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 H2O 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=10-6cm, and the collision time is =10-14sec. 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 Ne electrons, where Ne ~ 1023 . 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.