Grain growth self similarity

Statistical self-similarity holds when the normalized grain-size distribution and the number-of-sides distribution remain constant during growth. A grain structure at a later time then looks statistically similar to itself at an earlier time, except for a uniform magnification, and the structures are therefore scaled with respect to each other. 

Monte Carlo methods, direct tracking methods, and vertex models, where the evolution of the two-dimensional grain structure is described in terms of the motion of the vertices. After initial transients, all of these simulations exhibit statistical self-similarity during growth and an average grain area that increases linearly with time according to Eq. 15.35. 

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