DESCRIPCIÓN DE LAS ACCIONES REALIZADAS DURANTE EL PERIODO QUE SE REPORTA

This chapter offers an alternative to the traditional, dynamical approach to entropy production and starts the statistical approach for quantum systems. The first step is to be quite clear as to the relation between time-reversal and entropy production. As in the classical case, we consider this important for constructing nonequilibrium statistical mechanics. Specifically, we hope that the characterizations of (3.5) and (3.15) will be useful for deriving general fluctuation and response identities going beyond close to equilibrium.

We add some general remarks.

1. For reversible systems, the entropy production is a state function. It does not depend on the actual path but can be written as a difference of the same quantity evaluated at the initial and the final time, see equation (3.5) or (3.15). This can be different for driven or irreversible systems. A similar treatment for such systems in the classical regime was

discussed in reference [80]. One of the main consequences of the identities that correspond there to Rµˆ(ω) (3.15) is that certain symmetries in the

fluctuations of the entropy production rate are easily established. How this is realized in a specific quantum model is discussed by De Roeck and Maes [29]. For more general microscopic dynamics this was studied rigorously by Derezi´nski and De Roeck [30, 33, 34].

2. Nothing of the previous mathematical identities depends on the assump- tion that the system is large, of macroscopic size, containing a huge number of particles. Of course, to meaningfully discuss macroscopic variables or macrostates and their trajectories, one has in mind a clear separation between micro- and macroscales, but the results of sections 3.2 and 3.3 are true without. Yet again, their interpretation as thermo- dynamic entropy production and the relation with the second law can only be made when dealing with macroscopic systems, see also section 2.4.2.

3. The use of the projectionsPαin the construction of the path-space den-

sity matrix of section 3.3 refers to the so-called von Neumann measure- ments. One can imagine morefuzzy(or Kraus) measurements with more rounded-off macrostates. In that case, the corresponding decomposition of unity is

X

α

X⋆

αXα=1.

The treatment above was restricted to the case Xα =PαU. This ex-

tension is very much related to the dynamics of open systems, see, e.g., [3]. The involvement of measurements interrupting the unitary evolution already points to the interaction with the outside world. The question of isolation, even as an idealization, of a quantum system is much more subtle than for a classical system.

Aspects of quantum

irreversibility and

retrodiction

This chapter reviews a number of points with respect to time-reversal, re- versibility and irreversibility in quantum systems, that are not always empha- sized in the literature on the subject. The work in this chapter was previously published in the papers [60, 61].

4.1

The trouble with irreversibility

The discussion on time-reversal in quantum mechanics exists at least since Wigner’s paper [128] in 1932. Wigner was the first to explain how the op- eration of time-reversal can be implemented in a quantum system through complex conjugation of the wave function. If and how the dynamics of the quantum world is in fact time-reversible has been the subject of many contro- versies.

Some have seen quantum mechanics as fundamentally time-irreversible, see for example von Neumann [125, p. 358]:

Just as in classical mechanics therefore, [the unitary evolution] does not reproduce one of the most important and striking prop- erties of the real world, namely its irreversibility, the fundamen- tal difference between the time direction, “future” and “past”. [The collapse of the wavefunction] behaves in a fundamentally different fashion: the transition

U −→U′ = ∞ X

n=1

is certainly not prima facie reversible. We shall soon see that it is in general irreversible, in the sense that it is not possible in general to come back from a given U′ _{to its U by repeated}

applications of any process [of the unitary evolution].

– John von Neumann

Others have seen in the fundamental irreversibility of quantum mechanics the ultimate cause of time’s arrow and second law behaviour. In his best-selling book [94, chapter 8], Roger Penrose argues as follows:

It seems that we are indeed left with the conclusion that a “cor- rect quantum gravity” theory must be a time-asymmetric theory. [...] How is it that we can get a time-asymmetric theory out of two time-symmetric ingredients: quantum theory and general relativity? [...] If we wish to calculate the probability of a past state on the basis of a known future state, [i.e. if we want to do retrodiction], we get quite the wrong answers if we try to adopt the standard R _{procedure [of collapse of the wave function]. It}

is only for calculating the probabilities of future states on the ba- sis of past states that this procedure works - and there it works superbly well! It seems to me to be clear that, on this basis, the procedure R_{cannot be time-symmetric.}

– Roger Penrose

However, Erwin Schr¨odinger disagrees: his reply in 1950 to a letter from Max Born [102] is:

Irreversibility. It may seem an audacity if one undertakes to proffer new arguments in respect of a question about which there has been for more than eighty years so much passionate con- troversy, some of the most eminent physicists and mathemati- cians siding differently or favouring opposite solutions – Boltz- mann, Loschmidt, Zermelo, H. Poincar´e, Ehrenfest, Einstein, J. von Neumann, Max Born, to name only those who come to me instantly. But, to my mind, in this case, as in a few others, the “new doctrine” [of quantum mechanics] which sprang up in 1925/26 has obscured minds more than it has enlightened them. It is sometimes believed that only quantum mechanics, or some processes of thought borrowed from it, give the final clue to the problem. I wish to show here that this is wrong ...

– Erwin Schr¨odinger

Not so long ago, we read about yet another project in Physicalia, [26]: to ex- tend quantum mechanics into new fundamentally irreversible equations, thus proposing a new theory giving “... une description fondamentale irr´eversible de tout syst`eme physique”.

We will argue in chapter 5 that even in the standard formulation of quantum mechanics, using von Neumann’s collapse postulate, the microscopic reversibil- ity is compatible with the thermodynamic irreversibility. We have already ex- plained in sections 1.2.2 and 1.3.2 what is meant by mechanical reversibility and how it applies for the free evolution in quantum mechanics. The main conclusion there was that symmetry with respect to time-reversal amounts to having identical mechanical laws for prediction and for retrodiction. If the evolution is not free, i.e., it is interrupted by measurements, the quantum formalism is challenged by the problem of retrodiction. Related to that is the notion of statistical reversibility which is very similar to what is more commonly known as the condition of detailed balance, at least for stochastic processes describing the spatio-temporal fluctuations in equilibrium.