Principals aspectes que menys agraden vers la competició

We conclude with the following remarks:

1. If one includes the collapse of the wave function in the mechanics of the quantum world, then there is no reversibility even on the microscopic level. One cannot demeasure things. One can however associate fun- damental importance to that kind of microscopic irreversibility only up to the point where one considers the collapse as a fundamental input of quantum mechanics rather than as an effective device for all practical purposes.

2. If one understands the collapse of the wave function within the standard statistical interpretation of quantum mechanics, it is appropriate to ask for statistical reversibility, i.e., in terms of probabilities of histories. In that case, it is quite similar to the situation ofmicroscopic reversibility

ordetailed balancefor transition probabilities as obtained also classically from Hamiltonian dynamics. Statistical reversibility is satisfied within the standard theory of quantum mechanics.

3. Thermodynamic irreversibility is an emerging property in macroscopic behaviour for which the reasons are basically unchanged in the transition from classical to quantum dynamics. In particular, such macroscopic irreversibility can be expected and sometimes is obtained on appropriate time-scales for quantum unitary evolutions with respect to typical initial data.

As a final remark, we like to add that a fully mechanically reversible version of quantum mechanics exists, which is, for all we know, empirically equiva- lent with standard quantum mechanics: the Bohmian equations of motion.12

However, one does not necessarily need to resort to a modification of standard quantum practice; in a practical sense, standard quantum theory can account for both the macroscopic irreversibility and statistical reversibility, as present in our daily experience.

12_{They consist of an equation for the wave function, nothing else than the Schr¨odinger}

equation, complemented with an equation for the positions of all particles. We refer to reference [1] for a discussion on the issue of retrodiction in Bohm’s theory.

An extension of the Kac

ring model

In the previous chapters, it was mentioned that aspects of the fundamental irre- versibility of the measurement process in quantum mechanics do not necessarily propagate to the macroscopic world. Now, we turn to the opposite question: does the reversibility of the quantum unitary evolution prevent macroscopic ir- reversibility from typically being observed? This chapter shows via a toy model how the mechanisms responsible for thermodynamic irreversibility from a clas- sical, reversible dynamics, also apply in the quantum case. This work was previously published in the paper [27].

5.1

Relaxation to equilibrium

The Kac ring model was introduced by Mark Kac1 _{[66, 67] to clarify how}

macroscopic irreversible behaviour can be obtained from an underlying mi- croscopic, reversible dynamics. It explains via a simple model some of the conceptual subtleties in the problem of relaxation to equilibrium as for exam- ple are present in the derivation and the status of the Boltzmann equation for dilute gases. In particular, the Kac dynamics shares some basic features with a Hamiltonian time evolution like being deterministic and dynamically reversible. We go into more detail in section 5.2.

The aim of this chapter is to extend the Kac dynamics to a unitary evolution on a finite quantum spin system, see section 5.3. Again, the dynamics remains far from realistic, but it allows a precise formulation and discussion of some features of relaxation to equilibrium for a quantum dynamics. That is espe- cially useful and relevant as, in the quantum domain, the problem of relaxation is beset with even greater conceptual difficulties. In our framework, relaxation

1_{Pronounced [katz] and not [katsj], as vigorously denounced by S. Pirogov in a private}

to equilibrium becomes visible if one can select a small number of macroscopic variables that typically evolve via autonomous deterministic equations to take on values that correspond to equilibrium. Here, “typical” refers to a law of large numbers with respect to the initial data. Paradoxes are avoided by tak- ing seriously the fact that relaxation is a macroscopic phenomenon, involving a huge amount of degrees of freedom whose evolution is monitored over a re- alistic time-span.

This model was already discussed by Dresden and Feiock [35], as we have only recently learnt. Their discussion was limited to the mathematical properties of the extended Kac model, and they were surprised by the general oscillatory relaxation to equilibrium in the quantum case. A few years later, Tavernier [111] reformulated the Dresden-Feiock quantum variant and pointed out that the master equation approach introduced by Dresden and Feiock was flawed. With respect to their work, our work as presented in this chapter adds the im- portant discussion on the existence and status of a H-theorem, which will allow us to understand the oscillatory motion of the Dresden-Feiock formulation and bring the model back to its original context of Boltzmann’s mechanical deriva- tion of the second law, see section 5.2.1.

One should also keep in mind that relaxation to equilibrium goes beyond questions ofreturnto equilibrium, see [98], which are mostly related to stability of equilibrium states. In our set-up, the initial state is not merely a slightly disturbed equilibrium state, but can be far from equilibrium. A more general introduction to that and various related problems can be found in the book [106].