The probability density of landslide debris volumes observed in a mountain region is…

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

The probability density of landslide debris volumes

V

observed in a mountain region is found to obey a strict self-similarity relation: for every scale factor

b>0

there is a function

g(b)

, not depending on

V

, such that

p(bV) = g(b)\,p(V)

whenever

V

and

bV

both lie above the smallest volume the survey records,

V_{\min}>0

. Assume

p

is smooth above

V_{\min}

and that there is no upper cutoff. ((a)) Differentiate the relation with respect to

b

and set

b=1

to show that

p

obeys

V\,p'(V) = c\,p(V)

for some constant

c

, and solve this differential equation to find the general form of

p(V)

. ((b)) Substitute your solution back into

p(bV)=g(b)p(V)

to show that

g(b) = b^{c}

is forced, so above

V_{\min}

a power law is the only density satisfying the relation. ((c)) Surveys show that a landslide of ten times the volume of a reference landslide has one-hundredth of its probability density. Find the exponent

\alpha

p(V)\propto V^{-\alpha}

, and state the value of

c

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