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

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?

• Comments are not for extended discussion; this conversation has been moved to chat. Dec 29, 2015 at 12:25
• The definition of temperature that comes from Boltzmann equation does not depend on this kind of problems, but it is also thought in other context. Feb 3, 2016 at 11:33
• You basically count states using factorials, and, for such large numbers, the Stirling's approximation works extremely nicely. That approximation is continuous. Oct 29, 2017 at 22:46