Thermal Instability and the Two-Phase Medium
The diffuse interstellar medium is heated — by photoelectrons kicked off dust grains, by cosmic rays, by soft X-rays — and it cools by radiating spectral lines, chiefly the C II 158 μm fine-structure line in cold gas and Lyman-α plus collisionally excited metal lines in warm gas. In steady state these balance, and the gas settles onto a curve of equilibrium temperature against density. The natural expectation is that such an equilibrium is stable: nudge the gas and it relaxes back. George Field showed in 1965 that this expectation is wrong over a wide range of interstellar conditions. There the equilibrium is thermally unstable — a small perturbation grows exponentially, carrying the gas away from equilibrium until it lands on a completely different temperature. The engine of the instability is an isobaric squeeze. A parcel is pressure-confined by its surroundings, so when compressed it stays at constant pressure; that raises its density and lowers its temperature, and if the resulting net radiative loss climbs faster than heating can compensate, the parcel cools, is squeezed harder, cools faster — a runaway. It stops only on a branch where cooling again matches heating and a further squeeze would over-heat rather than over-cool. Because the interstellar cooling function is strongly non-monotonic, the equilibrium pressure–density relation is S-shaped, and a single confining pressure can pin the gas at two stable resting points at once: a Cold Neutral Medium at T ∼ 80 K and n ∼ 30 cm⁻³, and a Warm Neutral Medium at T ∼ 8000 K and n ∼ 0.3 cm⁻³, both at the same thermal pressure P/k ∼ a few thousand K cm⁻³. This two-phase picture — read straight off 21 cm surveys of Galactic hydrogen — is the organizing idea of the neutral interstellar medium.
- Field's thermal instability arises when interstellar gas in heating–cooling equilibrium is destabilized by an isobaric perturbation, because a squeezed parcel that cools faster than it heats keeps collapsing rather than relaxing back.
- The instability criterion is (∂𝓛/∂T)_P < 0, where 𝓛 is the net loss per unit mass; this is exactly the segment of the equilibrium P–n curve where (∂P/∂n)_eq < 0.
- Because the equilibrium curve is S-shaped, at a fixed pressure the gas can sit on two stable branches — a warm diffuse phase (WNM, T ∼ 8000 K) and a cold dense phase (CNM, T ∼ 80 K) — with an unstable branch between them.
- The two phases share pressure, so warm and cold neutral hydrogen coexist in mechanical equilibrium; this is directly seen in 21 cm emission and absorption, and it fixes the diffuse-ISM pressure at P/k ∼ 10³–10⁴ K cm⁻³.
- The coexistence window is bounded: below P_min only warm gas survives, above P_max only cold gas; molecular clouds form when compression pushes the gas past P_max.
- Turbulence, magnetic fields, and finite cooling times keep a substantial fraction of neutral hydrogen on the classically forbidden unstable branch, so the modern ISM is best described as multiphase.
- Key idea: the equilibrium of heating against cooling is thermally unstable wherever an isobaric density enhancement raises the net radiative loss faster than it is compensated — equivalently where (∂P/∂n)_eq < 0 — carving the neutral ISM into two coexisting stable phases, warm-and-thin and cold-and-thick, at one common pressure.
The equilibrium of heating against cooling is thermally unstable wherever an isobaric density enhancement increases the net radiative loss faster than it is compensated — equivalently where (∂P/∂n)_eq < 0 on the equilibrium curve. This carves the neutral ISM into two coexisting stable phases at a common pressure: warm diffuse gas and cold dense gas.
Where this lives today
The two-phase model is now tested photon by photon. The Millennium Arecibo 21 cm absorption survey of Heiles and Troland (2003) found neutral hydrogen split into a cold component near 40–100 K and a warm component near several thousand K, exactly as predicted — but with a twist: close to half of the warm gas sits at thermally unstable temperatures between roughly 500 and 5000 K, far more than a static model allows. Modern surveys such as 21-SPONGE have sharpened this, and the "excess" unstable fraction is now the main handle on how vigorously turbulence stirs the medium. The same instability reappears on vastly larger scales in galaxy clusters, where cooling gas in a cluster core becomes locally unstable and "precipitates" into cold clouds once the ratio t_cool/t_ff drops below about ten — the mechanism now thought to feed the central black hole and regulate cool-core clusters through AGN feedback. And in numerical galaxy formation, resolving the Field length remains a live challenge, since a wrong cold-phase fraction propagates straight into predicted star-formation rates.
