# Thermodynamic temperature and the Micro-canonical ensemble

We can identify $k \beta = \left(\frac{\partial S}{ \partial E}\right)_V$ with the thermodynamic $k \beta = \frac{1}{T}$.

This seems to be, at least to me, very natural in the context of the canonical ensemble in which our system is in contact with a heat bath.

My question is as follows: suppose we are working in the micro-canonical ensemble and we manage to obtain the entropy as a function of the energy. Is this identification still meaningful in a physical sense? We can always define $k \beta = \left(\frac{\partial S}{ \partial E}\right)_V$ but will it be nicely related to the thermodynamic temperature in the micro-canonical ensemble as well? Where we obtain the entropy $S$ according to the formula $S=k \ln[\Omega]$. $\Omega$ is the total number of microstates.

Is the temperature well defined if our system is not in contact with a heat bath? Or does thermodynamics, in a certain sense, assume that our system is in contact with a heat bath?

• Thanks for your reply. I know, but does that mean that it does not have a well defined temperature? If we take the derivative$\left(\frac{\partial S}{ \partial E}\right)_V$ calculated in the microcanonical ensemble is it not identifiable in any way with the thermodynamic temperature? – Ben Stokes Dec 6 '16 at 11:15