Recall the essential steps that allow statistical mechanics to derive things such as black-body radiation and other thermodynamic phenomena:
- There is an idealized system known as the microcanonical ensemble. This ensemble keeps its total energy fixed and its components (possibly amounting to an infinite number of degrees of freedom) interact only with itself and do not leave the system. However, it is tacitly assumed that the components do interact with itself enough to reach properties such as ergodicity or similar. In practice, this requires some sort of ability to impose "reflection" or restricting the spatial movement of the components trying to leave the system, and none of the components being essentially inert to the other.
- Another step is the canonical ensemble, which is an idealized system that does not exchange particles (components) with its surroundings, while it is allowed to transfer energy to its surroundings. The canonical ensemble is assumed to be a part of a much larger microcanonical ensemble, and from this all its properties are derived. This gives rise to the notion of temperature, for instance.
- Then there is the grandcanonical ensemble, whis can exchange both particles (in both directions) and energy while being a subsystem of a much larger canonical or microcanonical ensemble. This gives rise to the notion of a chemical potential.
Equilibrium thermodynamics assumes that we are able to somehow achieve at least some of these ensembles to some degree of approximation.
For example, electromagnetic black-body radiation is achievable because almost every molecule has an electric dipole moment, and radiation will react with it. Similarly, the free ions in plasma will generally scatter, absorb and re-radiate radiation very well. How much radiation interacts with a chunk of material is characterized by its optical thickness $\tau \in (0,1)$. Only bodies that retain a significant amount of a burst of radiation passing through them, $\tau \sim 1$, can be considered as reaching grandcanonical ensembles with EM radiation in their interiors. This is because energy gets exchanged at many points, and in many ways between the matter and radiation degrees of freedom. And only these bodies will give off a spectrum of photons that is close to the black-body radiation computed by the statistical-mechanics idealizations mentioned above. That is, you will get (roughly) the Planck-law distribution of radiation with temperature matching the Maxwell-Boltzmann distribution of the matter degrees of freedom near the surfaces of only optically thick media.
However, there are also optically thin media. One of them is air. Air would have to be closed in a cavity for a long time before it would reach equilibrium. In the vast majority of practical situations, a probe placed in the middle of air will not measure thermal radiation of the same spectral temperature as the surrounding air. We are thankful for this, since it is this property that also allows us to, like, see things in our everyday lives on Earth.
Many other interesting examples of optically thin media can be found in astrophysics. For example, the plasma in the accretion disk around the black hole in the center of our galaxy is assumed to have such a low density of particles that Coulombic collisions happen very little, radiation is produced rarely by the usual mechanisms, and the particles have a hard time staying in thermal equilibrium amongst themselves, let alone radiation. However, free electrons have a three orders of magnitude higher higher specific charge (charge per mass) than protons, and thus cool down separately by getting accelerated by random magnetic fields and produce synchrotron radiation. The result of this is a "two-temperature" plasma of electrons and ions of very different effective temperatures that produces EM radiation with a spectrum very far from the usual Planck's law. The radiation is random, has a broad distribution, but is not thermal in any sense of the word.
GW radiation should be likened to the second type of situations. The "GW thickness" (analogue of $\tau$) of essentially any known matter object is close to zero. GW radiation passes through matter almost freely, and the only way an object retains any of it is by having some sort of internal friction that captures the contractions or dilation due to the passage of the wave. (As in the famous sticky bead argument.) Let us compare the strength of interaction of a generic simple molecule (such as $\rm H_2 O$) with EM radiation and GW radiation. The dipole EM radiation power is
$$ P = \frac{\mu_0 p^2 \omega^4}{6 \pi c}$$
where $p$ is the charge dipole moment and $\omega$ is the molecule rotation frequency. The size of a molecule is $L\sim$ Angstroms $ = 10^{-10}$ meters. The dipole moment is typically a $\sim 0,1$ fraction of the elementary charge times the size of the molecule. The typical rotation frequency will be roughly given by the fact that equipartition energy $k_{\rm B}T/2$ is equal to rotation energy $\omega^2 I/2$, where the moment of inertia is roughly the mass of the molecule times size squared and a geometric prefactor, or $\omega^4 \sim k_{\rm B}^2 T^2 m^{-2} L^{-4}$. Mass is roughly few times the proton mass. Omitting geometrical prefactors we get
$$P \sim \frac{\mu_0 e^2 k_{\rm B}^2 T^2}{c \,m^2 L^2} \sim 10^{-20} \left(\frac{T}{100 \,\rm Kelvin} \right)^2 \,\rm W $$
You can see that this dipole will be able to radiate its rotational energy at the order $k_{\rm B} T/2$ in the order of $10^{-1}$ seconds at Earth-like temperatures ($T\sim 100$ K). To keep equilibrium, we would maybe require even stronger coupling, but it is admissible that this gas can be kept in equilibrium with EM radiation.
As for the GW radiation, the leading-order radiation is given by the quadrupole formula $$P \sim \frac{G}{c^2} \left(\frac{d^3 Q}{d t^3}\right)^2$$ where $Q$ is the molecule mass quadrupole. The quadrupole itself scales as $Q\sim m L^2$, and its third derivative will scale as $\dddot{Q} \sim m L^2 \omega^3$ with frequency being again $\omega^2 \sim k_{\rm B} T m^{-1} L^{-2}$. Now we have the GW power
$$P \sim \frac{G k_{\rm B}^3 T^3 }{c^2 m L^2} \sim 10^{-69} \left(\frac{T}{100 \, \rm Kelvin}\right)^3\,\rm W$$
Again, we can estimate that at Earth-like temperatures this would take $10^{47}$ seconds or $\sim 10^{30}$ times the age of the Universe to radiate energy of order $\sim k_{\rm B}T$. You would get similar rates for transfers of a Planck-type spectrum of GWs back into the gas. (And notice also that the $\sim \omega^3$ power will also make the spectral shape different from that of a Planck black-body spectrum in any situation of interest.) So yes, reaching GW thermodynamic equilibrium with normal Earth-like systems is just not going to happen.
As for black holes and thermal GW radiation. If we trust tree-level quantization of gravitational perturbations, then yes, a black hole will emit a thermal-like spectrum of gravitons (with the appropriate corrections due to BH greybody factors, of course). However, depending on who you ask, they may not trust even that about quantum gravity.
Note also that one is not in thermal equilibrium with the surrounding vacuum, the leakage of Hawking radiation into the surrounding is a non-equilibrium process that should lead to a true global thermal equilibrium at late times. There is much discussion about this. There is one gravitational system that is in thermal equilibrium by construction, and that is a black hole in an asymptotically Anti-deSitter (AdS) space-time. The AdS space-time has its temperature and a reflective boundary that does reflect massless particles back and forth (a property set by hand, other boundaries are admissible), so that one can talk about the entire universe being a microcanonical ensemble. The black hole can then be the small sub-part, a canonical ensemble. But you can have more fun interpreting these kinds of space-times, see black hole chemistry as proposed by my colleague David Kubiznak and Robert B. Mann.