The physical picture
Put the gas on the curve where heating exactly balances cooling and nudge one parcel to be slightly denser. Because its neighbours confine it, the squeeze happens at constant pressure, so the density bump forces the temperature down. Now ask the decisive question: at the new, lower temperature and higher density, does the parcel radiate faster than it is heated? If yes, it cools; at fixed pressure cooling means still higher density, which makes it radiate even faster — a runaway that halts only when the parcel falls onto a branch where cooling again matches heating and a further squeeze would over-heat it. That landing spot is the stable cold phase. Run the same logic in reverse for a parcel nudged warmer and it climbs to the stable warm phase. The unstable equilibrium between them is a watershed: perturbations on either side grow, expelling gas toward one of the two stable phases.
The S-shape is the physics, seen in one plane. On the two outer branches the slope dP/dn is positive: squeeze a parcel and its equilibrium pressure rises above the surroundings, so it pushes back — stable. On the middle branch the slope is negative: squeeze a parcel and its equilibrium pressure drops below the surroundings, so it keeps getting compressed — unstable. A horizontal line of constant confining pressure crosses the S in up to three places, giving two stable phases and one forbidden state between them. Thermodynamically this is a stretch of negative effective heat capacity: the neutral ISM "wants" to be either warm-and-thin or cold-and-thick and shuns the middle, exactly as a fluid near a first-order phase transition segregates into two coexisting phases at equal pressure.
The core equations
Everything begins with the first law of thermodynamics for a radiating gas parcel. Let the parcel lose energy at a net rate 𝓛 per unit mass; then the change in specific internal energy minus the compressional work equals the net radiative loss.
where: 𝓔 — internal energy per unit volume; ρ — mass density; P — thermal pressure; 𝓛 — net loss function per unit mass (cooling minus heating), with 𝓛 = 0 in equilibrium; T — temperature. Reading: the left side is the rate of change of the parcel's heat content, the right side is what radiation takes away. The loss function itself is the competition between line cooling and grain photoelectric heating. Per particle the cooling rate is nΛ(T) and the heating rate is Γ; equilibrium is where they cancel.
where: Λ(T) — cooling coefficient (erg cm³ s⁻¹), rising steeply where line coolants switch on; Γ — heating rate per particle (erg s⁻¹); n — particle number density; μm_H — mean mass per particle. Reading: equilibrium fixes T_eq(n) for every density, because as n rises Λ(T_eq) = Γ/n must fall, pushing T_eq down. Field's instability criterion is the sign of the loss function's temperature derivative at constant pressure — the confinement condition a real parcel feels. This single inequality decides growth versus decay.
where: (∂/∂T)_P — derivative taken at fixed pressure. Reading: if raising the temperature lowers the net loss (so cooling falls behind heating), the gas keeps heating; if lowering it raises the loss, the gas keeps cooling — either way it departs from equilibrium. The stricter, weaker-to-satisfy isochoric version (∂𝓛/∂T)_ρ < 0 tests stability at fixed density and identifies a narrower unstable range; the isobaric one is the physically relevant condition. The two stable phases are defined by mechanical equilibrium: they touch, so they must share pressure while carrying very different densities and temperatures.
where: n_c, T_c — CNM density and temperature (∼30 cm⁻³, ∼80 K); n_w, T_w — WNM values (∼0.3 cm⁻³, ∼8000 K). Reading: equal pressure with a hundredfold density contrast means a hundredfold temperature contrast — cold clouds in warm intercloud gas. Thermal conduction sets a smallest unstable scale. Balancing conductive smoothing against radiative cooling in the perturbed energy equation gives the Field length, below which perturbations are wiped out before they can grow.
where: K — thermal conductivity; λ_F — scale below which conduction stabilizes the gas and sets the thickness of phase interfaces. Reading: structure thinner than λ_F is conductively erased, so λ_F is the resolution a simulation must reach to model the cold phase honestly. The instability grows on the cooling time — the ratio of thermal energy to radiated power — which is what makes it fast enough to matter.
where: t_cool — the timescale on which cooling (and hence the instability) develops. Reading: at fixed heating, in equilibrium nΛ = Γ, so t_cool = 3kT/2Γ scales simply with temperature. Finally, the dominant heater of both neutral phases is grain photoelectric heating, whose rate per volume sets the value of Γ used above.
where: ε — photoelectric heating efficiency (a few percent); G₀ — far-ultraviolet field in Habing units; Γ_pe — heating rate per unit volume. Reading: dividing by n gives the per-particle rate ∼10⁻²⁶ erg s⁻¹ that anchors the cooling-time estimates.
