If the system in in thermal equilibrium, this means all its constituents take and give the same amount of energy at the same time, making them looking like the same for an observer looking at their (thermal radiation) only. Sizes, masses, (real visible colors), .. etc will all be indistinguishable.

This is in theory. In experiment, distinguishing between bodies (thought to be) in thermal equilibrium is possible if this equilibrium state was not reached yet. In fact, one can use this to identify non- equilibrium states of some closed system.

This is a really interesting and thought provoking question. I suppose the strong corollary to this would be that it is only possible to observe an object due to its non-equilibrium with respect to the measure that you are considering. I've actually been thinking and deleting comments here for a while as I flip-flop between different explanations. Some of the text below reflects this as I've gone back to refine my thinking - sorry if it doesn't flow so well.

As you discuss I think that it is fairly obvious that one will not be able to sense the presence of a black object via thermal means if it is in thermal equilibrium with its environment. This is presumably due to the assumption that a passive detector is sampling the emission spectrum and that this is therefore the only element of reality that we can sample from this system of interest. Equally because the detector is in thermal equilbrium with the source, it is generating as many photons (noise) as it is receiving. Hence you can either detect the object by the emitted photons from the object if it is hotter, or by the absorbance of the photons emitted by the detector if it is cooler.

I can't see the filter doing anything, as sooner or later it is going to heat up and reach thermal equilibrium with everything else, and hence be equally indistinguishable. Note also that because this is a reversible device, the amount of light in the wavelength band of interest is going to be the same for the detector and object, so again I can't see that it is going to help you.

However your question is not about black bodies per se, and I think that the most important point to consider is microstructing, and there has been some especially nice work done in the context of thermovoltaics. I saw a very nice talk from Marin Soljačić recently on thermovoltaic devices where the black body emission was combined with a photonic bandgap material. The photonic bandgap altered the density of states of the radiation field and hence then emission. In fact it was possible to demonstrate a spectrum that was apparently anti-thermal from this (I'm sorry, I don't have the reference now, although there is a short article here: http://www.mit.edu/~soljacic/TPV_Oyo_Buturi.pdf). I would assume that in this case the material would be distinguishable above the background because in this case the emission spectrum would be non-black body like in both spatial distribution and spectrum due to the modification of the radiation modes.

I think that this case is harder to understand, but I would guess in the true equilibrium case it will still be impossible to distinguish the system. This argument would follow from ray tracing every point back from the detector to an object and determining the spectrum of what was being sampled. Ultimately in a closed system all of the radiation modes still need to be black-body like and the apparent non-equilibrium distribution is going to come from some part of the system trying to sample something beyond the closed system. (I keep on thinking about an element that perturbs the radiation field such that it couple in modes from outside the structure - but all that that does is to create a non-equilibrium condition).

in fact the more I think about it, the reverse solution seems to be the correct one. Rather than thinking about the object as heating the detector, think about whether the detector can heat the source. Now the photons leave the detector and there will be some non-trivial interaction with the microstructured material, but ultimately the photons will be taken to some region of space that they are in thermal equilibrium with, and hence there will be no detection.

Now the second part of the question - non-idealities. i think that there is a strong clue in the above post by Selman in the time taken to reach equilibrium. I would wonder if driving a microstructured would provide such an effect (doing work on the radiation field). In fact I'm sure it would and I think that we can consider a simpler thought experiment than this.

imagine we have a Fabry-Perot cavity as our object and we are going to probe this from the output port of the leaky mirror. we will assume that the whole system is in thermal equilbrium, so that all of the mirrors are emitting as black bodies, and also the detector. Now we drive the output mirror, and we drive it fast. We know that such driving can create extra particles in the cavity and these are presumably not going to be in thermal equilibrium with the temperature of the system, but rather are governed by the drive that is applied to the mirror.

Note that I'm pretty sure that I'm cheating here. The canonical Maxwell daemon type argument here would have you concluding that even the drive must eventually reach some kind of thermal equilibrium and would therefore not give the relativistic photon creation that you want. Or alternatively, you could say that I'm cheating because the drive has created an active type of detector.

Nevertheless, I would guess that strongly perturbing the radiation modes as a function of time should give rise to a distinct possibility of detection when our of equilibrium.

Actually this moving case would seem to be far and away the easiest to understand (sorry, I said that this is stuck in my head!).

Imagine our object and our detector in one-dimension and in thermal equilibrium. The detector cannot perceive anything because the rate of photons hitting the detector is the same as the rate at which photons leaves the detector - so there is no imbalance to detect and there is just noise.

