Dynamic scaling

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 WWW, the Internet etc are also ever growing systems. Another example is polymer degradation process where degradation does not occur in a blink of an eye rather it happens over quite a long time.

Spread of biological and computer viruses too does not happen over night. In such system we find certain stochastic variable which assume values that depend on time. In such cases, we are often interested to know the distribution of at various instant of time i.e. . Now the numerical value of and the typical or mean value of 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 because the system is similar to itself at different time. The dynamic scaling is the litmus test of showing that an evolving system exhibits such self-similarity. In general a function is said to exhibits dynamic scaling if it satisfies

Here the exponent is fixed by the dimensional requirement . Now, the numerical value of should remain invariant despite the unit of measurement of is changed by some factor since is a dimensionless quantity. However, T. Vicsek and F. Family first proposed the idea of dynamic scaling in the context of diffusion-limited aggregation DLA of clusters in two dimensions.[1] The form of their proposal for dynamic scaling was

One way of verifying the dynamic scaling is to plot dimensionless variables as a function of of the data extracted at various different time. Then if all the plots of vs 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 theorem.[2] 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,[3] complex network described by Barabasi–Albert model,[4] kinetic and stochastic Cantor set.[5] The growth model within the Kardar–Parisi–Zhang (KPZ) class, one find that the width of the surface exhibits dynamic scaling.[6][7] The dynamic scaling sometimes also known as Family-Vicsek scaling.[8]

References

  1. ↑ T. Vicsek and F. Family, Phys. Rev. Lett. 52 1669 (1984)
  2. ↑ G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics (Cmpridge University Press, 1996).
  3. ↑ M. K. Hassan and M. Z. Hassan, "Emergence of fractal behavior in condensation-driven aggregation", Phys. Rev. E 79, 021406 (2009); M. K. Hassan and M. Z. Hassan, "Condensation-driven aggregation in one dimension", Phys. Rev. E 77, 061404 (2008). M. K. Hassan, M. Z. Hassan and N. Islam, "Emergence of fractal in aggregation with stochastic self-replication" Phys. Rev. E 88, 042137 (2013).
  4. ↑ M. K. Hassan, M. Z. Hassan and N. I. Pavel, "Dynamic scaling, data-collapseand Self-similarity in Barabasi–Albert networks" J. Phys. A: Math. Theor. 44 175101 (2011)
  5. ↑
    M. K. Hassan, N. I. Pavel, R. K. Pandit and J. Kurths, "Dyadic Cantor set and its kinetic and stochastic counterpart" Chaos, Solitons & Fractals 60 31–39 (2014).
  6. ↑ Kardar, Mehran; Parisi, Giorgio; Zhang, Yi-Cheng (3 March 1986). "Dynamic Scaling of Growing Interfaces". Physical Review Letters. 56 (9): 889–892. doi:10.1103/PhysRevLett.56.889. PMID 10033312..
  7. ↑ RAISSA M. D’SOUZA, International Journal of Modern Physics C, 8 941–951 (1997).
  8. ↑ Family, F.; Vicsek, T. (1985). "Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model". Journal of Physics A: Mathematical and General. 18 (2): L75–L81. doi:10.1088/0305-4470/18/2/005.
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