# Heat transfer between a common material and a non-thermalizable one?

I've read that some systems cannot reach equilibrium (page 15 of the book Selected Scientific Papers of Sir Rudolf Peierls: With Commentary or R. Peierls, “Zur kinetischen Theorie der Wärmeleitung in Kristallen” Ann. Phys. 395, 1055–1101 (1929) for instance, or in condensed matter/QM papers), and as such the meaning of a well defined temperature becomes meaningless.

I am wondering what would happen if such systems are placed into direct contact with a normal material (i.e. metal, alloy, semiconductor, insulator, etc.). More precisely, what would happen to the temperature of the normal material that is placed into contact with such exotic systems? Would their temperature change? If so, what would happen to the exotic systems/materials? And which law would describe the heat transfer? (cannot be Fourier's law I suppose since $$T$$ is not well defined).

Edit to address the answer written by Thorondor: As Thorondor points out, I might have misunderstood Peierls's point. However there is a bunch of papers claiming that thermalization is not a given, for a given system. Thus, as is, I cannot accept his answer as an answer to my question(s).

For example this paper claims

Under what conditions does a system fail to thermalize, thus evading the conventional classical fate even at long times? In contrast to a majority of experiments in solid state systems, these questions pertain to highly non-equilibrium states of matter with non-zero energy density that could translate to high and even infinite effective temperature. Can quantum effects survive at long times in many-body systems at such high energy densities? Answering these basic questions is a necessary step towards understanding a potentially very rich variety of new states of matter that can appear in highly non-equilibrium quantum systems.

And it isn't just pure theory that cannot exist. The abstract contains

Many-body localized (MBL) systems remain perfect insulators at non-zero temperature, which do not thermalize and therefore cannot be described using statistical mechanics.

and then

Experimentally, synthetic quantum systems, which are well-isolated from an external thermal reservoir, provide natural platforms for realizing the MBL phase. We review recent experiments with ultracold atoms, trapped ions, quantum materials, and superconducting qubits, in which different signatures of many-body localization have been observed.

There are other papers (e.g. this one) claiming

The headline from the past few years, however, is that the classical fate of a closed many-body system is not inevitable. There is at least one class of system that fails to thermalize and can retain retrievable quantum correlations to arbitrarily long times through the phenomenon of many-body localization (MBL).

Therefore, I am still awaiting an answer to my questions (title + body text). I thank Chemomechanics for sharing his interest in those questions and offering a bounty, and Thorondor for giving his point of view.

First of all, I think you are misunderstanding the point Peierls is trying to make. The relevant paragraph from “Zur kinetischen Theorie der Wärmeleitung in Kristallen” is

Genau so verhält es sich im Fall der festen Körper. Man weiß seit Born und Debye, daß das idealisierte Modell mit harmonischen Kräften zwischen den Atomen für die Untersuchung der Wärmeleitfähigkeit unbrauchbar ist, da es eine unendlich große Wärmeleitfähigkeit liefern würde. In dem Modell mit harmonischen Kräften kann man nämlich -- wie wir im einzelnen im 1. Abschnitt sehen werden -- die Bewegungen der Atome aus voneinander unabhängigen ebenen Schallwellen aufgebaut denken. Besteht einmal eine beliebige Verteilung der Energie auf diese Schwingungen, so bleibt sie für immer bestehen. Es stellt sich also kein thermisehes Gleichgewicht ein und man kann daher von einer Temperatur im allgemeinen gar nicht reden. Aber selbst wenn man die Wärmeleitung mit Hilfe des Energiegefälles statt des Temperatur- gefälles definiert, kommt man nicht zum Ziel, denn der Begriff der Wärmeleitfähigkeit setzt voraus, dab eine Proportionalität zwischen Energiestrom und Energiegradient besteht, die in diesem Modell nicht vorhanden ist. Man sieht das am einfachsten daran, daß es Zustände gibt, die einen Energiestrom, aber kein Gefälle haben, z. B. wenn nur eine einzelne Welle vorhanden ist. Man kann auch mit Debye sagen, daß die Wärmeleitfähigkeit dieses Modells unendlich groß ist.

The second to fifth sentences can be translated

It has been known since Born and Debye that the idealized model with harmonic forces between the atoms is unusable for the study of thermal conductivity, because it would yield an infinite thermal conductivity. That is, in the model with harmonic forces, one can -- as we will see in Section 1 -- think of the motions of atoms as built up from independent, smooth sound waves. Once an arbitrary energy distribution exists among these vibrations, it will continue to exist forever. Hence, there exists no thermal equilibrium and therefore one cannot in general speak of a temperature at all.

This doesn't say that there are any systems in real life that cannot reach thermodynamic equilibrium. Rather, it says that a simple idealized model discussed later in the paper cannot be true, precisely because the model predicts that certain systems cannot reach thermodynamic equilibrium.

Anyway, since this is supposed to be Physics SE rather than German SE, I should probably address your physics question as well. Do non-thermalizable materials actually exist and, if so, what would happen if such a material came into contact with an ordinary material?

