Statistical Theory of Diffusion-Limited Growth in Two Dimensions

A formalism of irreversible aggregation processes is presented in terms of statistical mechanics. Thermodynamical variables that include the degeneracy of the histories, which we call “history entropy,” are introduced by taking into account all possible growth histories. By considering the thermodynamic properties of the growth history and harmonic measure, we find the condition of the most probable history for diffusion-limited aggregation (DLA) as a function of mass fractal dimension of the resulting clusters. Families of fractal growth generators are utilized for the estimation of the history entropy, and the fractal dimension of observable DLA clusters in two dimensions is evaluated in accordance with numerical and experimental results.