Repeated random splitting of droplets in a microfluidic device is a multiplicative…

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 Heavy tails: power laws and how to measure them

Problem

Repeated random splitting of droplets in a microfluidic device is a multiplicative random process: the volume after many splits is a product of many independent random factors. By the central limit theorem applied to

\ln X

, the droplet volume

X

is lognormally distributed,

p(x) = \dfrac{1}{x\sigma\sqrt{2\pi}}\exp\!\left[-\dfrac{(\ln x - \mu)^2}{2\sigma^2}\right]

, with

\sigma = 4

in natural-log units (the mean

\mu

is not needed below). An analyst plots

p(x)

on log-log axes from

x_1 = 10\ \mathrm{\mu m^3}

x_2 = 1000\ \mathrm{\mu m^3}

(two decades), finds the curve looks straight, and writes: "the plot is straight over two decades, so

X

obeys a power law with exponent equal to the fitted slope." ((a)) By comparing

p(x)

with

e^{-\lambda x}

for a fixed but arbitrary

\lambda > 0

, decide whether

X

is a heavy-tailed distribution. ((b)) The local log-log slope of the lognormal density is

\dfrac{d\ln p}{d\ln x} = -1 - \dfrac{\ln x - \mu}{\sigma^2}

. Compute the magnitude of the change in this slope between

x_1

and

x_2

. ((c)) Explain, using the functional equation that defines scale invariance, why the analyst's conclusion is wrong even though the numerical slope change from part (b) is small.

Browse the problem library