Are entropy and temperature inverse to each other in microcanonical ensemble?

Question

I have a fundamental question about microcanonical ensemble. Its definition is: the ensemble in an isolated system with fixed $N$ and $E$. However, when we calculate $T$, we use:

$$\frac{\partial S}{\partial E} = \frac{1}{T} \quad . $$

How come we can take partial derivative $S$ respect to $E$ even though we set $E$ fixed?

Also, this equation seems like temperature and entropy are inverse to each other. However, I thought in high temperature, entropy is also high.

I am confused about these concepts. Thanks in advance.

Answer 1

There is no contradiction between the definition of the microcanonical ensemble as a fixed energy ensemble and the definition of temperature through $$ \frac{1}{T}=\frac{\partial S}{\partial E}.\tag{1} $$ Indeed, for each energy, we have a different entropy of the system, and we can use the dependence of the entropy from energy to evaluate the partial derivative.

The relationship between entropy and temperature cannot come from equation ($1$) only. It can be derived once we add the request (valid for usual laboratory systems) that entropy is a strictly monotonically increasing function of the energy.