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In my textbook, the definition of temperature begins by determining the maximum number of microstates for two systems in thermal equilibrium of energies $E_1$ and $E_2$ and microstates $\Omega_1(E_1)$ and $\Omega_2(E_2)$ with respect to a change in $E_1$. It does this by differentiation: $\frac{d}{dE} (\Omega_1(E_1) \Omega_2(E_2)) = 0$. This apparently assumes that there are no local minima or maxima which this could be identifying. Does this derive from the Ergodic principle?

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