Abstract: In this paper we provide a theoretical model of fractal growth which bases on potential theory and random walk. It is characterized by branch and growth and can be used to study or simulate the river branch, breakdown, discharge and even the growth.The fractal dimension D has been measured; the relationship between D and the probability of growth outwards P-out(indicates the potential condition u(r) has been discussed.We compared the computer simulations with some experiment facts, facts, the results were satisfactory.

Key words: fractal, fractal dimension, random, walk, branch, growth, discharge, Boltzmann equation