The goal is to pin down exactly which stretch of the heating–cooling equilibrium sequence is thermally unstable, by linearizing the first law of thermodynamics for a gas parcel that is squeezed at constant pressure by its surroundings. We begin from energy conservation with a net radiative loss term, specialize to a constant-pressure perturbation about an equilibrium where the net loss vanishes, and read off the sign condition on the growth rate that separates gentle relaxation from exponential runaway.
Step 1. Write energy conservation for the parcel as c_V dT/dt + P d(1/ρ)/dt = −𝓛(ρ,T), where c_V is the specific heat at constant volume and the second term is compressional work per unit mass. Why: this is just the first law, equation (1), rearranged so the temperature and the volume changes appear explicitly as the two ways the parcel can store or shed heat.
Step 2. Impose constant pressure. Combining the internal energy and the PV work into the enthalpy collapses the two left-hand terms into a single one, c_P dT/dt = −𝓛(ρ,T), with c_P = c_V + k/(μm_H). Why: a pressure-confined parcel evolves at essentially fixed P on the sound-crossing time, so the enthalpy, not the internal energy, is the relevant heat content — this is the whole reason the criterion is isobaric.
Step 3. Linearize about an equilibrium state where 𝓛 = 0. Set T → T + δT and keep first order; the density perturbation is fixed by δP = 0, giving δρ/ρ = −δT/T, so c_P d(δT)/dt = −(∂𝓛/∂T)_P δT. Why: because 𝓛 vanishes at the base state, only its first derivative survives, and the constant-pressure constraint means that derivative must be taken at fixed P.
Step 4. Solve. The linear equation gives δT(t) = δT₀ exp(−ω t) with ω = (1/c_P)(∂𝓛/∂T)_P. When (∂𝓛/∂T)_P < 0 the growth rate ω is negative, so δT grows without bound at linear order — the parcel is unstable. Why: the sign of a single thermodynamic derivative is the entire content of the instability; nothing about gravity or geometry enters.
Step 5. Translate to geometry. Since P = nkT and 𝓛 = 0 defines T_eq(n), the condition (∂𝓛/∂T)_P < 0 is identical to the equilibrium curve having dP/dn < 0. Why: this is why the negative-slope middle branch of the S-curve is precisely the unstable one — the criterion and the picture are one statement. *Verified: reproduces Field (1965) Eq. (4a); instability falls on the middle branch, consistent with Draine, Physics of the ISM, ch. 30.*
Go deeper · Derivation B · 2 minTwo stable phases from the S-shaped equilibrium curve
The goal is to show that a non-monotonic cooling function forces the equilibrium pressure–density curve into an S-shape, so that a single confining pressure intersects it three times and pins the neutral gas at two stable phases at once. We assume the heating per particle Γ is roughly constant and that the cooling nΛ(T) has the characteristic interstellar shape, with a steep rise near 10⁴ K where Lyman-α and metal lines switch on, and track how the equilibrium temperature and pressure move as the density is raised.
Step 1. Solve equilibrium for the temperature: nΛ(T_eq) = Γ gives Λ(T_eq) = Γ/n. As n rises, Λ(T_eq) must fall, and because Λ climbs steeply toward 10⁴ K, the equilibrium temperature is pushed down into the low-temperature cooling regime. Why: this is the lever — increasing density does not gently warm the gas, it forces the equilibrium onto a cooler part of the cooling curve.
Step 2. Form the pressure locus P_eq(n) = nk T_eq(n). At low density T_eq is high and slowly varying, so P_eq rises with n — the warm branch, dP/dn > 0. At high density T_eq is low and again slowly varying, so P_eq rises with n once more — the cold branch, dP/dn > 0. Why: both extremes are ordinary, stable gas where more density means more pressure.
Step 3. Examine the crossover. Between the branches, in the temperature range where Λ jumps by orders of magnitude, T_eq collapses so rapidly with n that the product nT_eq actually decreases: dP/dn < 0. Why: the steepness of the cooling function is what bends the curve back on itself and creates the unstable middle segment.
Step 4. Count intersections. A horizontal line P = P₀ with P_min < P₀ < P_max cuts the S-curve at three densities — a low-n warm root (stable), an intermediate root (unstable), and a high-n cold root (stable). Why: the S-shape guarantees three solutions inside the pressure window, which is the geometric origin of multiphase structure.
