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Realistic molecular dynamics and self-assembly is represented in a lattice simulation where water, water-hydrocarbons, and water-amphiphilic systems are investigated. The details of the phase separation dynamics and the constructive self-assembly dynamics are discussed and compared to the corresponding experimental systems. The method used to represent the different molecular types can easily be expended to include additional molecules and thus allow the assembly of more complex structures. This molecular dynamics (MD) lattice gas fills a modeling gap between traditional MD and lattice gas methods. Both molecular objects and force fields are represented by propagating information particles and all microscopic interactions are reversible. Living systems, perhaps the ultimate constructive dynamical systems, is the motivation for this work and our focus is a study of the dynamics of molecular self-assembly and self-organization. In living systems, matter is organized such that it spontaneously constructs intricate functionalities at all levels from the molecules up to the organism and beyond. At the lower levels of description, chemical reactions, molecular selfassembly and self-organization are the drivers of this complexity. We shall, in this chapter, demonstrate how molecular self-assembly and selforganization processes can be represented in formal systems. The formal systems are to be denned as a special kind of lattice gas and they are in a form where an obvious correspondence exists between the observables in the lattice gases and the experimentally observed properties in the molecular self-assembly systems. This has the clear advantage that by using these formal systems, theory, simulation, and experiment can be conducted in concert and can mutually support each other. However, a disadvantage also exists because analytical results are difficult to obtain for these formal systems due to their inherent complexity dictated by their necessary realism. The key to novelt simpler molecules (from lower levels), dynamical hierarchies are formed [2, 3]. Dynamical hierarchies are characterized by distinct observable functionalities at multiple levels of description. Since these higher-order structures are generated spontaneously due to the physico-chemical properties of their building blocks, complexity can come for free in molecular self-assembly systems. Through such processes, matter apparently can program itself into structures that constitute living systems [11, 27, 30].

We illustrate growth phenomena in two-dimensional cellular automata (CA) by four case studies. The first CA, which we call Obstacle Course, describes the effect that obstacles have on such features of simple growth models as linear expansion and coherent asymptotic shape. Our next CA is random-walk-based Internal Diffusion Limited Aggregation, which spreads sublinearly, but with a shape which can be explicitly computed due to hydrodynamic effects. Then we propose a simple scheme for characterizing CA according to their growth properties, as indicated by two Larger than Life examples. Finally, a very simple case of Spatial Prisoner’s Dilemma illustrates nucleation analysis of CA. In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e.g., Meakin [34] and more than 1300 references cited in the latter). Cellular automata (CA) growth models, by virtue of their simplicity and amenability to computer experimentation [25], have become particularly popular in the last 20 years, especially in physics research literature [40, 42]. Needless to say, precise mathematical results are hard to come by, and many basic questions remain completely open at the rigorous level. The purpose of this chapter, then, is to outline some successes of the mathematical approach and to identify some fundamental difficulties. We will mainly address three themes which can be summarized by the terms: aggregation, nucleation, and constraint-expansion transition. These themes also provide opportunities to touch on the roles of randomness, monotonicity, and linearity in CA investigations. We choose to illustrate these issues by particular CA rules, with little attempt to formulate a general theory. Simplicity is often, and rightly, touted as an important selling point of cellular automata. We have, therefore, tried to choose the simplest models which, while being amenable to some mathematical analysis, raise a host of intriguing unanswered questions. The next few paragraphs outline subsequent sections of this chapter. Aggregation models typically study properties of growth from a small initial seed. Arguably, the simplest dynamics are obtained by adding sites on the boundary in a uniform fashion.