Towards a Universal Measure of Complexity

Recently, it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and propose a universal measure of complexity that is based on Gell-Mann’s view of complexity. Our universal measure of complexity is based on a non-linear transformation of time-dependent entropy, where the system state with the highest complexity is the most distant from all the states of the system of lesser or no complexity. We have shown that the most complex is the optimally mixed state consisting of pure states, i.e., of the most regular and most disordered which the space of states of a given system allows. A parsimonious paradigmatic example of the simplest system with a small and a large number of degrees of freedom is shown to support this methodology. Several important features of this universal measure are pointed out, especially its flexibility (i.e., its openness to extensions), suitability to the analysis of system critical behaviour, and suitability to study the dynamic complexity.

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Introduction

10.1093/oso/9780198805014.003.0001 ◽

2018 ◽

Author(s):

Niels Engholm Henriksen ◽

Flemming Yssing Hansen

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Degrees Of Freedom ◽

State Level ◽

Boltzmann Distribution ◽

Theoretical Description ◽

Time Dependent ◽

Nuclear Dynamics ◽

Molecular Reaction ◽

Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

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Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators

Entropy ◽

10.3390/e23040408 ◽

2021 ◽

Vol 23 (4) ◽

pp. 408

Author(s):

Matteo Villani ◽

Guillermo Albareda ◽

Carlos Destefani ◽

Xavier Cartoixà ◽

Xavier Oriols

Keyword(s):

Single Particle ◽

Degrees Of Freedom ◽

Single Photon ◽

Open Quantum Systems ◽

Wave Functions ◽

Practical Application ◽

Light Matter Interaction ◽

Pure States ◽

Energy And Momentum ◽

Full Quantum

Without access to the full quantum state, modeling quantum transport in mesoscopic systems requires dealing with a limited number of degrees of freedom. In this work, we analyze the possibility of modeling the perturbation induced by non-simulated degrees of freedom on the simulated ones as a transition between single-particle pure states. First, we show that Bohmian conditional wave functions (BCWFs) allow for a rigorous discussion of the dynamics of electrons inside open quantum systems in terms of single-particle time-dependent pure states, either under Markovian or non-Markovian conditions. Second, we discuss the practical application of the method for modeling light–matter interaction phenomena in a resonant tunneling device, where a single photon interacts with a single electron. Third, we emphasize the importance of interpreting such a scattering mechanism as a transition between initial and final single-particle BCWF with well-defined central energies (rather than with well-defined central momenta).

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THE DYNAMIC RESPONSE OF OFFSHORE STRUCTURES TO TIME-DEPENDENT FORCES

Coastal Engineering Proceedings ◽

10.9753/icce.v9.29 ◽

1964 ◽

Vol 1 (9) ◽

pp. 29

Author(s):

William S. Gaither ◽

David P. Billington

Keyword(s):

Degrees Of Freedom ◽

Wind Waves ◽

Offshore Structures ◽

Time Dependent ◽

Wave Forces ◽

Ocean Bottom ◽

Single Wave ◽

Single Degree Of Freedom ◽

The Many ◽

Floating Objects

This paper is addressed to the problem of structural behavior in an offshore environment, and the application of a more rigorous analysis for time-dependent forces than is currently used. Design of pile supported structures subjected to wave forces has, in the past, been treated in two parts; (1) a static analysis based on the loading of a single wave, and (2) a dynamic analysis which sought to determine the resonant frequency by assuming that the structure could be approximated as a single-degree-of-freedom system. (Ref. 4 and 6) The behavior of these structures would be better understood if the dynamic nature of the loading and the many degrees of freedom of the system were included. A structure which is built in the open ocean is subjected to periodic forces due to wind, waves, floating objects, and due occasionally to machinery mounted on the structure. To resist motion, the structure relies on the stiffness of the elements from which it is built and the restraints of the ocean bottom into which the supporting legs are driven.

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Entanglement in massive coupled oscillators

Quantum Information and Computation ◽

10.26421/qic11.3-4-7 ◽

2011 ◽

Vol 11 (3&4) ◽

pp. 278-299

Author(s):

Nathan L. Harshman ◽

William F. Flynn

Keyword(s):

Coupled Oscillators ◽

Coherent States ◽

Degrees Of Freedom ◽

Classical Model ◽

Reduced Density Matrix ◽

One Dimension ◽

Atomic Position ◽

Pure States ◽

Non Gaussian ◽

Reduced Density

This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In general, when the trap frequency and the molecular frequency are very different, and when the atomic masses are equal, the atoms are highly-entangled for molecular coherent states and number states. Surprisingly, while the interatomic entanglement can be quite large even for molecular coherent states, the covariance of atomic position and momentum observables can be entirely explained by a classical model with appropriately chosen statistical uncertainty.

