Random walks and the FokkerPlanck equation SSP 8.3.1 8.3.5

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Lecture 6

• Random walks and the Fokker-Planck equation (SSP
  8.3.1 - 8.3.5)

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Topics

• Classical random walks
• Fokker-Planck equation
• Relation to stochastic differential equations
• Monte Carlo solution of such equations
• Example of Black-Scholes equation

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Simple Random Walk
Every unit of time, ?t, walker steps to right or
left by an amount ?x with probability T and T-
The sequence of steps is an example of a Markov
chain.
At any time t, the probability that she is at x
is p(x,t). The probability that he is at x at
time t?t is
p(x,t?t)p(x-?x,t)T
p(x?x,t)T-
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For case TT-½ show that
and in the limit ?t?0 gives the equation for
p(x,t)
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Diffusion equation
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Solution by simulation
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Analytical solution
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N-step random walk
Mean displacement 0 Root mean square
displacement ?xvN
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Brownian motion
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Random walk with decay
In addition to stepping probabilities TT- every
?t, walker may die with a probability ??t (where
? may be a function of position), i.e. TT-(1-
??t)/2
Show that then the equation satisfied by p(x,t) is
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Solution by simulation
Many walkers, each started from a position in
p0(x), at a later time tN?t will be distributed
as p(x,t)

• Select a starting point from p0(x)
• Each step (?t) annihilate walker with
  probability ??t
• if it survives step to left or right by
  ?x(2D?t)½
• Stop if Nt/?t
• Repeat M times to build up a histogram of x(t)

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• Difficulty!
• Because of annihilation probability, few,
  possibly none, of the walkers may survive until
  t, i.e. for N steps
• Population stabilisation algorithm - later

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Continuous steps
The symmetry of the step means that this process
is a martingale, the expectation of the location
after the step is the same as its position
beforehand a fair coin
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The solution above is true for any x0 and t
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For this simple case there is a natural
distribution from which the steps can be chosen,
i.e. a Gaussian with mean zero and variance 2D?t.
In a practical simulation a step would be
calculated using
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Wiener, or Brownian, process

• Notation
• In the limit ?t?0 an infinitesimal stochastic
  variable dW(t) has mean zero and variance dt

A realisation of this variable would be
where ? is a standard Gaussian variate.
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More general random walk

• The distribution from which the steps are chosen,
  T(?)d?, need not be symmetric, i.e. ?0.
• Its properties may depend on position, T(x,?)d?,
  so (x)
• But still Markovian, step independent of previous
  history of the walk

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The probability that the walker is at x at time
t?t
p(x,t?t)p(x-?x,t)T
p(x?x,t)T-
generalises to

Chapman-Kolmogorov equation
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p(x,t) then satisfies (for details see slides at
the end, or SSP8.3.5 for discrete steps)
Cases T symmetric ? b0, T independent of x ?
b,D const.
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Corresponding stochastic differential equation
dx(t)b(x)dt v(2D(x))
dW(t), known as the Langevin equation.
Two ways of looking at it, Stochastic variable
x(t) satisfies the stochastic differential
equation, or
deterministic quantity p(x,t) satisfies the
Fokker-Planck equation.
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Relation of pde to sde (stochastic differential
equation)

• pde

• sde

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Solution of Diffusion equation by simulation
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An example, simple diffusion D1/2, x00,
T2.5, ?t0.001
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Diffusion-Convection Equation

• pde

• sde

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If, in addition, annihilation is possible, i.e.
We have the Fokker-Planck (or backward
Kolmogorov) equation
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Relationship to macroscopic quantities

• Diffusion

• Convection (drift)

• Annihilation

• Production

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Fokker-Planck equation
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Examples
Density of positrons in a solid
Density of neutrons in a nuclear reactor
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Solution of the Fokker-Planck equation
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Its solution by simulated random walks.
The distribution of population of walkers after
Nt/?t steps will approximate the solution ?(x,t)
if

• Starting points of walkers selected from ?0(x)

• A walker, at x, is annihilated with probability
  ??t

• Else it steps by ?x(2D(x)?t)½?v(x)?t,
• where ? is a standard Gaussian variate

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Assignment IV

• The Fokker-Planck equation for the density in
  momentum space of cosmic ray particles diffusing,
  undergoing acceleration and escaping, has the form

The case of t and a constant and continuous
injection at a single value of momentum i.e.