Temperature from entropy

In equilibrium thermodynamics, temperature is not assumed a priori; it can be defined from how entropy changes with energy. Statistical mechanics implements this using microcanonical entropy.

Microcanonical entropy as a function of energy

For an isolated system with energy $U$, volume $V$, and particle number $N$, the Boltzmann (microcanonical) entropy is

where $\Omega(U,V,N)$ is a density of states (or phase-space volume of a thin microcanonical energy shell ), as described in the entropy–density-of-states construction .

Definition of temperature and inverse temperature

The thermodynamic temperature (as in thermodynamic temperature ) is defined by

Equivalently, the inverse temperature is

This definition is meaningful when $S(U,V,N)$ is differentiable (or at least has a well-defined slope in an appropriate thermodynamic limit).

Why this is the equilibrium temperature

Consider two weakly interacting subsystems (1) and (2) that can exchange energy, with total energy fixed:

If the interaction energy is negligible, the total entropy is approximately additive:

Equilibrium corresponds to maximizing $S_{\mathrm{tot}}$ with respect to $U_1$. Differentiating and setting the derivative to zero yields

Thus the equilibrium condition is equality of the quantities $1/T$, justifying the derivative definition.

Connection to the canonical ensemble

If subsystem (2) is a very large reservoir (“heat bath”), then expanding $S_2(U_{\mathrm{tot}}-U_1)$ around $U_{\mathrm{tot}}$ gives

with $\beta$ fixed by the bath’s slope. This leads to the canonical weight $\exp(-\beta H)$ for subsystem (1), connecting directly to the canonical ensemble and to entropy-maximization construction of thermal states .

Physical interpretation and sign of temperature

• Large $\partial S/\partial U$ means entropy increases rapidly with energy, corresponding to low temperature.
• Small $\partial S/\partial U$ corresponds to high temperature.
• If $S(U)$ decreases with $U$ over some range (possible when the energy spectrum is bounded above, e.g. certain spin systems), then $\partial S/\partial U<0$ and the resulting temperature is negative; this is captured cleanly by the sign of $\beta$ in the definition above.