Dynamic scaling

The dynamic scaling is the litmus test of showing that an evolving system exhibits self-similarity. In general a function is said to exhibit dynamic scaling if it satisfies

Here the exponent ${\displaystyle \theta }$ is fixed by the dimensional requirement ${\displaystyle [f]=[t^{\theta }]}$. Now, the numerical value of ${\displaystyle {\frac {f}{t^{\theta }}}}$ should remain invariant despite the unit of measurement of ${\displaystyle t}$ is changed by some factor since ${\displaystyle \varphi }$ is a dimensionless quantity. However, Tamás Vicsek and Fereydoon Family first proposed the idea of dynamic scaling in the context of diffusion-limited aggregation DLA of clusters in two dimensions. The form of their proposal for dynamic scaling was

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