1) [30 pts total] The following questions are based
on the concept of the biased random walk, and are meant to have you think
about the following question: We learned that the electrons that make up
the electric current traveling down a wire (or equivalently the H_{2}O
molecules in a pipe in which water is flowing) should not be thought of
as moving uniformly down the length of the wire (or pipe) -- rather they
are taking a biased random walk: they undergo seemingly random motion in
all directions, and it is only on averaging over many collisions does one
see that there is in fact a net average motion in one direction over the
other. This means that, following the path of a single electron, there
is always some probability that after a given time t has elapsed, the electron
has actually moved in the opposite directions as the average current is
traveling. Moreover, if this is true of a single electron, there similarly
is some finite probability that all the electrons, due to some random fluctuation,
might have moved a net distance opposite to that in which the average current
flows. If this is true, one would find that the instantaneous current,
over the time t, has flowed in a direction opposite to its average behavior.
The present question tries to have you think about how likely this is,
or put differently, how short must t be if this is to have any significant
probability to happen.

Consider electrons moving in a conducting wire. The
average distance between collisions (also called the "mean free path")
for the electrons is L=10^{-6}cm, and the
collision time is =10^{-14}sec.
The electrons are moving and colliding with the ions in the wire, and so
each electron can be viewed as if it is taking a random walk with step
length L and time
per step. Suppose a voltage is applied, so as to drive an electric current
down the length of the wire. Assume that as a result of this voltage each
electron now has a probability p=0.50005 to move to the right, and probability
1-p=0.49995 to move to the left, following each collision.

a) [5 pts] Find the mean distance that an electron has traveled after N collisions.

b) [5 pts] Find the standard deviation of the probability distribution for the distance traveled by an electron after N collisions.

c) [5 pts] How many collisions M are needed such that ? M is the number of collisions needed for the average displacement of the electron to equal the standard deviation of the displacement about the average. For N<<M, how does compare to ? For N>>M, how does compare to ? Give some physical discussion of what these results mean in terms of the likelihood to find the electron to the left or to the right of its starting position.

d) [5 pts] If M is the number of collisions in part
c, find the probability that after M collisions, the electron will be found
at some position to the *left* of its starting position (i.e. the
electron has moved a net amount in the direction *opposite* to the
motion of the average position).

e) [5 pts] Suppose the wire contains a total of N_{e}
electrons, where N_{e}
~ 10^{23} . What is the probability that after
M collisions, exactly half have moved a net amount to the left, while half
have moved a net amount to the right? You just have to give the correct
mathematical expression - you don't have to find the numerical value, but
if you can, go ahead!

f) [5 pts] Suppose that after M collisions, electron
number "i" has moved a distance .
Define the net distance moved by all the electrons as .
What is the probability that X<0, i.e. that there has been a net motion
of the electrons to the left? Give a detailed arguement. Such an occurance
would correspond to a net electric current flowing *opposite* to the
direction which the voltage is pushing in (i.e. opposite the direction
of the electrons average motion) - it is an extemely unlikely statistical
fluctuation, as your calculation should show.