Now accelerate the object away from the detector. The rate at which photons leave the detector is unchanged, but the rate at which photons arrive is reduced. There is also a redshift in the emission spectrum. Hence the object shook look cooler and the shift from equilibrium on the detector should be observable.

Conversely if the object accelerates towards the detector if will spectrally blue shift, and in addition, more photons per unit time will also reach the detector than are emitted. (Note of course that the shifting in the spectrum and the change in number of photons are different processes). Hence the detector will be heated up. Again, this departure from equilibrium dynamics should be detectable.

So on the basis of this argument, I would hold that any net relative accelerations should be detectable in the problem as described. The reason for introducing the word 'net' above is that if we consider the case where the object is a deformable membrane - for example the interior of a balloon. Then at thermal equilibrium I would expect the system to have random fluctuations in the object - detector relative motion. However, precisely because it is in thermal equilibrium, I would expect every 'local cooling' to be accompanied by a 'local heating' due to the random directions of the accelerations, and hence in this limit again I would not expect the system to show the non-equilibrium dynamics necessary for detection.

Kirchhoff first corollary defines emission-absorption of a body with fixed size and shape to be equally defined. If we assumed that this (body) is being a whole system of fixed geometry, then thermal equilibration is sustained. Although such a system could be in fact a closed system of many-body, having their own geometry. Hence, the (emission) detector, when in a thermal equilibrium with other constituents of the system, is actually just another body belonging to these constituents. If it can, by comparison to anything else in such system, detect any kind of sensible emission, then, logically, it is either out of equilibration, or the radiation being detected is coming from a body that is.

In fact, Changing the system's shape(s), or rearranging it's constituents via acceleration/deceleration or with fixed speed, alone, would bring the system out of thermal equilibrium, according to Kirchhoff. However, the system, as a whole, can have a motion in space and keep it's equilibrium unchanged.

Also, the term (thermal equilibrium) implies that emission/absorption both come equally, i.e., with the same time rate, with perfectly homogenous space (e.g., spherical) distribution, and with the same total amount. This leave the distinguishing between two (or more) bodies with a different shapes when being a part of the thermodynamically equilibrated system, a not fully logical choice, in theory.

However, in practice, thermal equilibrium is not an easy-to-reach state of nature. In fact it is an ideal case. I think this is what is required as an answer.

Thank you all for these very interesting comments on this very challenging question.. @Ahmed: In practice, how could we take advantage of these non-idealities for discrimination purpose between bodies ?

@ Lehtihet:·Thank you for this question.
One important application of is in astrophysics. In radio astronomy, as an example, the slight change of temperature of galactical centers for far galaxies (which can be approximated to be thermodynamically homogenous), allows us to find out vital information about the universe. Faint objects in far space in general can only be detected using the difference of their emission from (presumed) thermal equilibrium with space. These objects are not in balance, of course, with their surroundings thus their emissions are sensible using radio-wave telescopes. Regards.

What if the the 2 bodies are radioactive, and made of two different materials,both exhibiting radioactive decay, and by chance be at the same temperature? Wouldn't a geiger counter passively distinguish between the two bodies if the decays were different enough? Or is it limited to a measurement of wavelength only? In reality though, thermal equilibrium is never really reached, except at absolute zero.

@Ahmed: Interesting possibility. Thank you. I did not think at all about an application in astrophysics. After reading @Derek's question and the subsequent exchange you have had with @Andrew, I started asking myself about potential applications. However, I don't know why, my mind was rather oriented towards the other side of the scale.

Speaking of scale, if I am not too wrong, there must be some sort of a threshold "scale" for the non-idealities under which no discrimination would be possible. If the answer is yes, then how strongly would this threshold depend on the inherent parameters of the experiment?

Thank you Joachim.

Great thought experiment and thought provoking question.

If I may, with all due respect, Kirchhoff's radiation law indicates that the emissivity (radiant emittance) of a non-black body at a given temperature is equal to the absorptance at the same temperature (radiant emittance of a black body). According to Sprackilng, Thermal Physics, 'The non-black body emits radiation that has a quality and quantity such that, when combined with the reflected radiation, it produces an isotropic distribution of radiation from the surface which is identical with E (T)." Here E (T) is the radiant emittance as the energy a body emits per unit area per unit time.

'Thermal equilibrium' is too often thought of as a fixed and determinative state of matter, observable in our sensible real-world. In the thought experiment, in where the bodies, cavity, passive detector, are in equilibrium with the system, one must step back and redefine the system in classical terms, as this experiment is rooted in classical thermodynamics. The extended thought experiment of the case containing different sizes, material, emissivities, and observer filters is also addressed in classical thermodynamics. The first part appears to be suggesting a closed system in which the boundary is not permeable, maybe adiabatic. The second part suggests an open system otherwise the observer could not change the parameters of the system and evaluating equilibrium processes become trivial.

