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Foundations of Classical and Quantum Statistical Mechanics. Volume 19 in International Series in Natural Philosophy. Book • Edited by: D. TER HAAR.

**Table of contents**

- Statistical Mechanics
- STATISTICAL MECHANICS / — University of Bologna
- Statistical mechanics
- Course contents
- Foundations and History of Statistical Mechanics

The EPR paradox involves correlations between conjugate variables such as position and momentum of widely-separated, incommunicado particles, and it was designed to demonstrate that quantum mechanics was incomplete, but it was cast in the form of an abstract Gedankenexperiment with little possible connection to experimental reality. What Bell did was to treat correlations between widely-separated particles statistically and to derive an inequality that places an upper bound on possible specific classical statistical correlations but which allows quantum statistical correlations to exceed this upper bound under certain conditions.

There have been numerous refinements to and variants on Bell's original inequality, but perhaps the most straightforward is what is known as the CHSH inequality [18], which was specifically constructed with experimental testing in mind.

## Statistical Mechanics

It will be used for the following brief discussion for more details, see [15]. Alice and Bob, the standard information theory cartoon characters, are separated by an effectively infinite distance i. Pairs of correlated particles, say, an electron pair in a singlet state or a photon pair having opposite polarizations, are prepared, and one particle from each pair is sent to Alice, the other to Bob, who then proceed to make independent binary measurements on their particles.

And Q could be measurements with respect to a vertical axis and R with respect to a skewed axis. Bob's corresponding measurements on each of his particles are termed S and T.

After making many, many measurements in order to achieve statistical significance, Alice and Bob get together to compare statistical notes. The quantity of interest for them to compare is.

## STATISTICAL MECHANICS / — University of Bologna

Either way,. In other words, classical mechanics through this inequality places an upper bound on the possible statistical correlations for specific combinations of results obtained by presumably independent and randomly chosen measurements made on widely-separated particles. The parallel derivation for quantum mechanical qubits is similar, but the pairs are assumed to be produced in the entangled Bell singlet state,.

The first qubit from each ket is sent to Alice and the second to Bob, who proceed to make measurements as before, but on the following combinations of observables:. Here X and Z are the "bit flip'' and "phase flip'' quantum information matrices, corresponding to the Pauli s 1 and s 3 spin matrices, respectively. This leads to the CHSH quantum mechanical combination,. This is a larger value than was obtained in the classical inequality, which means that within the framework of this entangled system , quantum mechanics can produce greater statistical correlations than classical mechanics.

In other words, classical systems must obey Bell-type inequalities, whereas quantum systems can at times violate them. During the last several decades numerous Bell-type experiments have been made, as covered extensively by Bertlmann and Zeilinger [19], and they have consistently ruled in favor of quantum mechanics.

An example of this might be the following: Two electrons are emitted in a spin-singlet state, where their individual spin directions are unknown according to the Copenhagen interpretation, actually undefined until an observer makes measurements on them , but whatever direction the spin of one points, the spin of the other must be in the opposite direction. When Alice randomly measures the direction of her electron, say, with respect to a z axis and gets , this information is instantaneously conveyed somehow to Bob's electron, whose wave function immediately reduces to.

The small fraternity of "Bell inequality physicists'' is sharply divided into two camps, those who support Bell's theorem that violation of the inequality rules out classical behavior and those who strongly oppose it. Among the more compelling arguments against Bell's theorem is that it ignores relativistic and QED effects, so is simply no applicable to systems involving electron spin or photon polarization [20].

I adopt an alternative approach in this paper. In a sense this is attacking a theorem, which according to its detractors has already been shown to be irrelevant to many of the experimental results to which it has been applied. However, in a sense it is also a simpler alternative explanation, and, more important, it points out a slippage in the use of statistics that is rife in interpretations of quantum mechanics. In this nonlinear parallel, the problem lies not in the quantum mechanics derivations but in the classical ones.

Instead of a contest between quantum and classical mechanics, it is one between correlated versus uncorrelated statistics.

## Statistical mechanics

In the so-called classical derivation above, the particles were presumably prepared in correlated pairs, but these correlations were then tacitly ignored, whereas the use of the Bell entangled state necessarily retained the complete correlations. A codification of correlated statistics was introduced by Tsallis and his coworkers [21,22], when they introduced "nonextensive'' meaning nonadditive thermodynamics. Correlations can be expressed by a generalized, nonextensive entropy,. When the exponent q termed the "entropic index'' has a value of 1, this generalized entropy reduces, as it should, to the standard Boltzmann entropy,.

As q differs more and more from 1, the deviation from standard distributions becomes greater and greater, indicating that "long-range'' correlations become more and more important. In a nonlinear system, when such correlations are present, the entropy becomes nonextensive, with the total entropy becoming.

This concept of nonextensive entropy has found widespread applications in classical systems, ranging from the velocity distributions in tornadoes a good example of an emergent system to the energy distributions of cosmic rays, and, of course, it plays an important role in biological evolutions. It has also been the subject of several recent international conferences [23]. Pertinent to our discussion are the recent studies of systems "at the edge of quantum chaos [24].

Nevertheless, the crossover into classical would be similar for nonlinear systems, and both systems need contain long-range correlations. It thus seems clear that classical systems can indeed exhibit behavior in which long-range correlations play an important role. This does not necessarily imply long-range forces or action-at-a-distance, as has been shown clearly in the behavior of cellular automata and emergent systems.

This includes versions that do not directly involve inequalities, such as the GHZ formulation, [26] but which require statistical arguments when experimental justification is sought.