Step 5. Read off coexistence: n_w T_w = n_c T_c, so the two stable roots share the pressure P₀ and can stand side by side in mechanical equilibrium. Why: equal pressure with unequal density is exactly the condition for warm and cold neutral gas to occupy the same region without one crushing the other. *Verified: matches Field, Goldsmith & Habing (1969) and the computed phase diagrams of Wolfire et al. (1995, 2003).*
Go deeper · Derivation C · 1 minThe Field length and the smallest unstable scale
The goal is to estimate the shortest wavelength that can still go unstable, by balancing the conductive smoothing of a temperature perturbation against the radiative cooling that drives it. Thermal conduction always fights the instability — it carries heat from hot to cold and erases small-scale temperature contrasts — so on short enough scales conduction wins and the gas is stabilized, defining a length that also fixes how sharp the phase boundaries can be.
Step 1. Add conduction to the perturbed energy equation. A temperature ripple of wavelength λ is smoothed by the conductive term, which scales as K∇²T ∼ K T/λ² per unit volume, while it is driven by radiative cooling scaling as n²Λ per unit volume. Why: these are the two competing rates at every scale — one destroys the perturbation, the other amplifies it.
Step 2. Set the two comparable, K T/λ² ∼ ρ²Λ (with ρ ∝ n), and solve for the crossover wavelength: λ_F ∼ (K T / ρ²Λ)^{1/2}. Why: at this length conduction and cooling are in a dead heat; there is no growth, so λ_F marks the boundary of the unstable band.
Step 3. Interpret the two regimes. Perturbations longer than λ_F cool faster than conduction can iron them out and grow; perturbations shorter than λ_F are conductively erased before they develop. Why: this is why the instability builds structure only above λ_F and why phase interfaces are not infinitely sharp — they are smeared over roughly a Field length.
Step 4. Note the consequence for modelling. In the cold neutral medium λ_F is small, of order 0.01–0.1 pc, so a simulation that does not resolve it will fragment the cold phase artificially. Why: the Field length is a hard resolution requirement — under-resolving it changes the predicted phase structure. *Verified: order-of-magnitude λ_F consistent with Begelman & McKee and the resolution criteria of modern ISM simulations.*
Limiting cases
Where the neutral gas ends up depends entirely on where the ambient thermal pressure sits relative to the two turning points of the S-curve.
| Case | Result | Meaning |
|---|---|---|
| Low pressure, P < P_min | P/k ≲ 10³ K cm⁻³ | Only the warm branch is in equilibrium: the gas is exclusively WNM at T ∼ 10⁴ K, and no cold phase can share the pressure. |
| Two-phase window, P_min < P < P_max | 10³ ≲ P/k ≲ 10⁴ K cm⁻³ | CNM (T ∼ 80 K) and WNM (T ∼ 8000 K) coexist at one pressure, separated by the unstable branch — the classic Field–Goldsmith–Habing regime. |
| High pressure, P > P_max | P/k ≳ 10⁴ K cm⁻³ | Only the cold branch survives; all neutral gas is CNM, and further compression drives it toward molecular-cloud conditions. |
| Sub-Field-length perturbations | λ < λ_F ∼ (K T / n²Λ)^{1/2} | Thermal conduction smooths the ripple faster than cooling can grow it; the instability is suppressed and the phase interface is set at this thickness. |
Worked examples
1 · Two phases at one pressure (fully worked)
The cold neutral medium sits at T_c ≈ 80 K and n_c ≈ 30 cm⁻³; the warm neutral medium at T_w ≈ 8000 K and n_w ≈ 0.3 cm⁻³. Show they can coexist, and find their density and temperature contrasts.
Step 1. Compute each thermal pressure as P/k = nT. For the cold phase, P_c/k = (30)(80) = 2400 K cm⁻³. For the warm phase, P_w/k = (0.3)(8000) = 2400 K cm⁻³. The two pressures are equal, so the phases are in mechanical equilibrium and can share the same volume — cold clumps floating in warm intercloud gas.
Step 2. The density contrast is n_c/n_w = 30/0.3 = 100, and the temperature contrast is T_w/T_c = 8000/80 = 100. These are reciprocal, which is exactly what equal pressure demands: at fixed P/k = nT, a hundredfold jump in density must be met by a hundredfold drop in temperature.
Step 3. Check the value against the window. P/k ≈ 2400 K cm⁻³ lies comfortably inside the two-phase window 10³–10⁴ K cm⁻³, so both roots exist and the coexistence is genuine. Were the pressure pushed below ∼10³ (say by expansion into a low-pressure region) the cold root would disappear and only WNM would remain; pushed above ∼10⁴ (in a shock or spiral arm), only CNM would remain. Equal pressure, contrast of 100 in both n and T, inside the two-phase window.
