What is the Meaning of Temperature at Equilibrium?

[arcteus]

Heya!

I've done my statistical mechanics course, but still. Whenever I see the temperature, a question pops up in my head! So I'm curious to see if someone can help me with this obsession!

Temperature is defined as

T = (dS/dE)^-1

S: amount of accessible microstates
E: energy

1) temperature is defined at equilibrium. this should mean approximately constant energy. yet this definition implies a variation of energy. O,O

-> is the ghost of Heisenberg behind this (energy defined always as part of an interval)?

2) at equilibrium, I expect S to be at its maximum. so dS/dE should be equal to 0! and 1/0, well...

I'm very curious to see ideas about this!
Thanks for any ideas!

 

[nnnm4]

Well first of all this is the definition in the microcanonical ensemble. Because of this energy for the ensemble is constant. So when you speak dS/dE, what this means is how much the equilibrium entropy will change by changing the ensemble energy by dE.

In this ensemble the entropy is actually a slowly increasing function of energy, so it doesn't reach a maximum, but this isn't an issue because you contrict the available equilibrium states to those of a particular energy defined by the ensemble.