THE PUZZLE CONCERNING THE CONCEPT OF PROBABILITY IN SM

Throughout the book we have been preoccupied with a single simple scenario. We consider now an n-particle system M that is in a state st at t. We want to know whether st has a certain property A. The question that we are asking has a perfectly clear physical meaning. Nevertheless, the information that the answer requires may not always be available to us. It may be the case that the experiment needed to answer the question is too expensive or difficult to perform. It is also possible that there are theoretical reasons why we cannot perform the experiment, or, simply, the experiment has not been performed yet. In all of these cases the need for probabilities arises. We need to calculate the probability P(st ∈ A) and use our results to derive predictions concerning M. But, even if there is little doubt that we constantly use probabilities, there is a basic problem concerning the interpretation of the probabilities. It is difficult to define probabilities as observable parameters that depend only on the physical properties of M. This difficulty gives rise to a philosophical puzzle: If probabilities are not physical parameters that we discover by methods of observation and measurement only, how can we justify our willingness to be guided by them? How can we explain the utility of probabilities? In the previous chapters, we reviewed several attempts to solve this problem.

THE ATOMISTIC CONCEPTION IN LIGHT OF THE RESEARCH ON THE BROWNIAN MOTION

In 1909, William Ostwald, formerly one of the most vocal opponents of atomism, admitted that, “recently we have come into possession of experimental proof of the discrete or granular nature of matter, for which the atomic hypothesis had vainly sought for centuries, even millennia.” The proof consisted of the “isolation and counting of gas ions” and “the agreement of Brownian movements with the predictions of the kinetic hypothesis.” Ostwald commended J. J. Thomson and J. Perrin for the former and later parts of the experimental work, respectively.

The interest in atomism and the willingness to consider adopting it as a new theory of matter was not confined to the community of physicists. Due to the efforts of Perrin, the topic was discussed and debated in a wide range of semipopular publications and lectures. The result was a fairly wide-ranging acceptance of atomism, a change of world view that deserves to be called a conceptual revolution. On first blush, the willingness of the educated world to reverse its opinion on the issue of atomism seems rather inexplicable. Only nine years earlier, Boltzmann was practically the only physicist on the continent who espoused the atomistic conception of matter. One should remember, though, that the climate of opinion at the time, especially on the continent, favored new and extreme views. Indeed, both Mach's radical phenomenalism and Perrin's strange new world of “real” atoms exerted considerable fascination at the time.

Probabilities have a puzzling character within the context of classical physics. Here, unlike in quantum mechanics, the use of probabilistic reasoning is not based on the existence of indeterminacy or objective lawlessness. Indeed, indeterminacy, in general, has no place in classical physics. In the very foundation of classical physics we find the assumption that, given the precise state of the world in one instance, the laws of physics determine its future states completely. Therefore, the introduction of probabilities into classical physics presents a problem. How are we to interpret statements from statistical physics whose abstract form is “The probability of A is p.”? Do probabilistic statements form an integral part of the description of the physical world? Do they merely reflect our ignorance with respect to the precise state of the world? Can they be deduced from nonprobabilistic statements? Can they be understood in terms of frequencies? This book is dedicated to the study of these questions.

Readers who are not very familiar with statistical mechanics may want a few examples of physical statements using probabilities. The following is a list of statements, written in plain English, expressing physical truths (classified under the heading of thermodynamics) whose precise formulation requires probabilistic concepts:

1. Water freezes at zero centigrade.

2. When two bodies are in contact with one another, the heat flows from the warmer body to the colder one.

3. When you heat a container full of gas, the pressure on the walls of the container will increase.

4. […]

THE INTRODUCTION OF STATISTICS INTO PHYSICS

In 1859, J. C. Maxwell presented his “Illustrations of the Theory of Gases” to the British Society for the Advancement of Science. In the paper he applied the theory of errors to the problem of finding “the average number of particles whose velocities lie between given limits, after a great number of collisions among a great number of particles.” The conclusion that Maxwell reached was that “the velocities are distributed among the particles according to the same law as the errors are distributed in the theory of the method of least squares.” In the second part of the paper, Maxwell distinguishes between the motion of translation of the system as a whole and the motion of agitation. The latter is characterized as a state where “the collisions are so frequent that the law of distribution of molecular velocities, if disturbed in any way, will be reestablished in an appreciably short time.” This last statement may serve to explain why “Illustrations” is considered to be the beginning of modern statistical mechanics.

The discussions on the theory of errors itself started more than 100 years earlier. Simpson discussed the idea of averaging the results of various observations in 1757. Bernoulli introduced the assumption that large errors should be regarded as less probable than small errors in 1777, and Lagrange introduced the least-squares method in an undated manuscript written in the 1770s. Other important contributors to the early stages of the theory of errors are Gauss and Legendre, but it was Laplace who systematized the theory and showed how to apply it in various contexts.

IS THE PROBABILISTIC FRAMEWORK OF EQUILIBRIUM SM TOO RESRICTIVE?

The main objective of this book has been to analyze the concept of probability as it is defined and used in the context of statistical mechanics. Other topics that arise in the discussion on the foundations of statistical mechanics have been given much less attention and emphasis. A case in point is nonequilibrium statistical mechanics (henceforth NESM) in general and the issue of irreversility in particular. These issues are, no doubt, extremely important, so much so that, from certain points of view, most of the nontrivial aspects of statistical mechanics necessitate the introduction of the framework of NESM. What distinguishes this framework, some writers maintain, is the presence of time-dependent probabilities.

