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Similarity and self-similarity in random walk with fixed, random and shrinking steps

Author

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  • Mitra, Tushar
  • Hossain, Tomal
  • Banerjee, Santo
  • Hassan, Md. Kamrul

Abstract

In this article, we study a class of random walk (RW) problem for fixed, random, linearly decreasing and geometrically shrinking step sizes and find that they all obey dynamic scaling which we verified using the idea of data-collapse. We show that the full width at half maximum (FWHM) of the probability density P(x,t) curves is equivalent to the root-mean square (rms) displacement which grows with time as xrms∼tα/2 and the peak value of P(x,t) at x=0 decays following a power-law Pmax∼t−α/2 with α=1 in all cases but one. In the case of geometrically shrinking steps, where the size of the nth step is chosen to be Rnn, with Rn being the nth largest number among N random numbers drawn within [0,1], we find α=1/2. Such non-linear relation between mean squared displacement and time 〈x2〉∼tα with α=1/2 instead of α=1 suggests that the corresponding Brownian motion describes sub-diffusion.

Suggested Citation

  • Mitra, Tushar & Hossain, Tomal & Banerjee, Santo & Hassan, Md. Kamrul, 2021. "Similarity and self-similarity in random walk with fixed, random and shrinking steps," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    • Handle: RePEc:eee:chsofr:v:145:y:2021:i:c:s0960077921001429
    DOI: 10.1016/j.chaos.2021.110790
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