A microscope tracks swimming E

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

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

Problem

A microscope tracks swimming E. coli to measure the effective run-and-tumble diffusion coefficient. Runs have an exponentially distributed duration with mean

\tau=1\,\mathrm{s}

and speed

v=20\,\mu\mathrm{m\,s^{-1}}

; each run's contribution to the measured mean-square displacement,

\ell^2 = v^2t^2

, is an independent random quantity whose relative fluctuation (standard deviation divided by mean) is

\sqrt5 \approx 2.24

, because

\langle t^4\rangle = 24\tau^4

while

\langle t^2\rangle = 2\tau^2

for an exponential run time. Using the relative-fluctuation law for a block average of

|B|

independent parts,

\sigma_{\bar x}/\langle x\rangle=(\sigma_x/\langle x\rangle)\,|B|^{-1/2}

, estimate: ((a)) the number of independent runs that must be pooled to measure

D_{\rm eff}

to a relative precision of

5\%

; ((b)) if a single continuous recording follows one bacterium for

2

minutes, contributing about

120

runs, the number of independently tracked cells whose recordings must be pooled to reach that precision.

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