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For a binary fluid mixture, the compositional order parameter $\varphi (\mathbf{r})$ may be taken as the local concentration difference between the two species. In a closed systemis fixed by the total amount of each species. This is the sense in which $\varphi$ is conserved; locally it changes through a current and obeys a continuity equation.

For a symmetric mixture, the Landau-Ginzburg theory free energy isA term $h{\int }_V\varphi ,d\mathbf{r}$ is constant on the allowed configurations and therefore changes neither equilibrium probabilities nor the dynamics. Equivalently, it merely shifts the chemical potential by a spatial constant, whose gradient vanishes.

The two binodal compositions ${\varphi }_1<0<{\varphi }_2$ are the contact points of the common-tangent construction. Equivalently, they solveThe first equality is equality of chemical potential; the second is equality of pressure. Heretogether with the intercept equation determines the two densities. As $C\downarrow a$, both contact points approach zero continuously with scale $|{\varphi }_{1,2}|\propto \sqrt{C-a}$, so this mean-field onset of phase separation is continuous.

Chemical equilibrium in $n+{\nu }_e\leftrightarrow p+e^{-}$ givesThe negligible lepton chemical potentials therefore imply ${\mu }_n\simeq {\mu }_p$. Nonrelativistic Maxwell--Boltzmann number densities satisfyNeutrons and protons have equal spin degeneracy and nearly equal masses, so their prefactor ratio is approximately one. Hence

A phonon is a bosonic quantum of a crystal's normal modes. Because phonon number is not conserved in thermal equilibrium, its chemical potential is zero.

Chemical equilibrium of $n+{\nu }_e\leftrightarrow p+e^{-}$, with negligible electron chemical potential, gives the stated neutron-to-proton ratio. Positive electron-neutrino degeneracy reduces the neutron fraction, whereas negative degeneracy increases it. The primordial helium mass fraction then responds through neutron-limited helium synthesis while protons outnumber neutrons; a sufficiently neutron-rich mixture instead becomes proton limited, so that trend must not be extrapolated indefinitely. Extra neutrino energy density also accelerates cosmic expansion, changing weak freeze-out and the time available for neutron decay. These effects can compete; a quantitative Big Bang nucleosynthesis calculation must account for the altered reaction rates as well as the altered expansion rate.

The standard conditions for baryogenesis from a symmetric initial ensemble are baryon number violation, both C violation and CP violation, and departure from thermal equilibrium. Conserved baryon number cannot develop a net value from zero; an exact conjugation symmetry pairs production with equal antiproduction. In thermal equilibrium, CPT symmetry makes the equilibrium expectation of net baryon number vanish for zero imposed chemical potentials for charges odd under CPT symmetry. These conditions assume the usual local quantum-field framework and do not alone guarantee a large asymmetry.

For one particle species in chemical equilibrium under $2\varphi \leftrightarrow 3\varphi$, chemical balance gives $2{\mu }_{\varphi }=3{\mu }_{\varphi }$, hence ${\mu }_{\varphi }=0$. Elastic $2\varphi \leftrightarrow 2\varphi$ imposes no such constraint. In an expanding universe the number-changing reaction must remain fast compared with the Hubble parameter; after chemical freeze-out a nonzero chemical potential can develop even if elastic collisions maintain kinetic equilibrium.