k^{delta } for small delta, then delta>=q 4 (for the CHC dynamics). We find that experimental data indicates delta = 4. We also show that the CHC equation is sometimes inadequate for describing phase ordering dynamics. An alternative to the CHC model by Oono, Kitahara and Jasnow is examined. We find that many features of phase ordering dynamics are robust with respect to changing the dynamics.