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

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

Chaos, Solitons & Fractals, 2021, vol. 145, issue C

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.

Keywords: Diffusion; Random walk; Scale-invariance; Data-collapse; Dynamic scaling; Self-similarity (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:145:y:2021:i:c:s0960077921001429

DOI: 10.1016/j.chaos.2021.110790

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