Algorithmic Complexity and Thermodynamics of Fractal Growth Processes

A statistical mechanical formalism is developed using the computer information concepts of algorithmic complexity and Kolmogorov universal probability. This formalism provides a thermodynamic description of microstates of a system. For an isolated classical system, the algorithmic complexity is equal to the thermodynamic entropy. This approach does not rely on probabilistic ensemble concepts and can be applied to non-ergodic systems. An H-function is developed from the Kolmogorov universal probability that satisfies the properties of an entropy function. Using this approach, the thermodynamics of irreversible growth processes far from equilibrium can be investigated. Entropy functions for fractal growth processes far from equilibrium are developed from the algorithmic complexity and are seen to be similar to Flory-type equilibrium functions. This development does not require energy minimization procedures associated with equilibrium arguments. Using this approach, constraints can be put on the types of mean field models that will yield fractal structures. The algorithmic complexity of a system can also be used to explore the role of fluctuations and criticality in highly ordered systems.