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Adding $d\varphi$ to a free-energy density adds a constant at fixed total compositional order parameter. In the common-tangent construction for phase coexistence, it shifts the tangent slope by $d$ while preserving its points of contact. Thus coexistence compositions are unchanged, whereas their chemical potential shifts. This does not assert unchanged equilibrium composition at a fixed externally imposed chemical potential.

At fixed temperature, choose a coexistence pressure $p_{\ast }$ such that the gas and liquid have equal chemical potentials. Because $d\mu =vdp$, this gives ${\int }_{v_l}^{v_g}vdp=0$, or equivalently ${\int }_{v_l}^{v_g}[p(v,T)-p_{\ast }]dv=0$. The equal areas replace the unstable part of an isotherm by a horizontal coexistence segment.

For a dilute nonrelativistic gas with $x=m/T\gg 1$ and zero chemical potential, the Bose-Einstein distribution approaches the Maxwell-Boltzmann distribution. Expanding $E=m+p^2/(2m)+\cdots$ in the number density integral gives $n\sim g(mT/(2\pi ){)}^{3/2}{e}^{-x}$. Since $\rho =mn+3nT/2+\cdots$ and $P=nT$, the entropy density at zero chemical potential gives $s=n(x+5/2+\cdots )$ and hence the displayed leading expression. This includes the internal multiplicity $g$; a single real scalar field has $g=1$.

With negligible chemical potential, the Fermi-Dirac distribution has $3/4$ of the boson number density per degree of freedom at equal temperature. A species with $g_{\nu }$ populated states therefore has $n_{\nu }/n_{\gamma }=(3g_{\nu }/8)(T_{\nu }/T_{\gamma }{)}^3$, since photons have two states. This remains the number-density ratio after the neutrino becomes massive if its comoving particle number is conserved.

The positive order-parameter mobility $M$ relates the diffusive transport flux to the chemical potential gradient: $\mathbf{J}=-M\nabla \mu$. Combining this law with local conservation gives the Cahn--Hilliard equation when $\mu$ is the functional derivative of a composition free energy.

In Ostwald ripening, the higher interfacial chemical potential of small droplets drives diffusion toward larger droplets, increasing the characteristic domain size. The Gibbs--Thomson relation supplies the curvature-dependent surface value, while conserved order-parameter dynamics controls transport through the surrounding phase.