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AIP conference proceedings

Decoherence, Wave-Function Collapses, Non-Ordinary Statistical Mechanics and the Second Principle

Paolo Allegrini

Published Jan 1, 2002• 0 citations
Publisher

Abstract

We show that the derivation of Lévy statistics from a Liouville‐like approach is still an unresolved problem, even though a satisfactory derivation resting on the Generalized Central Limit Theorem (GCLT) has been obtained. We discuss a quantum relaxation that according to the supposed equivalence between decoherence theory and wave function collapses is expected to be equivalent to the characteristic function of a Lévy process. We notice that this way of proceeding is equivalent to establishing a finite Kolmogorov‐Sinai (KS) entropy, and consequently, a condition compatible with the Second Principle of Thermodynamics even though the connection between this kind of entropy and the Clausius entropy is not yet known. The quantum treatment, based on density rather than the collapse‐induced symbolic sequences, conflicts with this conclusion.

DOI
10.1063/1.1523784

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We show that the derivation of Lévy statistics from a Liouville‐like approach is still an unresolved problem, even though a satisfactory derivation resting on the Generalized Central Limit Theorem (GCLT) has been obtained. We discuss a quantum relaxation that according to the supposed equivalence between decoherence theory and wave function collapses is expected to be equivalent to the characteristic function of a Lévy process. We notice that this way of proceeding is equivalent to establishing a finite Kolmogorov‐Sinai (KS) entropy, and consequently, a condition compatible with the Second Principle of Thermodynamics even though the connection between this kind of entropy and the Clausius entropy is not yet known. The quantum treatment, based on density rather than the collapse‐induced symbolic sequences, conflicts with this conclusion.
In this second part of the paper, the quantum-mechanical equation of change with time is applied to an ensemble, and the quantum-mechanical analog of the classical Liouville theorem is derived. The conflict between the quantum-mechanical equation of change of an isolated system, and the second law of thermodynamics is explained. The connection between entropy and the uncertainty principle and the effect of measurement on entropy are discussed in detail. It is pointed out that in quantum mechanics, the uncertainty principle forbids the complete specification of the boundaries of a truly isolated system. The effects of random disturbances at the boundary, within the limits permitted by the uncertainty principle, are shown to cause a rapid spontaneous approach to maximum entropy of any “isolated” system. Applications to quasi-stationary irreversible effects are briefly discussed, and it is indicated how the concept of random fluctuations of the boundary leads to a deduction of the Fokker-Planck equation for irreversible processes.
The probability operator is derived from first principles for an equilibrium quantum system. It is also shown that the superposition states collapse into a mixture of states giving the conventional von Neumann trace form for the quantum average. The mechanism for the collapse is found to be quite general: it results from the conservation law for a conserved, exchangeable variable (such as energy) and the entanglement of the total system wave function that necessarily follows. The relevance of the present results to the einselection mechanism for decoherence, to the quantum measurement problem, and to the classical nature of the macroscopic world are discussed.
Entropy Beyond the Second Law presents a coherent formulation of all aspects of thermodynamics and statistical mechanics with entropy as the unifying theme. This includes formulating equilibrium theory and explaining the role of the Second Law in establishing the equilibrium state. It also establishes the foundations of non-equilibrium theory by generalizing the concept of entropy in an appropriate way. Finally, it addresses the relationship between classical and quantum statistical mechanics by invoking the collapse of the wave function into entropy states. A secondary purpose is to increase the understanding of entropy and the Second Law, and to correct misapprehensions and errors that have arisen over the years.
It is well known that the statistics of closed classical systems evolves according to the Liouville theorem. Here we study the dynamics of the marginal statistics of classical systems coupled to external degrees of freedom, by developing a time-local equation of motion using Green's functions and a series expansion method. We also compare this equation of motion with its supposed quantum counterpart, namely the quantum master equation, which we hope could shed some light on quantum-classical analogy (QCA) from the perspective of "open system dynamics". We notice an apparent exception to QCA in this case, as the first-order classical equation of motion derived herein contains a term that does not appear to have a quantum analogue. We also propose possible ways of getting around this tension, which may help re-establish QCA (in first perturbative order). We do not draw a definitive conclusion about QCA in the context of open system dynamics but hope to provide a starting point for investigations along this line.
Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.
The Kullback ‐ Leibler inequality is a way of comparing any two density matrices. A technique to set up the density matrix for a physical system is to use the maximum entropy principle, given the entropy as a functional of the density matrix, subject to known constraints. In conjunction with the master equation for the density matrix, these two ingredients allow us to formulate the second law of thermodynamics in its widest possible setting. Thus problems arising in both quantum statistical mechanics and quantum information can be handled. Aspects of thermodynamic concepts such as the Carnot cycle will be discussed. A model is examined to elucidate the role of entanglement in the Landauer erasure problem.