Thermal equilibrium may be more appropriately considered a condition of a system at some instant in time, say an equilibrium state. As such a state in our sensible real-world is not 'ad infinitum', a system in thermal equilibrium will undergo a process even when infinitesimally close to the equilibrium state (quasistatic, Sprackling). In this respect we do not have a closed, isolated system and the thermodynamic processes go from equilibrium state to equilibrium state. So we do not have the idealized state of a black body and are not obstructed from 'seeing' in theory, which in turn does not violate the second law which is formative for non-equilibrium thermodynamics.

The second, thought experiment, above would result in thermodynamic equilibrium processes continuing from equilibrium state to equilibrium state, even quasistatic, and the determination of temperature differences, our 'seeing' of discrete conditions of state, would be dependent on the precision of the measuring device, whether we are measuring 'bulk' surfaces (macroscopic) or distinguishable molecular states (microscopic). At some 'point', or range of points, along this continuum (micro-meso-macro), non-equilibrium processes become prominent in our view and a pure, fixed thermodynamic equilibrium state is an idealized condition, a starting point for evaluating condtions of condensed matter.

The point or range of points, I mention, was addressed by Boltzmann and Planck with greater clarity and precision than I, resulting in our understanding of entropy, S = k ln Omega. With this classical thermodynamics is 'bridged' through statistical mechanics to molecular behavior. Non-idealized, or real, experiments to distinguish these bodies, (mentioned above), have been performed for specific heat (Schottky, MacDonald), lattice vibrations (Debye, Brillouin, Born, Blackman, MacDonald), anharmonicity and thermal expansion (Gruneisen), to suggest a few.

Is it possible to distinguish thermal bodies in equilibrium? Yes and No, depending on the position one takes concerning an ideal or a non-ideal condition of state. Since the question presupposes non-ideal conditions with variable parameters, detection devices, and observers, I would suggest the affirmative.

@Lehtihet: There is a scale for detection threshold, yes, just like any other radiation detection experiment. The threshold of detection in experimental physics is obvious and well-known, it is the sensitivity of the detector. It is a rather general expression, it might count for detection efficiency as a whole, or energy resolution of the detector.

It would be useful if you told us what thoughts you had in mind concerning thermal signals detection application for systems near equilibrium.

In astrophysics, such application is preferred since we're speaking about large scales of distance. Here, a sort of homogenous distribution can be safely assumed to eliminate small finite differences in temperature between adjacent regions. If these differences were of an order smaller than the detection sensitivity, the experiment would give us results that make us (think) that this region sustains equilibrium. Of course, this is valid only when talking about thermal radiation only, hence the application mentioned above can be argued as an example.

@Ahmed. Thank you for your answer. I was not really talking about the detection threshold of the detector itself. I was just trying to understand the scale of the non-idealities mentioned in Derek's question.

Concerning your question, I do not have any specific application in mind but it seems to me there must be some potential applications in the fields of nano-technology or gas-leak detection, etc. Also, at low temperature, one can think about cryogenic-particle detection but I do not know if this is still within the scope of Derek's question. Finally, I have read somewhere an article about spectrally-selective thermal detection devices, which might relate to the filters mentioned by Derek.

I totally agree with Don Hargis.

Answering Derek question,
If all bodies (including detector) are in thermal "quasi static" equilibrium, probably cannot be distinguished. But the detecting process itself (depending on design of detector) can be energetically not neutral, for example the temperature of detector may change.

When the system: bodies and inactive detector is in equilibrium, starting the detection process theoretically can put the system bodies-detector out of the equilibrium state. In the transient period between detection start, and new equilibrium establishing, whole system is in fact out of equilibrium, and potentially the observed bodies can be distinguished. The question is the demanded sensitivity.

So, more general, the bodies can be potentially distinguished by permanent changes the temperature (or other energetic parameter) of the passive detector.

What is interesting, Derek wrote "... with a passive detector that is also in equilibrium with the system". In this context, the word "passive" takes completely new meaning. If passive detector is not in equilibrium with the system, in fact it became "quasi-active". Because, as wrote Don Hargis, in practice there is nothing like perfect static thermal equilibrium -> there is no passive detectors :).

@ Artur. To clarify: when I used the word 'passive' in my question, i merely meant that there is no external power source for the detector (otherwise the external power is breaking equilibrium). Something like a sheet of phosphor material could be eligible as a passive detector in this context.