How is temperature and pressure related to when a tire rolls on the ground? Actual observation: Here in Canada it is known that even when 'dry' snow is on the ground, and when a car drives over it that small amount of dry snow instantly becomes ice (NOT GOOD). Anyways as an inspiring physicist I am determined to understand why that is. I know that Pressure and Temperature have a direct correlation, and it makes logical sense to me that has the car drives over a cross-sectional area of 'dry' snow the increased pressure should cause the temperature to increase, thus melting the snow. Then with the extreme external temperature (~ -20C) that melted snow instantly freezes again and thus makes ice, and I have noticed that the amount of ice on roads has a direct relationship to drivers ability of perceived car-handling skills, he. 
Anyways, I am trying to create a very simplistic model for such scenerio in order to prove to myself that the temperature does indeed increase, and that is the reason for my - well - reasoning. The problem is, is that all laws that relate temperature and pressure is found in gas laws and dealing with fluids. 
I was wondering if using the Ideal Gas Law would hold true in this scenerio consdering the cross-section at question is indeed a volume of (let's assume) pure $H_2O$ and then can be calculated the number of moles etc.. and showing the temperature of the cross section after pressure is applied. 
Does my reasoning make sense, and is that in fact the reason why seemly dry snow turns to ice immediately after a car drives over it?
 A: Don't try using gas laws on ice crystals - it wasn't meant for that, and it doesn't work.
Instead, think about making a snowball. As you try to squish down the ice, you know it becomes more sticky, but if you start with truly "dry" snow, you can't get a good snowball. The reason is that it take a lot of energy to melt even a little bit of snow temporarily - long enough to make water which then re-freezes and makes the snow stick together and become ice.
Now think about the snow below a tire. If the snow starts out fluffy, it will be compacted by the car. As it becomes compact, the force of friction between the crystals will cause local heating. Because your car produces a lot of pressure, there is enough friction in the snow to cause some real heating and melting.
Whether this happens or not will depend on the composition of the snow - and on the shape of the tire and the weight of the car. But the fact that you observe it tells me that there is a combination of conditions in which the snow does melt - and the only way to supply the energy needed for this is through the friction / compression of the snow.
Note that the tires themselves also heat up a little bit - but they will heat very little on snow, because the major heating mechanism in a rolling tire is due to the mismatch in area between the tire "in the air" and the tire "touching the ground" (see for example this answer). When the road surface is slippery, that mechanism is less effective; instead, the friction that the tire experiences is precisely due to the fact that the snow is being compressed: in effect, the tire is always "rolling uphill", and it has to do work. Yet the center of mass never goes higher - instead, the snow is being compressed. And that is the source of the work, and the heating of the snow. Which will cause "sintering" under the right conditions (no, I'm not sure that you can use that word with snow, but it seems appropriate to describe the phenomenon). You can estimate the amount of work done against the snow by seeing how much harder it is to drive in soft snow than on a flat surface - the difference is the work done to compress snow.
I don't have good numbers for that - but I do know that when I was driving a 4x4 on a beach (soft sand), it was getting very poor gas mileage: on the order of 5 mpg for a car that would normally get around 20. This means that the engine had to work almost 4x harder than usual, close to its peak power. If we say that it was using 75 hp just to push through the soft stuff, at a velocity of 20 mph, then we get the work done per meter covered as about 6300 J; assuming that it's the front tires doing the compressing, with a width of 25 cm each we compressed an area of 0.5 m$^2$ meaning that there is about 12000 J / m$^2$ or 1.2 J / cm$^2$ available.
The latent heat of fusion of ice is about 330 J/g, so all that power, distributed like that, could melt only a few mg of snow per square cm (or 1 gram per meter for each of the two tire tracks). That doesn't seem enough to explain what you are observing: there just isn't enough power in an average car to melt the snow it drives over. 
A: From the Maxwell's relations you can easily see that each of these respective variables are held constant in relation to a free energy.
I think from a classical point of view considering the model of kinetic energy and temperature, along with the definition of pressure, you could conclude that the increase in temperature is from a constant pressure of the tires, providing a consistent level of pressure on the ground, in the form of kinetic and static friction.
This static friction over the journey of a vehicle would create the increase in temperature of the tires.
This is represented by the equation:
$ F_{s} = \mu_{s}F  $
where typically in this instance the force would be equal to the gravitation force of the car:
$ F = ma$, where $ a = g = 9.81 m s^{-2} $
