A Quantum Mechanical Maxwellian Demon
A Quantum Mechanical Maxwellian Demon
Meir Hemmo[1] and Orly Shenker[2]
1. Introduction
J. C. Maxwell devised his so-called Demon in 1867 to show that the Second Law of thermodynamics cannot be universally true if classical mechanics is universally true. Maxwell’s Demon is a way of demonstrating that the laws of mechanics are compatible with microstates and Hamiltonians that lead to an evolution which violates the Second Law of thermodynamics by transferring heat from a cold gas to a hot one without investing work.[3] That is, the Demon is not a practical proposal for constructing a device that would violate the law of approach to equilibrium, but rather a statement to the effect that the Second Law of thermodynamics cannot be universally true if the laws of mechanics are universally true.
Since then, many attempts have been made to disprove Maxwell’s intuition. David Albert, in his book Time and Chance (2000, Ch. 5), used a phase space argument to demonstrate that a classical mechanical macroscopic evolution that satisfies Liouville’s theorem can be entropy decreasing. This is the beginning of a proof that Maxwell’s Demon is compatible with classical mechanics. However, Albert’s proof was incomplete since he did not complete the thermodynamic cycle. The missing link to complete the cycle required proving that erasing the Demon’s memory is not dissipative, that is one needs to show that the so-called Landauer-Bennett thesis is mistaken. In our (2010, 2011, 2012, 2013, 2016) we provided this missing link. By this we have shown that classical mechanics does not rule out a dynamical evolution that takes all the points in an initial macrostate of the universe to macrostates with a smaller Lebesgue measure, i.e. with lower total entropies.
The situation just described is in the context of classical mechanics. There is a vast literature (see Leff and Rex 2003) on the question of Maxwell’s Demon also in the context of quantum mechanics. Most of it is based on the classical Landauer-Bennett thesis which we have disproved (see the discussion by Earman and Norton 1999 of Zurek 1984).
In his Time and Chance (2000, Ch. 7) Albert proposes an approach to statistical mechanics in which the only kind of probabilistic statements are those derived from the micro-dynamics. His proposal replaces standard quantum mechanics with the dynamics proposed by Ghirardi, Rimini and Weber (1986) (see Bell 1987), which postulates that for any initial quantum state j0 of a system S, the state will evolve for some time Dt (probabilistically determined by the GRW temporal constant and size of the system) according to the Schroedinger equation to another quantum state j1, and then the evolved state will collapse spontaneously into a third state j2, where j2 is a Gaussian superposition of positions centered around point X, with probability fixed by the amplitude of X in j1. Although the GRW spontaneous localizations are in position, one can as usual also talk about other observables. Suppose that the eigenvalues {ai} of the observable A are such that eigenvalue a1 corresponds to a far-from-equilibrium state of the system, a2 is a little closer to equilibrium, and a3 corresponds to equilibrium. For every quantum state the probabilities for the {ai} eigenvalues are given by the Born rule. If, given a certain Hamiltonian H (discussed below) y(t0) has a high probability for a1 while y(t1) (t1 t0) has a low probability for a1 and higher ones for a2 and a3, then we say that y(t0) is a thermodynamic quantum state relative to H (Albert uses the term “thermodynamically normal”). If, given the same dynamical evolution, y(t1) gives a higher probability to a1, we shall say that y(t0) is an anti-thermodynamic quantum state relative to H. Since in the above description of states that are “thermodynamic relative to a Hamiltonian” no GRW collapses are involved, but only Schroedinger evolutions under a given Hamiltonian H, and since the Schroedinger equation is time-reversal invariant, then if there are thermodynamic quantum states relative to a given H, there are also anti-thermodynamic quantum states relative to the same H.
In his proposal Albert adds the following dynamical hypothesis to the GRW dynamics which we divide into two parts: (a) There exists a Hamiltonian H such that every initial y(t0) has high Born probability to collapse under the GRW dynamics to another quantum state y(t1) which is thermodynamic relative to H, regardless of whether or not y(t0) itself was thermodynamic relative to H. (b) H is the actual Hamiltonian that governs thermodynamic evolutions. The term “exists” in part (a) above means that such a Hamiltonian is (by hypothesis) possible according to fundamental physics, that it is compatible with fundamental physics to say that there could be such a Hamiltonian. Albert provides no proof of either (a) or (b), and his plausibility argument for their conjunction is based on the fact that observed systems are actually thermodynamic. It seems to us that it would be as compatible with the fundamental physics, as far as Albert’s argument is concerned, if, for example, the actual Hamiltonian of a given system would entail high probability for anti-thermodynamic evolutions, so that, for example, the system would be a Maxwellian Demon. Thus Albert’s proposal is not a proof but only a conjecture that there are no Demons (see also Sklar 2015; Uffink 2002; Callender 2016.)
It is an achievement of Albert’s approach that whichever Hamiltonian turns out to be the case in our world the probabilities for thermodynamic (or anti-thermodynamic) behavior are only the Born probabilities. This means that the probabilities for thermodynamic (or anti-thermodynamic) behavior involve no ignorance over quantum “initial” states. All the other theorems concerning thermodynamic behavior in both classical and quantum statistical mechanics are valid only for most of the initial quantum states (given a measure; see Shenker 2017). This also holds for our own proposals in Hemmo and Shenker (2001, 2003, 2005) in which the initial conditions lead to environmental decoherence, and according to prevalent decoherence models these results hold at best for “most” quantum states of the universe, given the right measures.
