= Zero chemical potential from number-changing reactions
{title2=$\mu_\phi=0$}
For one particle species in <chemical equilibrium> under $2\phi\leftrightarrow3\phi$, chemical balance gives $2\mu_\phi=3\mu_\phi$, hence $\mu_\phi=0$. Elastic $2\phi\leftrightarrow2\phi$ 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>.
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