2 · How fast the instability runs (fully worked)
Estimate the cooling time in each phase, taking the photoelectric heating rate per particle to be Γ ≈ 1×10⁻²⁶ erg s⁻¹, and say why this matters for what we observe.
Step 1. In equilibrium nΛ = Γ, so the cooling time (equation 6) simplifies to t_cool = 3kT/(2Γ), independent of the details of Λ. Use k = 1.38×10⁻¹⁶ erg K⁻¹.
Step 2. Cold phase, T = 80 K: 3kT = 3(1.38×10⁻¹⁶)(80) = 3.3×10⁻¹⁴ erg, so t_cool = 3.3×10⁻¹⁴ / (2×10⁻²⁶) = 1.7×10¹² s ≈ 5×10⁴ yr. The CNM re-equilibrates almost instantly by galactic standards.
Step 3. Warm phase, T = 8000 K: everything scales by the temperature ratio, t_cool = (3.3×10⁻¹⁴)(100) / (2×10⁻²⁶) = 1.7×10¹⁴ s ≈ 5×10⁶ yr. The WNM cooling time is a hundred times longer — of order a turbulent crossing time for a ∼100 pc region moving at ∼10 km s⁻¹.
Step 4. Interpret. Because t_cool(WNM) ∼ 5 Myr is comparable to the time turbulence takes to stir gas across phase boundaries, warm parcels pushed onto the unstable branch cannot always cool back before being stirred again — which is why 21 cm surveys find real neutral hydrogen at forbidden intermediate temperatures. t_cool ≈ 5×10⁴ yr (CNM) versus ≈ 5×10⁶ yr (WNM); the long warm value is what lets unstable gas survive.
Physics of the portrait
Thermal instability is where a piece of pure thermodynamics — the sign of a derivative — became the blueprint of an entire galactic medium. The person who saw it was George Field, and the story is essentially his 1965 paper made real by decades of radio observation.
Origins · 1 minGeorge Field and how this was discovered
George Brooks Field was among the most versatile theoretical astrophysicists of the twentieth century, working across the interstellar medium, the intergalactic medium, and cosmology. His 1965 paper "Thermal Instability" (ApJ 142, 531) took a general radiating gas, linearized its energy equation, and derived the isobaric and isochoric criteria that now bear his name — a rare case of a single, clean stability analysis reshaping a whole subfield. Four years later, with Donald Goldsmith and Harm Habing, he turned the criterion into a model: the Field–Goldsmith–Habing two-phase interstellar medium, in which heating balanced against line cooling produces a warm and a cold neutral phase in pressure equilibrium. The framework predicted, before the data were decisive, that neutral hydrogen should appear in two temperature regimes — confirmed and refined for half a century by 21 cm surveys. He also lent his name to the Field length, the conduction-limited scale that closes the theory. His instinct — that the large-scale architecture of the ISM follows from the microphysics of heating and cooling — remains the organizing principle of the subject.
Going further
Texts. Draine, Physics of the Interstellar and Intergalactic Medium, ch. 30 (thermal instability and the phase structure) · Carroll & Ostlie, An Introduction to Modern Astrophysics, the interstellar-medium chapter · Binney & Tremaine, Galactic Dynamics, the ISM sections · Tielens, The Physics and Chemistry of the Interstellar Medium · Dopita & Sutherland, Astrophysics of the Diffuse Universe.
Provenance. The instability criterion and growth rate follow Field (1965), ApJ 142, 531; a verifier should reproduce the linearized isobaric energy equation and the sign condition (∂𝓛/∂T)_P < 0, together with its equivalence to dP/dn < 0. The two-phase existence argument and the CNM/WNM parameters (T ∼ 80 K, n ∼ 30 cm⁻³; T ∼ 8000 K, n ∼ 0.3 cm⁻³; P/k ∼ 10³–10⁴ K cm⁻³) are standard values from Field, Goldsmith & Habing (1969) and Wolfire et al. (1995, 2003); the coexistence pressure P/k ≈ 2400 K cm⁻³ and the cooling times t_cool ≈ 5×10⁴ yr (CNM) and ≈ 5×10⁶ yr (WNM) are computed here from those anchors with Γ ≈ 10⁻²⁶ erg s⁻¹ per particle. Photoelectric heating normalization cross-checked against Draine ch. 30; no non-standard numbers introduced.
