= Solution
<heating and cooling balance in the interstellar medium> depends on <density>, <ionization>, radiation and shielding. Write net cooling per volume as $\mathcal C=n^2\Lambda(T)-n\Gamma$. Ultraviolet <grain photoelectric heating> is important in diffuse neutral gas; <photoionization> heats ionized regions, <cosmic rays> penetrate shielded gas, and <shock waves> from <stellar winds> and <supernovae> supply hot gas. Cooling includes collisionally excited atomic lines, fine-structure metal lines at lower <temperatures>, molecular lines in shielded clouds, and continuum radiation and dust emission where appropriate. The heating efficiency and cooling abundances matter as much as a nominal <temperature>.
A <multiphase interstellar medium> can contain a <cold neutral medium> near $10^2\ \mathrm K$, a <warm neutral medium> and <warm ionized medium> near $10^4\ \mathrm K$, a <hot ionized medium> near $10^6\ \mathrm K$, and still colder <molecular clouds>. Cold and warm atomic equilibrium branches can coexist at similar thermal <pressure> over a finite <pressure> interval; this is a calculated heating/cooling balance, not a requirement that all galactic gas have the same <pressure>.<The Neutral Atomic Phases of the ISM in the Galaxy by Mark G. Wolfire and others>[Abstract: cold and warm neutral equilibrium temperatures and coexistence between minimum and maximum pressures]{ref}
For net cooling per unit <mass> $\mathcal L=\mathcal C/\rho$, an isobaric perturbation at equilibrium obeys $c_p\partial_t\delta T=-(\partial\mathcal L/\partial T)_p\delta T$. Thus \b[a negative constant-pressure cooling <derivative> is thermally unstable]: a cool perturbation cools further and condenses. Sound crossing must be fast enough to maintain <pressure> balance; <thermal conduction> suppresses sufficiently short <wavelengths>. Supernova-driven motions, <magnetic pressure>, finite cooling times and interfaces make the actual medium dynamic, so equilibrium branches describe useful phases rather than a static collection of perfectly separated components.
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