A motile alga swims by a run-and-tumble strategy in three dimensions: it moves in…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Non-equilibrium statistical mechanics and stochastic thermodynamics

Many Parts, One Behaviour, Ch Emergence and self-organisation: collective order from local rules

Problem

A motile alga swims by a run-and-tumble strategy in three dimensions: it moves in straight runs at speed v=100 μm s−1v=100\,\mu\mathrm{m\,s^{-1}}, after each of which a tumble randomises its heading completely (no memory of the previous direction). Unlike the exponential runs of the text, its run durations are spread uniformly between 00 and 2τ2\tau, with τ=10 s\tau=10\,\mathrm s, so the mean run time is still τ\tau. ((a)) Starting from the random-walk result ⟨r2⟩=N⟨ℓ2⟩\langle r^2\rangle=N\langle \ell^2\rangle for NN independent steps of length ℓ=vt\ell=vt, find ⟨t2⟩\langle t^2\rangle for this run-time distribution and derive the effective diffusion coefficient DeffD_{\rm eff} of the three-dimensional walk. Compare it with v2τ/3v^2\tau/3 for exponential runs of the same mean, and say why the two differ. ((b)) Evaluate DeffD_{\rm eff} for the alga's data, in μm2 s−1\mu\mathrm{m^2\,s^{-1}}. ((c)) The alga's body is roughly a sphere of radius a=5 μma=5\,\mu\mathrm m. Using the Stokes-Einstein relation D=kBT/(6πηa)D=k_{\rm B}T/(6\pi\eta a) for water at T=300 KT=300\,\mathrm K with viscosity η=1.0×10−3 Pa s\eta=1.0\times10^{-3}\,\mathrm{Pa\,s}, compute the purely thermal diffusion coefficient and the ratio Deff/DD_{\rm eff}/D. ((d)) State, with its value in seconds, the time scale that separates the short-time ballistic regime (⟨r2⟩∝t2\langle r^2\rangle\propto t^2) from the long-time diffusive regime (⟨r2⟩∝t\langle r^2\rangle\propto t) of this random walk. Use kB=1.38×10−23 J K−1k_{\rm B}=1.38\times10^{-23}\,\mathrm{J\,K^{-1}}.

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