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This is covered in more detail in a previous paper. With a value of q in the vicinity of 2, one would expect something closer to an exponential rather than a Gaussian statistical distribution. If so, then Bell-type arguments are moot in ruling out the existence of "local reality'' in quantum mechanics. Because of space limitations, it is not possible to cover the other nonlinear parallels in any detail.

### Course contents

However, the two covered in the previous sections should raise pertinent questions in the mind of the reader, even if they lack quantitative proof. Indeed, the basic object of this paper is to raise such questions rather than to attempt convincing, quantitative proof. In a sense we are still in a "quasi-botanical mode,'' seeking out and collecting specimens rather than offering detailed analysis. Nonlinear dynamics applies to essentially every other scientific discipline, so why is quantum mechanics exempt from nature's preferred feedback and nonlinearities?

Perhaps, as Mielnik suggested, we have been unconsciously using a form of scientific "newspeak,'' which has prevented us from expressing any nonlinear "thoughtcrimes. The founders and developers of early quantum mechanics did not have access to modern nonlinear dynamics and chaos theory, so they were forced to deal with the idiosyncrasies of quantum mechanics within a strictly linear, if perturbative framework.

This framework has worked beautifully insofar as quantitative, practical applications are concerned. Indeed, until recent years scarcely anybody bothered to question the validity of the Copenhagen interpretation or worry that a point might be reached where the application of increasingly peculiar concepts could possibly break down.

After all, who really cares if it requires thirteenth-order perturbation theory [27] or parameters in a variational calculation [28] to yield a precise value for the ground-state energy of the He atom?! With a glimpse of the possibility of quantum computing, however, this changes somewhat, for quantum computing depends critically on linear superpositions of qubits. But that remains an unanswered question for the future. What does seem plausible, however, is that there have been many naive applications of statistics in interpretations of quantum mechanics.

Statistics, including probabilities and wave-functions, rightfully apply only to large ensembles. When one speaks of a single wave-function of a single electron reducing to a specific expectation value, one must use extreme caution. Finally, perhaps Einstein and Bohr were basically right in their debates. Chaos in quantum mechanics has nothing to do with hidden variables, but it directly provides the fundamental determinism so dear to Einstein's heart.

### Foundations and History of Statistical Mechanics

On the other hand, for all practical purposes it yields indeterminate results that can only be interpreted statistically, as the Copenhagen interpretation insists. It is interesting to speculate how the Einstein-Bohr debates would have progressed had modern chaos theory been available. Einstein, B. Podolsky, and N. Rosen, Phys. Bohr, Phys. Bohm, B. Khrennikov, Ed. Bonifacio, Vitali, Eds. B: Quantum Semiclass. Weinberg, Ann. NY , Gisin, Helv. A , 1 Mielnik, Phys. Czachor, H. Throughout I assume rather than prove the basic irreversibility features of statistical mechanics, taking care to distinguish them from the conceptually distinct assumptions of thermodynamics proper.

I give a brief account of the way in which thermodynamics and statistical mechanics actually work as contemporary scientific theories, and in particular of what statistical mechanics contributes to thermodynamics over and above any supposed underpinning of the latter's general principles.

In doing so, I attempt to illustrate that statistical mechanics should not be thought of wholly or even primarily as itself a foundational project for thermodynamics, and that conceiving of it this way potentially distorts the foundational study of statistical mechanics itself. I contrast two possible attitudes towards a given branch of physics: as inferential i. I contrast these attitudes in classical statistical mechanics, in quantum mechanics, and in quantum statistical mechanics; in this last case, I argue that the quantum-mechanical and statistical-mechanical aspects of the question become inseparable.

Along the way various foundational issues in statistical and quantum physics are hopefully! I discuss classical and quantum recurrence theorems in a unified manner, treating both as generalisations of the fact that a system with a finite state space only has so many places to go.

Along the way I prove versions of the recurrence theorem applicable to dynamics on linear and metric spaces, and make some comments about applications of the classical recurrence theorem in the foundations of statistical mechanics. I review the role of probability in contemporary physics and the origin of probabilistic time asymmetry, beginning with the pre-quantum case both stochastic mechanics and classical statistical mechanics but concentrating on quantum theory. I argue that quantum mechanics radically changes the pre-quantum situation and that the philosophical nature of objective probability in physics, and of probabilistic asymmetry in time, is dependent on the correct resolution of the quantum measurement problem.

I provide an overview of the various asymmetries in time "Arrows of time" found in contemporary physics, predominantly but not exclusively in statistical mechanics and thermodynamics. I attempt to get as clear as possible on the chain of reasoning by which irreversible macrodynamics is derivable from time-reversible microphysics, and in particular to clarify just what kinds of assumptions about the initial state of the universe, and about the nature of the microdynamics, are needed in these derivations.

I conclude that while a "Past Hypothesis" about the early Universe does seem necessary to carry out such derivations, that Hypothesis is not correctly understood as a constraint on the early Universe's entropy. I discuss the statistical mechanics of gravitating systems and in particular its cosmological implications, and argue that many conventional views on this subject in the foundations of statistical mechanics embody significant confusion; I attempt to provide a clearer and more accurate account.

In particular, I observe that i the role of gravity in entropy calculations must be distinguished from the entropy of gravity, that ii although gravitational collapse is entropy-increasing, this is not usually because the collapsing matter itself increases in entropy, and that iii the Second Law of Thermodynamics does not owe its validity to the statistical mechanics of gravitational collapse.