Temperature and resistance? Why does resistivity increase with temperature?
The explanations I have heard so far are that increasing temperature increases vibrations in the lattice structure resulting in the number of collisions increasing.
But the vibrations from thermal energy are random, so around half of the vibrations would result in the atoms moving out of the electrons way and half would result in atoms moving into the way of the electrons.
And since half of the vibrations result in increasing the drift speed and half reduce it, shouldn't the average drift speed remain the same?
 A: The increase in resistance in metals is mainly due to the increasing velocity of the thermic electron motion (I think $v_{\rm therm}\sim \sqrt{kT}$). This shortens the time $\tau$ of free motion of the electron. If $l$ is the mean free path length we have $\tau=\frac{l}{v_{\rm therm}}\sim \frac{1}{\sqrt{kT}}$. If the electron bumps into the next atom/ion it looses all its directed velocity $v_{\rm drift}$. That means it has only the time $\tau$ to accelerate into the field-direction and we have a mean drift velocity $v_{\rm drift} = -\frac{eE}{2m_e}\tau\sim\frac1{\sqrt{kT}}$. This is surely simplified. The change in the mean free path length also must be taken into account.
See also Section 2.1.3 "Temperature Dependence of Conductivity" in http://www.newagepublishers.com/samplechapter/002014.pdf.
But, also see http://en.wikipedia.org/wiki/Classical_and_quantum_conductivity#Flaws_in_classical_conductivity.
A: Let us begin by taking a real life example.
You are in your room and at the door, there is a large block that blocks almost the entire way( the space between the ions is very small ). Say it is motionless. Now, you would not find it hard to pass the block. All you have to do is to squeeze a bit to get through.

Now, let us say that the block is vibrating. It will be difficult for you to get through. You've got to go extremely fast in order to have a chance to get through. Remember, electrons drift very slowly, slower than you could imagine. They travel approximately 1 meter every hour.

You might say that as a result of thermal agitation, the electrons gain velocity and so the gain and loss balance each other. The question is, in what direction does it gain speed? Thermal agitation results in randomness. So, it is not right to say that the electrons gain speed in a particular direction. Electrons can gain speed in any random direction due to thermal agitation. There is a chance for an electron to gain speed in a particular direction but, there is also a chance for another electron to slow down( equal amount ) in that direction. So, the average velocity as a result of thermal agitation remains zero. Furthermore, the electric field created by the voltage source stabilizes everything and the electrons drift in a particular direction with an average velocity of 1 m/hr( approx ). 
In summary, the number of collisions increases as the temperature increases. This is due to the lattice vibrations. The electric field tends to set the electrons in drift motion against thermal agitation. The vibrational kinetic energy of the ion increases which results in a greater number of collisions.
A: Yes, there will be increase in velocity of free electrons, when temperature increases. But the electrons will not be accelerated in a particular direction.   
Consider a conductor, when potential difference is applied across the two ends of the conductor, an electric field is set up. Under the effect of electric field, the free electrons accelerate and acquire a velocity component in a direction opposite to the direction of electric field in addition to their thermal velocity. Due to electric field, electrons gain drift velocity in a  direction opposite to it. If you increase the temperature of the conductor now, the electrons velocity increases in the direction other than due to electric field. Of course there will be some electrons which will be accelerated in the direction opposite to that of electric field, but there will be greater number of electrons which will be accelerated in other directions. As a consequence, current should decrease.
A: I think you have some of the right intuition, but the orders of magnitude of the problem need to be considered.  Look at Electricity and Magnetism by Purcell.  http://www.amazon.com/Electricity-Magnetism-Vol-II-Berkeley/dp/0070049084 They have a chapter on these concepts.
The idea is that conductivity in a metal is primarily the motion of free electrons.  However, remember they are small (scattering cross-section is small and therefore they don't hit each other often) and their mean free path can be quite long.
So, as the temperature increases, the kinetic energy increases and the velocity of electrons increases.  This reduces the time between collisions (not necessarily the path length or mean free path).  Therefore, it is less likely an electron will flow from the high electrical potential to the low electrical potential, increasing the electrical resistance.
