A walker on the same one-dimensional lattice

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems

Computational Physics, Ch Drawing numbers: random walks and Monte Carlo

Problem

A walker on the same one-dimensional lattice, sites

a_{\rm lat}

apart with one hop every

\tau

, is now biased: it hops to the right with probability

q

and to the left with probability

1-q

, where

q

is a little above

\tfrac12

. Let

c(x,t)

be the expected concentration, as in the derivation of the diffusion equation. ((a)) Write the hopping rule for

c(x,t+\tau)

in terms of

c(x-a_{\rm lat},t)

and

c(x+a_{\rm lat},t)

. ((b)) Expand that rule once to first order in

\tau

and once to second order in

a_{\rm lat}

, exactly as the unbiased rule was expanded, and show that it becomes

\partial c/\partial t = -v\,\partial c/\partial x + D\,\partial^2c/\partial x^2

. Give

v

and

D

in terms of

q

a_{\rm lat}

and

\tau

. ((c)) Evaluate

v

and

D

for

q=0.52

with

a_{\rm lat}=1

and

\tau=1

, and say what the biased cloud does that the unbiased one did not.

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