Why is the logarithm of the number of all possible states of a system differentiable?

Question

Temperature of a system is defined as $$\left( \frac{\partial \ln(\Omega)}{ \partial E} \right)_{N, X_i} = \frac{1}{kT}$$ Where $\Omega$ is the number of all accessible states (ways) for the system. $ \Omega $ can only take discrete values. What does this mean from a mathematical perspective? Many people say we have $10^{23}$ particles so $\Omega$ is almost continuous function of energy. Why is $10^{23}$ a nice number but $1000$ is not? When can one be sure they can differentiate $\ln(\Omega)$?

If you agree with me, do you know an alternative accurate definition for temperature?

Answer 1

From a mathematical perspective this means that it is not differentiable. The problem is that you need the discreteness to be able to count states. If you replace the discreteness by something smooth you get something differentiable, but your definition of entropy no longer makes sense. This is just one of the points where mathematicians cringe, but it works perfectly fine in practice for the amount of particles dealt with in statistical physics.

Of course if you look at systems where few particles are involved this definition would not work. This is due to the fact that temperature is a macroscopic variable (as is for example pressure). You just don't know what the temperature or pressure of a single particle are.

Answer 2

Mathematically speaking, you don't have to differentiate. You can replace the partial with the difference operator and take the discrete derivative (provided that you know the exact energy states). You will get the right answer until the point where the concepts of entropy and temperature start to loose their meanings, namely around a few states. In that sense, even 1000 is a nice number, but 5 is not! Thermodynamics is a statistical theory. You can do statistics with 1000 "samples", you cannot with 5.