This preview shows pages 1–3. Sign up to view the full content.
PHYS 445 Lecture 3  Random walks at large
N
3  1
© 2001 by David Boal, Simon Fraser University.
All rights reserved; further resale or copying is strictly prohibited.
Lecture 3  Random walks at large
N
What's Important:
•
random walks in the continuum limit
Text
: Reif
1D Random walks at large
N
As the number of steps
N
in a walk becomes very large, the individual step sizes in
n
R
become relatively small.
Further, the difference in values of
W
(
n
R
) between successive
values of
n
R
becomes small as well.
The proof of how the difference scales with
N
is
given in Reif.
For our purposes, the important point is that we can regard
n
R
and
W
(
n
R
)
as continuous at large
N
.
To obtain a continuous description of
W
(
n
R
), consider its form near its most likely value
at
˜
n
R
˜
n
R
At the peak of the distribution, its derivative with respect to
˜
n
R
vanishes
dW
dn
R
˜
n
R
=
0
or
d
ln
W
dn
R
˜
n
R
=
0
Let's expand
W
(
n
R
) or its logarithm around
˜
n
R
using
as an expansion parameter,
where
n
R
=
˜
n
R
+
.
Choosing the logarithm, for example, we have
ln
W
(
n
R
)
=
ln
W
(
˜
n
R
)
+
B
1
+
1
2
B
2
2
+
...
(3.1)
where
B
i
=
d
i
ln
W
dn
R
i
˜
n
R
At the maximum in
W
, the coefficient
B
1
= 0 by definition.
The coefficients
B
i
can be
obtained from the discrete expression for
W
in terms of factorials. Start with the
definition
ln
W
(
n
R
) = ln
N
!  ln
n
R
!  ln(
N

n
R
)! +
n
R
ln
p
+ (
N

n
R
)ln
q
,
so
W
(
n
R
)
n
R
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentPHYS 445 Lecture 3  Random walks at large
N
3  2
© 2001 by David Boal, Simon Fraser University.
All rights reserved; further resale or copying is strictly prohibited.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 DavidBoal
 Physics

Click to edit the document details