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

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 x(rms) similar to t(alpha/2) and the peak value of P(x, t) at x = 0 decays following a power-law Pmax similar to t(-alpha/2) with alpha = 1 in all cases but one. In the case of geometrically shrinking steps, where the size of the nth step is chosen to be R-n(n) , with Rn being the nth largest number among N random numbers drawn within [0,1], we find alpha - 1/2 . Such non-linear relation between mean squared displacement and time < x2 > similar to t(alpha) with alpha = 1/2 instead of alpha = 1 suggests that the corresponding Brownian motion describes sub-diffusion. (c) 2021 Elsevier Ltd. All rights reserved.