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PHYS 445 Lecture 2  Random walks  II
2  1
© 2001 by David Boal, Simon Fraser University.
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Lecture 2  Random walks  II
What's Important:
•
expectations
•
tiptotail distributions
Text
: Reif
"Tiptotail" displacement
In the first lecture, we determined the probability
W
N
(
n
R
,
n
L
) for a random walk with
N
steps to have
n
R
steps to the right and
n
L
steps to the left:
W
N
(
n
R
,
n
L
)
=
N
!
n
R
!
n
L
!
p
n
R
q
n
L
.
(2.1)
In this lecture, we show how to use this distribution to evaluate observables such as
ensemble averages.
In the random walk, the "tiptotail" distance, or equivalently the
displacement of the walker, is one observable of interest.
For a walk with equal length
steps
a
, the displacement
x
is
x
=
ma
= (
n
R

n
L
)
a
where
m
=
n
R

n
L
= (
n
R
 [
N

n
R
] )
a
= (2
n
R

N
)
a
.
(2.2)
Eq. (2.2) demonstrates that
m
changes by 2 units as a function of
n
R
, and can range
from 
N
to +
N
.
Quick review of mean values
The function
W
N
(
n
R
) tells us the probability that, in an ensemble of walks, there are
walks with
n
R
steps to the right.
From this distribution, we can extract quantities such
as the average number of steps to the right, or to the left
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This note was uploaded on 09/07/2009 for the course PHYS 445 taught by Professor Davidboal during the Spring '08 term at Simon Fraser.
 Spring '08
 DavidBoal
 Physics

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