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: Here the exponent
should remain invariant despite the unit of measurement of
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.
The litmus test of such self-similarity is provided by the dynamic scaling.
The term "dynamic scaling" as one of the essential concepts to describe the dynamics of critical phenomena seems to originate in the seminal paper of Pierre Hohenberg and Bertrand Halperin (1977), namely they suggested "[...] that the wave vector- and frequency dependent susceptibility of a ferromagnet near its Curie point may be expressed as a function independent of
[3] Later Tamás Vicsek and Fereydoon Family proposed the idea of dynamic scaling in the context of diffusion-limited aggregation (DLA) of clusters in two dimensions.
[2] The form of their proposal for dynamic scaling was: where the exponents satisfy the following relation: In such systems we can define a certain time-dependent stochastic variable
We are interested in computing the probability distribution of
The question is: what happens to the corresponding dimensionless variables?
If the numerical values of the dimensional quantities change, but corresponding dimensionless quantities remain invariant then we can argue that snapshots of the system at different times are similar.
One way of verifying dynamic scaling is to plot dimensionless variables
obtained at different times collapse onto a single universal curve then it is said that the systems at different time are similar and it obeys dynamic scaling.
The idea of data collapse is deeply rooted to the Buckingham Pi theorem.
Many phenomena investigated by physicists are not static but evolve probabilistically with time (i.e. Stochastic process).
Similarly, growth of networks like the Internet are also ever growing systems.
Spread of biological and computer viruses too does not happen over night.
Many other seemingly disparate systems which are found to exhibit dynamic scaling.