Statistical definition of temperature

[WrongMan]

hello everyone. i need help understanding this statement:
d(lnΩ)/dE = 1/k_bT
so Ω are the posible microstates for energy E, and the derivative of Ω w.r.t E is 1/k_bT.
why?
what i understand so far is: looking at the division of energy of two "connected" systems the energy will divide itself in a way that maximizes the total possible microstates, and since the total number of microstates is: Ω₁(E₁)*Ω₂(E₂)
(where "Ω₁(E₁)" means posible microstates at E1 for system 1) this would mean:
E_t=E₁+E₂ and
Ω_t(E_t)=Ω₁(E₁)*Ω₂(E₂)
and since i want to maximize this i say:
dΩ_t/dE_t=0
so now what? i feel that I am close but i can't get there

 

[Ssnow]

Hi, I want remember you that $k_{b}\ln{\Omega}$ is a well know quantity called Entropy...

Ssnow

 

[Khashishi]

Energy wants to move from a higher temperature to a lower temperature, until the temperatures are equal.
The statistical mechanics definition of temperature tells you that by moving energy from a higher temperature to a lower temperature, the total number of microstates is increased.

 

[WrongMan]

i understand the theory part of it, i just don't understand the calculus part of it, where does the 1/kbT comes from? what is meant by "d(lnΩ)/dE" is the rate at which lnΩ chages w.r.t E right? but i was never given an expression that relates mictostates with energy, i just sort of "count" ( or calculate using combinatory "formulas").

 

[Khashishi]

Ok, how do you take the derivative of a discrete quantity? Well, you can't. The statistical mechanics definition of temperature is really only valid for large systems where you can smooth over the quantum steps. You need to imagine a system with a whole bunch of closely spaced states. Then you imagine a small region of energy which overlaps many states. You have to calculate a density of states, which is a function that tells you how many states are in a small region of energy around any particular energy. Your small region has to be much bigger than the distance between adjacent states, or it fails. I'm not sure if there's a more mathematically rigorous way to define it.