Consider a random walk in which the -th step has a fixed

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Physical foundations of the living cell

Life at Thermal Energy, Ch Random walks, diffusion and the crowded cell

Problem

Consider a random walk in which the ii-th step has a fixed, known length ℓi\ell_i but a random, independent sign: each step is +ℓi+\ell_i or −ℓi-\ell_i with equal probability, independently of every other step. After NN such steps the walker's position is x=s1+s2+⋯+sNx = s_1 + s_2 + \cdots + s_N. ((a)) Show that ⟨x2⟩=∑i=1Nℓi2\langle x^2\rangle = \sum_{i=1}^{N} \ell_i^2, and state exactly where independence of the steps is used. ((b)) A molecule takes N=1000N = 1000 such steps, each of duration Δt=1.0×10−6 s\Delta t = 1.0\times10^{-6}\,\mathrm{s}, alternating between two step lengths: 500500 steps of ℓA=2.0 nm\ell_A = 2.0\,\mathrm{nm} and 500500 steps of ℓB=4.0 nm\ell_B = 4.0\,\mathrm{nm}. Calculate the root mean square displacement after all 10001000 steps, and the effective diffusion coefficient DeffD_{\rm eff} defined by ⟨x2⟩=2Defft\langle x^2\rangle = 2D_{\rm eff}t, where t=NΔtt = N\Delta t is the total time taken.

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