Pressure from entropy

In the microcanonical viewpoint, a macrostate is specified by conserved quantities such as energy $U$, volume $V$, and particle number $N$. Its entropy is often taken to be the microcanonical (Boltzmann) entropy defined from the phase-space volume of the energy shell .

Definition (pressure from the entropy).
Given an entropy function $S(U,V,N)$, define the temperature via the entropy derivative with respect to energy and then define the pressure by the conjugate entropy derivative with respect to volume:

Equivalently,

In terms of the inverse temperature $\beta=1/(k_B T)$ (with Boltzmann’s constant $k_B$),

Key identity and interpretation.
These definitions are encoded in the fundamental differential form

so $p/T$ measures how rapidly the number of accessible microstates grows under an infinitesimal expansion at fixed $U$ and $N$. In equilibrium, this pressure matches the mechanical pressure that balances an external constraint; it is the coefficient of the reversible $p\,dV$ work term in the first law written in entropy variables.

Connection to free energy and partition functions.
After constructing the Helmholtz free energy by Legendre transforming the entropy , the same pressure can be obtained at fixed $T$ from

which matches the canonical expression described in pressure from the partition function .