Chaos: Possible underpinnings for quantum mechanics?

Alternative, parallel explanations for a number of counter-intuitive concepts connected with the foundations of quantum mechanics can be constructed in terms of nonlinear dynamics. These include ideas as diverse as the statistical exponential decay law and spontaneous symmetry breaking to decoherence itself and the inference from violations of Bell’s inequality that local reality is ruled out in hidden variable extensions of quantum mechanics. Such alternative explanations must not be taken as demonstrations of nonlinear underpinnings for quantum mechanics, but they do raise the possibility of their existence. In this article I delve a bit into ideas connected with the exponential decay law and with Bell’s inequality as demonstrations. Then an investigation of the Klein-Gordon equation shows that it should not come as a complete surprise that quantum mechanics just might contain fundamental nonlinearities.     

Cosmology and celestial mechanics

The similarities between cosmology and celestial mechanics are discussed from the scientific and historical points of view and the scientific aims of these two fields are compared. Newton's and Poincaré's contributions to celestial mechanics, dynamics, and cosmology are presented. The recently established instability of triple stellar configurations is discussed to relate results of this classical, nonintegrable problem of celestial mechanics to cosmology and to offer an example for order out of chaos. It is shown that the presently emphasized reasons for limited predictability in dynamical systems are closely related to some of the existing basic difficulties in cosmology.     

Thermal mechanics: A quantum mechanical analogue of nonequilibrium statistical thermodynamics

A formal but not conventional equivalence between stochastic processes in nonequilibrium statistical thermodynamics and Schrödinger dynamics in quantum mechanics is shown. It is found, for each stochastic process described by a stochastic differential equation of Itô type, there exists a Schrödinger-like dynamics in which the absolute square of a wavefunction gives us the same probability distribution as the original stochastic process. In utilizing this equivalence between them, that is, rewriting the stochastic differential equation by an equivalent Schrödinger equation, it is possible to obtain the notion of deterministic limit of the stochastic process as a semi-classical limit of the “Schrödinger” equation. The deterministic limit thus obtained improves the conventional deterministic approximation in the sense of Onsager-Machlup. The present approach is valid for a general class of stochastic equations where local drifts and diffusion coefficients depend on the position. Two concrete examples are given. It should be noticed that the approach in the present form has nothing to do with the conventional one where only a formal similarity between the Fokker-Planck equation and the Schrödinger equation is considered.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

Nearly-integrable dissipative systems and celestial mechanics

The influence of dissipative effects on classical dynamical models of Celestial Mechanics is of basic importance. We introduce the reader to the subject, giving classical examples found in the literature, like the standard map, the Hénon map, the logistic mapping. In the framework of the dissipative standard map, we investigate the existence of periodic orbits as a function of the parameters. We also provide some techniques to compute the breakdown threshold of quasi-periodic attractors. Next, we review a simple model of Celestial Mechanics, known as the spin-orbit problem which is closely linked to the dissipative standard map. In this context we present the conservative and dissipative KAM theorems to prove the existence of quasi-periodic tori and invariant attractors. We conclude by reviewing some dissipative models of Celestial Mechanics. Among the rotational dynamics we consider the Yarkovsky and YORP effects; within the three-body problem we introduce the so-called Stokes and Poynting–Robertson effects.     

Equations-of-motion approach to quantum mechanics: application to a model phase transition

We present a generalized equations-of-motion method that efficiently calculates energy spectra and matrix elements for algebraic models. The method is applied to a five-dimensional quartic oscillator that exhibits a quantum phase transition between vibrational and rotational phases. For certain parameters, 10 x 10 matrices give better results than obtained by diagonalizing 1000 x 1000 matrices.     

Shape-invariant hypergeometric type operators with application to quantum mechanics

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape-invariant operators. These operators can be analysed together and the mathematical formalism we use can be extended in order to define other shape-invariant operators. All the shape-invariant operators considered are directly related to Schrödinger-type equations.     

Dissipative quantum dynamics and optimal control using iterative time ordering: an application to superconducting qubits

We combine a quantum dynamical propagator that explicitly accounts for quantum mechanical time ordering with optimal control theory. After analyzing its performance with a simple model, we apply it to a superconducting circuit under so-called Pythagorean control. Breakdown of the rotating-wave approximation is the main source of the very strong time-dependence in this example. While the propagator that accounts for the time ordering in an iterative fashion proves its numerical efficiency for the dynamics of the superconducting circuit, its performance when combined with optimal control turns out to be rather sensitive to the strength of the time-dependence. We discuss the kind of quantum gate operations that the superconducting circuit can implement including their performance bounds in terms of fidelity and speed.     

A unified quantum theory of mechanics and thermodynamics. Part I. Postulates

A unified axiomatic theory that embraces both mechanics and thermodynamics is presented in three parts. It is based on four postulates; three are taken from quantum mechanics, and the fourth is the new disclosure of the existence of quantum states that are stable (Part I). For nonequilibrium and equilibrium states, the theory provides general original results, such as the relation between irreducible density operators and the maximum work that can be extracted adiabatically (Part IIa). For stable equilibrium states, it shows for the first time that the canonical and grand canonical distributions are the only stable distributions (Part IIb). The theory discloses the incompleteness of the equation of motion of quantum mechanics not only for irreversible processes but, more significantly, for reversible processes (Part IIb). It establishes the operational meaning of an irreducible density operator and irreducible dispersions associated with any state, and reveals the relationship between such dispersions and the second law (Part III).     

Significance analysis and statistical mechanics: an application to clustering

This Letter addresses the statistical significance of structures in random data: given a set of vectors and a measure of mutual similarity, how likely is it that a subset of these vectors forms a cluster with enhanced similarity among its elements? The computation of this cluster p value for randomly distributed vectors is mapped onto a well-defined problem of statistical mechanics. We solve this problem analytically, establishing a connection between the physics of quenched disorder and multiple-testing statistics in clustering and related problems. In an application to gene expression data, we find a remarkable link between the statistical significance of a cluster and the functional relationships between its genes.     

Nonlinear wave mechanics,information theory,and thermodynamics

A logarithmic nonlinear term is introduced in the Schrödinger wave equation, and a physical justification and interpretation are provided within the context of information theory and thermodynamics. From the resulting nonlinear Schrödinger equation for a system at absolute temperatureT>0, the energy equivalence,kT 1n 2, of a bit of information is derived.     

A unified quantum theory of mechanics and thermodynamics. Part IIa. Available energy

Part II of this three-part paper presents some of the most important theorems that can be deduced from the four postulates of the unified theory discussed in Part I. In Part IIa, it is shown that the maximum energy that can be extracted adiabatically from any system in any state is solely a function of the density operator associated with the state. Moreover, it is shown that for any state of a system, nonequilibrium, equilibrium or stable equilibrium, a unique propertyS exists which is proportional to the total energy of the system minus the maximum energy that can be extracted adiabatically from the system in combination with a reservoir. For statistically independent systems, propertyS is extensive, it is invariant during all reversible processes, and it increases during all irreversible processes.Part I of this paper appeared inFound. Phys. 6(1) (1976). The numbering of the sections, equations, and references in this part of the paper continues from those in Part I.