The issue is that one may enter an upper level course on statistical and thermal physics with only an appreciation of the classical macroscopic view never having attached quantum mechanics to anything thermodynamic. Statistical mechanics makes accurate predictions about systems which may contain 1023 or more particles. A sense of disbelief may be partly responsible for the widely held, but, incorrect view that statistical mechanics has to be difficult. A true understanding of entropy only comes from the microscopic treatment of the quantum approach made by statistical mechanics. The more fundamental, issue is that classical statistical mechanics is internally inconsistent, and can only be made to work by the importation from quantum theory of non-classical ideas such as indistinguishability. Gibbs’ paradox, over which the classical books agonize at length, vanishes into thin air in the quantum approach.
The issue is that one may enter an upper level course on statistical and thermal physics with only an appreciation of the classical macroscopic view never having attached quantum mechanics to anything thermodynamic. Statistical mechanics makes accurate predictions about systems which may contain 1023 or more particles. A sense of disbelief may be partly responsible for the widely held, but, incorrect view that statistical mechanics has to be difficult. A true understanding of entropy only comes from the microscopic treatment of the quantum approach made by statistical mechanics. The more fundamental, issue is that classical statistical mechanics is internally inconsistent, and can only be made to work by the importation from quantum theory of non-classical ideas such as indistinguishability. Gibbs’ paradox, over which the classical books agonize at length, vanishes into thin air in the quantum approach.
Many quantum predictions seem to defy “common sense” and appear beyond the comprehension of some of the greatest human minds. Richard Feynman states that “ I think I can safely say that nobody understands quantum mechanics ” in his book entitled The Character of Physical Laws . 1 Many pioneers of quantum theories have constructed examples to highlight this situation. These examples are now known as quantum paradoxes . Moreover, the common understanding that only microscopic systems behave quantum mechanically is incorrect. Macroscopic systems, e.g., superconducting systems and Bose-Einstein condensate, also exhibit quantum behaviour. These systems do not satisfy the properties of orthodox quantum systems. 2
This book presents an innovative unified approach to the statistical foundations of entropy and the fundamentals of equilibrium statistical mechanics. These intimately related subjects are often developed in a fragmented historical manner which obscures the essential simplicity of their logical structure. In contrast, this book critically reassesses and systematically reorganizes the basic concepts into a simpler sequential framework which reveals more clearly their logical relationships. The inherent indistinguishability of identical particles is emphasized, and the resulting unification of classical and quantum statistics is discussed in detail. The discussion is focused entirely on fundamental concepts, so applications are omitted. The book is written at the advanced undergraduate or beginning graduate level, and will be useful as a concise supplement to conventional books and courses in statistical mechanics, thermal physics, and thermodynamics. It is also suitable for self-study by those seeking a deeper and more detailed analysis of the fundamentals.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.
Many-body theory stands at the foundation of modern quantum statistical mechanics. It is introduced here to graduate students in physics, chemistry, engineering and biology. The book provides a contemporary understanding of irreversibility, particularly in quantum systems. It explains entropy production in quantum kinetic theory and in the master equation formulation of non-equilibrium statistical mechanics. The first half of the book focuses on the foundations of non-equilibrium statistical mechanics with emphasis on quantum mechanics. The second half of the book contains alternative views of quantum statistical mechanics, and topics of current interest for advanced graduate level study and research. Unique to textbooks on this subject, this book contains a discussion of the fundamental Gleason theorem. Quantum entanglements are treated in application to quantum computation and the difficulties arising from decoherence. The relativistic generalization of the Boltzmann equation is derived, and modern transport applications to reservoir ballistic transport are developed.
It is widely accepted that the papers by Bose in 1924 and by Einstein in 1924-1925 definitely opened the path to
the quantum concept of ‘indistinguishability’. Since then, distinction in physics between ‘identity’ and
‘indistinguishability’ of particles has come to be required.
However, even before the rise of quantum physics, Maxwell, Boltzmann and Gibbs, among others, had to tackle
the difficulties and paradoxes that appear when counting identical particles. Later, after Sackur’s and Tetrode’s
first quantum derivation of the canonical partition function of an ideal gas, much confusion arose around its
dependence on the number of molecules in the system. Many other physicists, such as Planck, Lorentz or
Schrodinger took part in those debates. In 1921, Ehrenfest and Trkal devoted a whole paper to this subject.
Clearly, in the days before Bose’s seminal work, embroiling and misunderstanding were commonplace.
In considering the emergence of the distinction between ‘identical’ and ‘indistinguishable’ particles, we will go
through the so-called statistical route to quantum mechanics. The first developments in statistical mechanics
became immediately confused and entangled with the irruption of quantum physics. In the first quarter of the
twentieth century the mutual influence between both disciplines was profuse.
We will pay special attention to the story of the so-called ‘Gibbs Paradox.’ We will argue that what is now often
known by this name in some textbooks of statistical mechanics did not exist as such in those foundational days.