Macroscopic order from reversible and stochastic lattice growth models

Abstract

This thesis advances the understanding of how autonomous microscopic physical processes give rise to macroscopic structure. A unifying theme is the use of physically motivated microscopic models of discrete systems which incorporate the constraints of locality, uniformity, and exact conservation laws. The features studied include: stochastic nonequilibrium fluctuations; use of pseudo-randomness in dynamical simulations; the thermodynamics of pattern formation; recurrence times of finite discrete systems; and computation in physical models. I focus primarily on pattern formation: transitions from a disordered to an ordered macroscopic state. Using an irreversible stochastic model of pattern formation in an open system driven by an external source of noise, I study thin film growth. I focus on the regimes of growth and the average properties of the resulting rough surfaces. I also show that this model couples sensitively to the imperfections of various pseudorandom number generators, resulting in non-stochastic exploration of the accessible state space. Using microscopically reversible models, I explicitly model how macroscopic dissipation can arise. In discrete systems with invertible dynamics entropy cannot decrease, and most such systems approach fully ergodic. Therefore these systems are natural candidates for models of thermodynamic behavior. I construct reversible models of pattern formation by dividing the system in two: the part of primary interest, and a "heat bath". We can observe the exchange of heat, energy, and entropy between the two subsystems, and gain insight into the thermodynamics of self-assembly. I introduce a local, deterministic, microscopically reversible model of cluster growth via aggregation in a closed two-dimensional system. The model has a realistic thermodynamics. When started from a state with low coarse grained entropy the model exhibits an initial regime of rapid nonequilibrium growth followed by a quasistatic regime with a well defined temperature. The growth clusters generated display a rich variety of morphologies. I also show how sequences of conditional aggregation events can be used to implement reusable logic gates and how to simulate any digital logic circuit with this model.