Quantum Thermodynamics

A quantum system can have many states, differing in (typically microscopic) internal details, in which the properties amenable to external observation are substantially the same. When the externally observable properties are exactly the same, such states are said to be (exactly) degenerate; but limitations on precision of measurement mean that even (sufficiently accurate) approximate equality of external observables can lead to such (approximately degenerate) states being effectively indistinguishable by external observation. Such degeneracy among possible states is particularly common when the system is made up of a large number of identical sub-systems. Such a system may have some extreme states that are uniquely (or nearly so) identified by their gross observable properties, with states closer to such extremes being subject to less degeneracy.

Our knowledge of the system comprises its measured properties; when there are many states consistent with those properties, we have no knowledge of which such state the system is in. A given description of the internal details of the system many give us a discrete set of states consistent with our measurements; but, in quantum mechanics, these states can be mixed in arbitrary proportions (and relative phases) to produce infinitely many possible states with the measured properties; and a different description of the internal details would be apt to produce a different set of discrete states, each of which would be expressible (in terms of the first description's states) by some such mixture. The official term for such mixtures (when properly formalised) is superpositions. The system itself cares not how we describe it – only how we measure it – so is in no way constrained to be in a state which fits nicely with our description. However, this very lack of constraint implies that the system is most probably in one of the states as disinterested as possible in how they are described – and there are plenty of such states. These most probable allowed states can be represented as superpositions, of the allowed states of a given description, in specific proportions (with arbitrary phases) determined by the observed properties of the system and the details of the chosen description.

It is usual for one of a system's observable properties to be energy; and it is usual for the system's state with lowest energy to be quite isolated – there will be little or no degeneracy in this ground state. While an isolated system will generally change in such manner as will reduce its energy, hence cause it to decay towards its ground state, to do so it must shed the surplus in one manner or another – if it can do that, it's not truly isolated, but our standard meaning of isolated only addresses impact from outside as opposed to impact on its surroundings; the universe is leaving the system alone, but the system is not obliged to leave the universe alone. However, in the real world, just as the system tends to lose energy to its surroundings, it surroundings will be tending to lose energy to theirs – of which the system is a part. Thus the system will settle on an equilibrium in which it is shedding energy (by decaying into lower-energy states) as fast as it is being inundated with energy. This need not be the system's ground state; and the more surplus energy the system has, typically, the more states available to it with (at least approximately) that energy.

Thus, in practice, real material systems generally have observed properties which leave ample ambiguity as to (the internal details of) their state, hence ample scope for random superposition of possible states. When two such systems are in a position to exchange energy, determining which will shed more to the other than it gets back depends on understanding their respective states and the effect, on each, of gaining and losing energy. The study of such questions is known as thermodynamics. One simple case that's fairly tractable to analyse is a model of a solid in terms of each molecule vibrating in three directions, modelling each direction's vigration as an independent simple harmonic oscillator.

In so far as the internal states of molecules support a meaningful temperature, one may study how they exchange energy – both internal and gross (external) kinetic – in collisions: this we may understand as micro-thermodynamics. On the larger scale, we can consider how bodies, each comprising many molecules, exchange energy when they interact: this macro-thermodynamics is what is usually meant by thermodynamics.

The observable formerly known as heat

The first law of thermodynamics says heat is (a form of) energy (also known as work or mass). This is a standard observable of quantum systems; and the formalism for handling quantum superpositions of highly degenerate systems is all predicated on the observables of the sysem; so let's see what rôle heat gets to take in the solution.


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