In this appendix we shall argue that the concept of probability in NESM is still ill understood. We maintain that those aspects of NESM that can be given a coherent and rigorous treatment can be discussed in a framework that is only a slight generalization of the framework of equilibrium statistical mechanics. Conversely, those aspects of NESM that transcend the equilibrium framework do not have, as of yet, secure foundations. The main object of our discussion will be the conception of NESM that was developed by the so-called Brussels school. We shall argue that the beautiful conceptual analysis of NESM that the members of this school developed do not yield, at present, a substantial generalization of the framework of equilibrium statistical mechanics.

THE NATURALIST CONCEPTION OF PHYSICS

In the previous chapters we were occupied with the dual theme of the centrality and the problematic character of the frequentist's conception of probabilities in the context of SM. On the positive side, the frequentist's conception links the probabilities directly with the laws of physics, because the calculation of frequencies is effected through an integration along the trajectories formed by the successive solutions of the equations of motion. However, because there are in general many such trajectories, to each there corresponds a different value of the frequencies.

We saw in Chapter 3 how, in the case of a system with a single degree of freedom, there is only a single possible trajectory that the system may follow, and hence a single way of calculating the frequencies. In this case, which is the “best-case scenario” from the frequentist's point of view, it is possible to reduce the probabilities to relative frequencies. However, we have seen that this case is exceptional from a physical point of view, and so we have determined it necessary to reformulate the frequentist's standpoint.

One reaction to the failure of the simple frequentist's conception is the idea that, rather than calculating the exact frequencies along individual trajectories, we should calculate the average of the frequencies among the members of a set of trajectories. Having done that, we may ask whether these averages are different relative to the choice of different sets of trajectories (ergodic systems have the property that the different sets will yield the same averages).

Introduction

In spite of its wonderful agreement with every experiment performed so far, standard quantum theory (SQT) fails to describe events and to define the circumstances under which such events occur [171]. It is the linearity of Schrödinger evolution that lies, as we have seen, at the root of the problem. A state vector always evolves to become a linear superposition of the states corresponding to several possible outcomes of a measurement, and it is only when an experiment is actually done that one of these possible outcomes is realized at a particular instant. Thereupon the state has to be changed to the one corresponding to the particular outcome in order to follow its subsequent evolution. This additional information is not contained in the theory and has to be obtained from outside. This means that the theory is unable to predict when an event will occur. All it can predict is that if an event occurs, the possible outcomes and their probabilities are such and such. Since events do occur in every experiment, there is something missing from the theory.

Since a state vector can be written as the linear sum of a complete set of basis states and these basis states can be chosen in a number of ways, each of which corresponds to a different set of outcomes, the theory also fails to tell us how to choose the preferred basis.

Introduction

Ever since 1916/17 when Einstein argued [45] [46] that spontaneous emission must occur if matter and radiation are to achieve thermal equilibrium, physicists have believed that spontaneous emission is an inherent quantum mechanical property of atoms and that excited atoms inevitably radiate. This view, however, overlooks the fact that spontaneous emission is a consequence of the coupling of quantized energy states of atoms with the quantized radiation field, and is a manifestation of quantum noise or of emission ‘stimulated’ by ‘vacuum fluctuations’. An infinity of vacuum states is available to the photon radiated by an excited atom placed in free space, leading to the effective irreversibility of such emissions. If these vacuum states are modified, as for example by placing an excited atom between closely spaced mirrors or in a small cavity (essentially the Casimir effect), spontaneous emission can be greatly inhibited or enhanced or even made reversible. Recent advances in atomic and optical techniques have made it possible to control and manipulate spontaneous emission. A whole new branch of quantum optics called ‘cavity QED’ has developed since 1987 utilizing these dramatic changes in spontaneous emission rates in cavities to construct new kinds of microscopic masers or micromasers that operate with a single atom and a few photons or with photons emitted in pairs in a two-photon transition.

Introduction

In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons.

Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [a, b]. Three kinds of time have been defined in this context. One, called the transmission time τT(a, b), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τR(a, b) is the average time spent within the barrier region by the particles that are eventually reflected.

Introduction

The quantum theory of atoms and molecules had its origin in the famous 1913 paper of Bohr [186] in which he suggested that the interaction of radiation and atoms occurred with the transition of an atom from one stationary internal state to another, accompanied by the emission or absorption of radiation of a frequency determined by the energy difference between these states. These transitions came to be known as ‘quantum jumps’. Since all experiments until the late 1970s used to be carried out with large ensembles of atoms and molecules, these jumps were masked and could not be directly observed; they were only inferred from spectroscopic data. In fact, with the advent of quantum mechanics they eventually came to be regarded as artefacts of Bohr's simple-minded semi-classical model. But with the availability of coherent light sources and single ions prepared in ion traps [187], [188], and optically cooled [189], [190], the issue has been reopened with the experimental demonstration of quantum jumps in single ions [191], [192], [193]. These experiments have also opened up the contentious issue of wave function collapse or reduction on measurement, as they can be regarded as making collapse visible on the oscilloscope screen [191].

Evidence of the discrete nature of quantum transitions in a single quantum system had been accumulating from the observation of photon anti-bunching in single-atom resonance fluorescence [194], the tunneling of single electrons in metal–oxide-–semiconductor junctions [195] and spin-flips of individual electrons in a Penning trap [196].