Chemical potential from entropy

In microcanonical statistical mechanics, a macrostate is characterized by $(U,V,N)$, and its entropy is typically the microcanonical (Boltzmann) entropy $S(U,V,N)$ built from the density of states (or the phase-space volume of the energy shell ).

Definition (chemical potential from the entropy).
Given $S(U,V,N)$, define temperature via the energy derivative of the entropy and define the chemical potential $\mu$ as the conjugate to particle number:

Equivalently, using $\beta=1/(k_B T)$ from inverse temperature ,

Key identity and interpretation.
Together with the volume derivative, this is summarized by

so $-\mu/T$ is the marginal entropy change when adding particles at fixed $U$ and $V$. Physically, $\mu$ is the intensive “cost” (in energy units) that controls particle exchange: if two subsystems can exchange particles while keeping total $U$ and $V$ fixed, maximizing total entropy forces equality of $\mu/T$ between them, and (when they also share energy) equality of $\mu$ as the familiar chemical potential equilibrium condition.

Connection to the grand canonical ensemble.
When $N$ fluctuates due to contact with a particle reservoir, the equilibrium weighting is governed by $\mu$ in the grand canonical ensemble . This construction is reflected in the factor $\exp(\beta\mu N)$ that enters the grand canonical partition function construction , and it is the Legendre-conjugate variable used when passing from $F(T,V,N)$ to the grand potential via a Legendre transform .