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If two otherwise isolated systems freely exchange energy while $E_1+E_2$ remains fixed, their total entropy isAt an interior maximum, $\partial S_1/\partial E_1=\partial S_2/\partial E_2$. Since thermodynamic temperature satisfies $1/T=\partial S/\partial E$, the systems have equal temperatures at thermal equilibrium.

In the microcanonical ensemble, the microcanonical energy-shell entropy is the Boltzmann entropyFor a macroscopic system, changing the small energy resolution by a fixed factor changes $\log \Omega$ only by a nonextensive additive term, while $S=O(N{k}_B)$. Its relative effect therefore vanishes in the thermodynamic limit.

The thermodynamic temperature, pressure, and chemical potential are defined in the entropy representation byThusor, equivalently, the first law of thermodynamics in fundamental form is

The Carnot theorem states that no engine between two fixed heat reservoirs can be more efficient than a reversible engine, and that all reversible engines between those reservoirs have the same efficiency. Thus the heat ratio of a reversible engine depends only on the two reservoirs. Its multiplicative consistency for three reservoirs permits a state variable $T$, unique up to an overall scale, such thatFixing one reference value defines the thermodynamic temperature scale, and the reversible efficiency is