Dynamic scaling

Dynamic scaling (sometimes known as Family-Vicsek 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:

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

Here the exponent $\theta$ is fixed by the dimensional requirement $[f]=[t^{\theta }]$. Now, the numerical value of ${\frac {f}{t^{\theta }}}$ should remain invariant despite the unit of measurement of $t$ is changed by some factor since $\varphi$ 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:

$f(x,t)\sim t^{-w}x^{-\tau }\varphi \left({\frac {x}{t^{z}}}\right).$

Many phenomena which physicists often investigate are not static but rather evolve probabilistically with time. The universe is perhaps one of the best examples which is expanding ever since the Big Bang. Similarly, growth of networks like the Internet are also ever growing systems. Another example is polymer degradation where degradation does not occur in a blink of an eye but rather over quite a long time. Spread of biological and computer viruses too does not happen over night. Many of these evolves 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.

In such system we find certain stochastic variable $x$ which assume values that depend on time. In such cases, we are often interested to know the distribution of $x$ at various instants of time i.e. $f(x,t)$. Now the numerical value of $f$ and the typical or mean value of $x$ may well be very different at every different instant measurement. The question is: What happens to the corresponding dimensionless variables? If the numerical values of the dimensional quantities are different, however, corresponding dimensionless quantities remain invariant then we can argue that the snapshot of the system at different times are similar. When this happens we conclude that the system is self-similar.

One way of verifying the dynamic scaling is to plot dimensionless variables $f/t^{\theta }$ as a function of ${\frac {x}{t^{z}}}$ of the data extracted at various different time. Then if all the plots of $f$ vs $x$ 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. Essentially such systems can be termed as temporal self-similarity since the same system is similar at different times.

There have many seemingly disparate systems which are found to exhibit dynamic scaling e.g., kinetics of aggregation described by Smoluchowski coagulation equation, complex network described by Barabasi–Albert model, kinetic and stochastic Cantor set. The growth model within the Kardar–Parisi–Zhang (KPZ) class, one find that the width of the surface $W(L,t)$ exhibits dynamic scaling. The area size distribution of the blocks of weighted planar stochastic lattice (WPSL) too exhibits dynamic scaling.