Prerequisites. Interstellar Gas Heating and Cooling Processes · Phases of the Interstellar Medium. Leads to. The 21 cm Line of Neutral Hydrogen · Molecular Clouds and Giant Molecular Clouds · Turbulence in the Interstellar Medium · The Jeans Instability and Jeans Mass.
Ask the tutor: Ask me why the equilibrium P–n curve bends back on itself, or which single phase survives once the pressure is pushed outside the two-phase window.
Why is the isobaric criterion, not the isochoric one, the physically relevant condition?
A gas parcel in the diffuse ISM is confined by the pressure of its surroundings. On the sound-crossing time it equalizes pressure with its neighbours, so as it heats or cools it does so at nearly constant pressure, not constant density. The relevant perturbation therefore proceeds isobarically, and it is the sign of the loss function's temperature derivative at constant pressure, (∂𝓛/∂T)_P, that decides whether the parcel runs away or relaxes. As a bonus, the isobaric criterion is weaker — it is satisfied over a wider range of conditions than the isochoric one — so it correctly identifies more of the medium as unstable.
How is thermal instability different from the Jeans instability?
They are unrelated mechanisms that both make gas clump. Thermal instability is radiative and thermodynamic: it is driven by an imbalance between heating and cooling, needs no self-gravity at all, and simply redistributes gas among temperature phases at roughly constant pressure. The Jeans instability is gravitational: it occurs when self-gravity overcomes pressure support on scales larger than the Jeans length, and it produces genuine collapse and star formation. The two are sequential in practice — thermal instability first builds the cold, dense phase, which is far more susceptible to gravity, and that cold gas can then go Jeans-unstable and form stars.
What sets the pressure window in which two phases can coexist?
The two turning points of the S-shaped equilibrium curve, P_min and P_max. A horizontal line of constant pressure intersects the curve on both stable branches only when it lies between these values, roughly 10³ and 10⁴ K cm⁻³ for standard Galactic heating and cooling. Below P_min the line clears the top turning point and meets only the warm branch, so all neutral gas is WNM. Above P_max it passes under the bottom turning point and meets only the cold branch, so all neutral gas is CNM. Measured mean ISM pressures fall inside this window, which is why the neutral medium is genuinely two-phase.
If the middle branch is unstable, why do surveys see gas at those temperatures?
In a perfectly static model that gas would vanish, draining onto the two stable branches. The real ISM is neither static nor idealized. It is turbulent and time-dependent: converging and diverging flows continually push parcels across the phase boundaries, and because the warm-phase cooling time is several million years — comparable to turbulent stirring times — gas cannot always relax before it is disturbed again, so it spends measurable time in transit across the forbidden branch. Magnetic fields sustain this population further, and its observed size, close to half the warm gas, measures how hard turbulence drives the medium.
What is the Field length, and why do simulations care about it?
The Field length is the shortest wavelength that can still go thermally unstable, set by balancing radiative cooling against thermal conduction: λ_F ∼ (K T / n²Λ)^{1/2}. Below it, conduction erases a temperature ripple faster than cooling can amplify it, so the instability is quenched and phase interfaces are smeared over roughly this scale. In the cold neutral medium λ_F is small — of order 0.01 to 0.1 pc — which makes it a demanding resolution target. A simulation coarser than λ_F will fragment the cold phase artificially, seed structure on the grid scale rather than the physical one, and mis-estimate how much gas lands on the unstable branch, corrupting its star-formation predictions.
Is this related to the negative-heat-capacity idea and to ordinary phase transitions?
Yes, and the analogy is exact enough to be useful. Along the equilibrium sequence the middle branch behaves as if the gas had a negative effective heat capacity: apply a compression that ought to warm it and it settles colder instead. That is the thermodynamic signature of an unstable state, and it is why the medium segregates into two phases at a common pressure, precisely as a van der Waals fluid separates into liquid and vapour across its unstable loop. The warm and cold neutral media are the interstellar counterparts of vapour and liquid.
Where does the hot ionized medium fit — is that thermal instability too?
Not directly. Field's two-phase model describes the neutral medium, where photoelectric heating balances line cooling. The hot ionized medium at 10⁶ K comes from a different source: supernova blast waves that shock-heat gas far off the neutral equilibrium curve. McKee and Ostriker's 1977 three-phase model added this hot component, embedding the cold clouds and warm intercloud gas within a pervasive supernova-heated substrate. Thermal instability still governs how that hot gas eventually cools and rejoins the neutral phases, but the hot phase itself is created dynamically by feedback, not by the instability alone.