A marine bacterium, Vibrio alginolyticus

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Complex and collective systems

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

Problem

A marine bacterium, Vibrio alginolyticus, swims in straight runs at speed

v = 25\,\mu\mathrm{m\,s^{-1}}

. Its run times are exponentially distributed with mean

\tau = 0.8\,\mathrm{s}

, and at the end of each run it tumbles instantaneously to a new direction uncorrelated with the old one, exactly as in the run-and-tumble model. ((a)) Using the random-walk result

\langle \ell^2\rangle = 2v^2\tau^2

for the mean-square length of one exponentially distributed run, find the effective diffusion coefficient

D_{\rm eff}

of a single bacterium's long-time trajectory in three dimensions. ((b)) A passive sphere of radius

a = 1\,\mu\mathrm{m}

in water at

T = 300\,\mathrm{K}

with viscosity

\eta = 1.0\,\mathrm{mPa\,s}

has the Stokes-Einstein diffusion coefficient

D_{\rm thermal} = k_{\rm B}T/(6\pi\eta a)

. Evaluate

D_{\rm thermal}

and the ratio

D_{\rm eff}/D_{\rm thermal}

. ((c)) A colleague argues: "Because

D_{\rm eff}

only takes this large, well-defined value once many run segments have accumulated, and because it is hundreds of times the thermal value, it must be an emergent property of the bacterium's motion." Using the three-part working definition of emergent behaviour (defined only for the whole collection of parts, not for any one of them; in principle determined by the rules for the parts; and possessed by no single part), with the bacteria as the parts, decide whether the colleague is right, and say which parts of the definition fail. Use

k_{\rm B}=1.38\times10^{-23}\,\mathrm{J\,K^{-1}}

Browse the problem library