Ch1b rubber band
1
Behavior of one dimensional random walk  calculation of average endtoend distance
Assume each unit goes to the right or left with equal probability (1/2)  independent of what
other units have done, or do. The step size length in either direction is
l
.
Start at the origin. Let
x
i
(n)
be position of the n
th
unit in the i
th
molecule. Then:
x
i
(
n
)
=
x
i
(
n
"
1)
±
l
The “+” step direction applies to ~half of the units, and the ““ applies to the other ~half.
The mean displacement of the polymer is given by:
x
(
n
)
=
1
M
x
i
i
=
1
M
"
(
n
)
=
1
M
x
i
i
=
1
M
"
(
n
#
1)
±
l
=
1
M
x
i
i
=
1
M
"
(
n
#
1)
±
1
M
l
i
=
1
M
"
last summation goes to 0 since equal numbers of + and 
=
x
(
n
#
1)
on average (averaged over many molecules), the position of the n
th
unit is the same as the
position of the n1
th
unit, .
..., is the same as the position of the first unit  ie, the average position
of all units is where the molecule is centered (reflects equal numbers of right and left moves).
OK  this is just like the average velocity of gas molecules along an axis is 0  what about
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 Winter '08
 Lewis
 Probability, Probability theory, Endtoend

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