Albert’s dynamical hypothesis entails that a Maxwellian Demon is strictly impossible. In this sense, Albert’s proposal concerning the quantum mechanical foundations of statistical mechanics seems, at first sight, to be a way to exorcise the Demon, the same Demon he re-introduced to classical physics in the argument we mentioned above. In this paper we propose an alternative dynamical hypothesis called Maxwell’s Demon according to which the Second Law in its mechanical probabilistic version is false (even in a GRW world). We shall prove that our hypothesis is compatible with quantum mechanics. Of course the question of which Hamiltonian is actually the case in our world, Maxwell’s Demon or Albert’s dynamical hypothesis, is a question of fact, which we don’t address here. Our point rather is that given everything we know from experience and from theory the answer to this question is as yet open. It turns out that the situation in quantum statistical mechanics is the same as in classical statistical mechanics: Maxwellian Demons are compatible with the fundamental theory, and there is even no proof that the actual dynamics in our world is not Demonic.
Our discussion is in the framework of standard quantum mechanics, by which we mean quantum mechanics in von Neumann’s (1932) formulation, with no hidden or extra variables in addition to the quantum state and without the projection postulate in measurement (we set aside questions concerning the physical interpretation of such a theory). In addition we shall formulate our argument within a quantum theory with the projection postulate (and with no hidden variables, again, setting aside questions of interpretation).
2. A Quantum Mechanical Demon
We consider a thought experiment along the lines of Szilard’s and Bennett’s particle-in-a-box, in a quantum mechanical context (see our 2011 for a classical analysis). Since we are interested in the question of whether or not thermodynamics is consistent with quantum mechanics as a matter of principle, we consider the experiment in a highly idealized framework, disregarding practical questions (some of which we shall mention along the way).[4]
Consider the setup in the Figure. At a particle is placed in a box. At a partition is inserted exactly at the center of the box so that the particle is trapped in the left-hand side L or the right-hand side R of the box. At a measurement of the location of the particle, left or right, is carried out and the outcome of the measurement – 0 or 1, respectively – is registered in the memory state of the measuring device. At the partition is replaced by a piston (in accordance with the measurement outcome), which is subsequently pushed by the particle at . The piston is coupled to a weight located outside the box which is raised during the expansion. At the particle is again free to move throughout the box and the weight is at its maximal height. At the memory of the device is erased and returns to its initial standard state. The particle returns to its initial energy state by receiving from the environment the energy it lost to the weight. The cycle of operation is thus closed. By this last statement we mean that everything returns to its initial state except for the energy transfer from the heat bath (environment) to the weight (see our 2010).
We will now show that according to quantum mechanics this setup can be a Maxwellian Demon. We start by assuming a quantum mechanical dynamics without the projection postulate, i.e., a dynamics that satisfies the Schrödinger equation at all times. We will subsequently consider the projection postulate and comment on the GRW dynamics in our set up.
At the quantum state of the entire setup is the following:
(2.1) ,
where is the initial state of the particle in a one-dimensional box; is the initial standard (ready) state of the measuring device; is the initial state of the weight which is positioned at some initial height we denote by down; and is the initial state of the environment, which we assume does not interact with the particle or the weight (or the partition). In particular, this means that the quantum state of the particle and the partition do not undergo a decoherence interaction with the environment.
The initial energy state of the particleis some superposition of energy eigenstates which depend on the width a of the box where the amplitudes in give a definite expectation value for the energy of the particle. We assume for simplicity that the box is an infinite potential well so that all the energy eigenstates at the initial time have a node in the center of the box, and the quantum mechanical probability of finding the particle exactly in the center of the box is zero. In general the quantum state of the particle will be a superposition of energy eigenstates of the form: ,
with an eigenvalue
,
where n is an even number and m is the mass of the particle. In standard quantum mechanics such a superposition is not interpreted as expressing ignorance about the energy eigenstates of the particle. It is the fine-grained quantum-mechanical microstate of the particle and not a macrostate in any standard sense. However, our preparation applies to any such superposition and in this sense it is a macroscopic preparation in the usual sense of the term in classical statistical mechanics (see Appendix).
is the standard ready state of the measuring device which is an eigenstate of the so-called pointer observable in this setup. For simplicity we take a spin-1 particle to represent the measuring device with in the z-direction. The two other spin-1 eigenstates, and in the z-direction, correspond to the two possible outcomes of the measurement. Later we consider pointer states that are only approximately orthogonal.
At we insert a partition in the center of the box, where the wavefunction is zero, so that the expectation value for the energy remains completely unaltered. This is highly idealized, to be sure, creating many technical questions. For example, given the quantum mechanical uncertainty relations, one cannot insert the partition exactly at a point since in this case its momentum would be infinitely undetermined. To avoid this, one may assume that the partition and consequently also the particle are fairly massive, but that nonetheless they are kept in isolation from the environment. Alternatively, if there is a certain amount of decoherence, we may assume that the degrees of freedom in the environment are controllable in the subsequent stages of our experiment. Of course if the partition is massive, the idealization of zero width may also be problematic and consequently the wavefunction may change when the partition is inserted. What we need is a way of inserting a partition in a way that will not change the expectation values of the energy of the particle. In this sense we assume here that the effects of the above issues are negligible and we continue with our idealization. We are not concerned here with the experiment’s feasibility, but rather with its consistency with thermodynamics.
We therefore take it that ideally at immediately after the insertion of the partition in the center of the box the quantum state of the setup becomes:
(2.2) ,
but the expectation value of the energy of the particle is unaltered (On each side of the partition the superposition involves both odd and even eigenstates, but the width of the box is now a/2.) Although the expectation value for the particle’s energy does not change in this interaction, the wavefunction of the particle now becomes a superposition of two components: