Dynamic scaling

Dynamic scaling (sometimes known as Family–Vicsek scaling^[1][2]) is a litmus test that shows whether an evolving system exhibits self-similarity. In general a function is said to exhibit dynamic scaling if it satisfies:

${\displaystyle f(x,t)\sim t^{\theta }\varphi \left({\frac {x}{t^{z}}}\right).}$

Here the exponent ${\displaystyle \theta }$ is fixed by the dimensional requirement ${\displaystyle [f]=[t^{\theta }]}$. The numerical value of ${\displaystyle f/t^{\theta }}$ should remain invariant despite the unit of measurement of ${\displaystyle t}$ is changed by some factor since ${\displaystyle \varphi }$ is a dimensionless quantity.

Many of these systems evolve in a self-similar fashion in the sense that data obtained from the snapshot at any fixed time is similar to the respective data taken from the snapshot of any earlier or later time. That is, the system is similar to itself at different times. The litmus test of such self-similarity is provided by the dynamic scaling.