First of all, let's discuss why ordinary materials reach thermodynamic equilibrium. Intuitively, one can imagine a vast space of possible microstates available to the collection of atoms in an object. As atoms collide, they move the object's state in a random direction in this space. Therefore, over time, the object tends to take on all possible microstates with roughly equal probability (i.e. the object is ergodic). It follows that almost all of the time, the object will be in a microstate with high entropy, because by definition there are many microstates with high entropy and only a few with low entropy. Since entropy is a measure of randomness, high-entropy states tend to distribute energy pretty much evenly throughout the object; after all, it wouldn't be very random if one corner got all the heat. That's why ordinary materials reach thermodynamic equilibria with well-defined temperatures, pressures, and so on.

What sorts of things might prevent this from happening? Well, the last paragraph wasn't exactly mathematically rigorous and we've left a few loopholes open. In particular, two major assumptions stand out: that atomic interactions move the object's state in a random direction, and that states with high entropy are thermodynamic equilibria. While these assumptions are true for all normal materials, they can be violated in certain cases. Here are a couple of examples:

• Many plasmas are nearly collisionless, i.e. their behavior is governed by long-range interactions rather than short-range ones. Since interactions between individual atoms are highly correlated rather than random, an ideal plasma does not move randomly through the space of microstates and does not in general reach thermodynamic equilibrium. As a result, observing plasmas (both in space and here on Earth) often yields strange results like different temperatures for electrons and ions.

What happens when you put a plasma into thermal contact with an ordinary material: Obviously there are two possible outcomes. Either the ordinary object evaporates and becomes part of the plasma, or the plasma condenses and reaches a state of thermodynamic equilibrium.

• Ideal superfluids exhibit infinite thermal conductivity, leading to exactly the effect Peierls described in his paper: "thermal waves" that slosh around in the material without ever settling down into an equilibrium state. (Real superfluids like helium II do show this effect, also known as "second sound," but the waves eventually die out because the fluid doesn't actually have infinite thermal conductivity.) They also have quantized vortices; as you spin up a superfluid, nothing changes until a threshold level is reached, after which a vortex forms and continues to exist until angular momentum is removed by an outside force.

Superfluids can exist in a non-equilibrium state because the entropy of a superfluid is zero for all microstates. Thus, there is no particular reason to expect a superfluid to have a constant temperature everywhere; particles in different regions can have different energies without impacting the entropy. (One could also make a case for long-range interactions in superfluids, especially considering that many are also Bose-Einstein condensates. Either way, though, the important point is that the usual statistical argument for objects reaching thermodynamic equilibrium, as given above, does not apply.)

What happens when you put a superfluid into thermal contact with an ordinary material: Ordinary materials and superfluids are able to coexist peacefully; superfluid helium, for example, can be safely stored in a very cold ordinary bottle as long as you seal it tightly. As discussed above, heat follows ordinary thermodynamics in the ordinary material but travels in thermal waves when it is transferred to the superfluid. Of course, adding too much heat to a superfluid simply causes it to boil away, once again resulting in a state of thermodynamic equilibrium.

Non-equilibrium thermodynamics is a very active area of research and it would be impossible to review the entire field in one answer, but hopefully this gives you some idea of the possibilities.

Edit to address the edit to the question: You're right, many-body localized systems are another class of non-thermalizable materials. Indeed, MBL systems are the only known class of real macroscopic materials that do not thermalize when left alone in isolation for a long time. Unfortunately, though, they're rather fragile because the crucial localization property is necessarily destroyed when the system is put into thermal contact with an ordinary material. Dynamically, since many-body localization is a quantum phenomenon, the process of heat transfer and relaxation of observables must be described in terms of the quantum state using the Lindblad equation:

$$\dot{\rho} = -i[H,\rho] + \gamma \sum_i \left( L_i \rho L_i^{\dagger} - \frac{1}{2}\{L_i^{\dagger} L_i, \rho \} \right)$$

where $$\rho$$ is the density matrix of the system, $$H$$ is the Hamiltonian, and the $$L_i$$ are jump operators representing the coupling to the ordinary material. Solving this equation is very difficult, but the Fischer et al. paper linked to above includes some tools for approaching the problem quantitatively.

In cases of weak thermal coupling, a limited amount of MBL-like behavior such as variable-range hopping can be preserved, but note that the system still eventually reaches ergodicity and thus thermodynamic equilibrium.

Caveat: MBL systems are an extremely hot research topic right now, so take the last few paragraphs with a grain of salt. Everything I've written might very well be obsolete in a few years.

• Thanks for your input. I've edited my question by adding more details, because I think your answer does not treat many-body localization, Anderson localization, etc. and the like, and the fact that many recent papers claim that some systems (not purely theoretical ones, but also ones realized in the lab.) do not reach thermalization. – AccidentalBismuthTransform Dec 10 '18 at 16:56
• @Kentucker_Filled_Turkey MBL systems are a fascinating topic and I added a brief discussion to my answer. The problem is that they can only exist in isolation, as any amount of thermal contact with an ordinary material eventually destabilizes the localization property. If you're interested in reading more, I'd recommend this paper and this paper as good starting points. – Thorondor Dec 10 '18 at 18:25
• It's worse than "destroyed by the bath" - even a single thermal inclusion can destroy localisation arxiv.org/abs/1707.00004. You're right though that there are a lot of people thinking about the problem though. – user213887 Dec 11 '18 at 2:22