Force & Temperature Basics
I have a rigid rod suspended (tied to and hanging from a horizontal beam) with a weight pulling it down. Is there tension on that rod? If there is no dx, how can F = dE/dx define that tension?
The correct way to think of the force is that it is the
gradient of a potential. So when you see the form F = dE/dx the "E" must always be a potential energy, not a kinetic energy. When you say "If there is no dx ..." the implication is that you want dx to represent a motion, something moving over a distance
dx, but that is a conceptual mistake. Nothing is moving (just because a force is applied does not require anything to move, just push your car with the brake set and see what happens). In this case
dx is just an incremental length of string and
dE is the increment of the total energy stored in the string, that is stored in that particular
dx of string, or rod. Pull on the rod with a stronger force, and any
dx increment of the rod will carry a larger
dE increment of internal, potential energy. Pull hard enough to break the rod (or string) and the potential energy will become kinetic.
"Work" is defined as a force applied over a distance (
force times distance in common parlance). Now something has to move. Push on your car with the brake set until you collapse in exhaustion, but you have still done no work because the car has moved zero distance.
In the case of Newtonian mechanics, just note that
momentum ("
P") is mass times velocity ("
mv") and force is mass times acceleration ("
ma"). But
ma =
m dv/dt so the force is the derivative of the momentum, or
F = dP/dt. This is the common way to describe force in Newtonian mechanics, but you have to remember that the concept of a field, and therefore the gradient of a potential, did not yet exist when Newton came up with his laws (we owe the concept of fields primarily to James Clerk Maxwell). So in old style Newtonian mechanics the force is defined in terms of its effect on moving bodies, as the first derivative of the momentum. But in a more modern formalism that includes particles and fields, we can define the force independently of its effect on things, and define it instead in terms of what it is intrinsically or where it comes from, and that is the gradient of a potential.
As for the temperature, the correct full form of the equation is
dE = TdS - pdV where E is internal energy, T is temperature, S is entropy, p is pressure and V is volume. So the form T = dE/dS is valid only if the volume is held constant, so that dV = 0 (there are a few similar looking definitions using enthalpy, Helmholtz functions, Gibbs functions, and maybe more I am unaware of). Basically, the equation tells you that for the given increment in energy, a small increment in entropy represents a larger temperature, while a larger increment in entropy represents a smaller temperature (since the volume is constant the system can do no work expanding itself, so all of the energy has to become internal energy).
It's easier to understand by going back to the original definition of thermodynamic
beta as the inverse temperature (beta =
1/kT =
d/dE (log[N]) where
N is the number of microstates, or the number of ways that one can arrange the quanta of energy in the system,
E is the energy,
T is the temperature and
k is Boltzmann's constant (a number you can look up)). Now you can see the same thing, the rate of change as a function of energy, of the ways one can arrange the quanta of energy internal to the system.
N is just a number, the count of available microstates, but it is a function of energy. Now suppose the system "saturates", so that even as more energy is pumped in, there are no more changes possible in microstates. The rate of change then becomes zero, beta becomes zero, and the temperature necessarily becomes infinite.
Now we all know that if you stick a mercury thermometer into what ever it is, the mercury column will not streak to infinity, and if we stick a digital thermometer into what ever it is, we won't see a sideways 8. That's what some people are thinking when they insist that the temperature cannot be infinite, and to that extent they are correct. But of course, those instruments don't measure temperature anyway, they measure something else and that is in turn interpreted as a temperature, over some limited and defined set of conditions. Temperature, like anything else in real physics, is whatever the equations define it to be. As you can see from the example above, the temperature becomes infinite, but there is nothing even close to "physically infinite" in the system.
Maybe that helps clarify things a bit? It's 1:45 AM here, so I might have slipped somewhere.