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Sliding Mode Control of a Three Degrees of Freedom Anthropoid Robot by Driving the Controller Parameters to an Equivalent Regime

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.1318353 ◽

2000 ◽

Vol 122 (4) ◽

pp. 632-640 ◽

Cited By ~ 47

Author(s):

M. Onder Efe ◽

Okyay Kaynak ◽

Xinghuo Yu

Keyword(s):

Degrees Of Freedom ◽

Sliding Mode ◽

Initial Conditions ◽

Tracking Error ◽

Variable Structure ◽

Mathematical Representation ◽

Dynamic Complexity ◽

Variable Structure Systems ◽

Sliding Motion ◽

Three Degrees Of Freedom

Noise rejection, handling the difficulties coming from the mathematical representation of the system under investigation and alleviation of structural or unstructural uncertainties constitute prime challenges that are frequently encountered in the practice of systems and control engineering. Designing a controller has primarily the aim of achieving the tracking precision as well as a degree of robustness against the difficulties stated. From this point of view, variable structure systems theory offer well formulated solutions to such ill-posed problems containing uncertainty and imprecision. In this paper, a simple controller structure is discussed. The architecture is known as Adaptive Linear Element (ADALINE) in the framework of neural computing. The parameters of the controller evolve dynamically in time such that a sliding motion is obtained. The inner sliding motion concerns the establishment of a sliding mode in controller parameters, which aims to minimize the error on the controller outputs. The outer sliding motion is designed for the plant. The algorithm discussed drives the error on the output of the controller toward zero learning error level, and the state tracking error vector of the plant is driven toward the origin of the phase space simultaneously. The paper gives the analysis of the equivalence between the two sliding motions and demonstrates the performance of the algorithm on a three degrees of freedom, anthropoid robotic manipulator. In order to clarify the performance of the scheme, together with the dynamic complexity of the plant, the adverse effects of observation noise and nonzero initial conditions are studied. [S0022-0434(00)01704-4]

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Effective 1D Time-Dependent Schrödinger Equations for 3D Geometrically Correlated Systems

Materials ◽

10.3390/ma13133033 ◽

2020 ◽

Vol 13 (13) ◽

pp. 3033

Author(s):

Devashish Pandey ◽

Xavier Oriols ◽

Guillermo Albareda

Keyword(s):

Degrees Of Freedom ◽

Equations Of Motion ◽

Three Dimensional ◽

Longitudinal Direction ◽

Time Dependent ◽

Ab Initio Molecular Dynamics ◽

Quantitative Accuracy ◽

Dimensional Simulation ◽

Transport Problems ◽

Computational Resources

The so-called Born–Huang ansatz is a fundamental tool in the context of ab-initio molecular dynamics, viz., it allows effectively separating fast and slow degrees of freedom and thus treating electrons and nuclei with different mathematical footings. Here, we consider the use of a Born–Huang-like expansion of the three-dimensional time-dependent Schrödinger equation to separate transport and confinement degrees of freedom in electron transport problems that involve geometrical constrictions. The resulting scheme consists of an eigenstate problem for the confinement degrees of freedom (in the transverse direction) whose solution constitutes the input for the propagation of a set of coupled one-dimensional equations of motion for the transport degree of freedom (in the longitudinal direction). This technique achieves quantitative accuracy using an order less computational resources than the full dimensional simulation for a typical two-dimensional geometrical constriction and upto three orders for three-dimensional constriction.

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A new method for calculating time-dependent quantum correlation functions for systems with many degrees of freedom

Chemical Physics Letters ◽

10.1016/0009-2614(89)87172-6 ◽

1989 ◽

Vol 155 (4-5) ◽

pp. 376-380 ◽

Cited By ~ 13

Author(s):

Andrew E. Depristo ◽

Kenneth Haug ◽

Horia Metiu

Keyword(s):

Correlation Functions ◽

Quantum Correlation ◽

Degrees Of Freedom ◽

Time Dependent ◽

New Method

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On a Conjecture regarding Fisher Information

Advances in Mathematical Physics ◽

10.1155/2015/120698 ◽

2015 ◽

Vol 2015 ◽

pp. 1-4 ◽

Cited By ~ 4

Author(s):

Angelo Plastino ◽

Guido Bellomo ◽

Angel Ricardo Plastino

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Probability Distributions ◽

Free Particle ◽

Quantum Probability ◽

Time Dependent ◽

Theoretical Physics ◽

One Dimension ◽

Pure States ◽

Time Dependent Solution

Fisher’s information measureIplays a very important role in diverse areas of theoretical physics. The associated measuresIxandIp, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The productIxIphas been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimensionIxIp≥4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifiesIxIp→0fort→∞.

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Singular Vectors and Time-Dependent Normal Modes of a Baroclinic Wave-Mean Oscillation

Journal of the Atmospheric Sciences ◽

10.1175/2007jas2364.1 ◽

2008 ◽

Vol 65 (3) ◽

pp. 875-894 ◽

Cited By ~ 5

Author(s):

Christopher L. Wolfe ◽

Roger M. Samelson

Keyword(s):

Degrees Of Freedom ◽

Channel Model ◽

Normal Modes ◽

Time Dependent ◽

Inner Product ◽

Time Interval ◽

Baroclinic Wave ◽

Singular Vectors ◽

Mean Oscillation ◽

Local Attractor

Abstract Linear disturbance growth is studied in a quasigeostrophic baroclinic channel model with several thousand degrees of freedom. Disturbances to an unstable, nonlinear wave-mean oscillation are analyzed, allowing the comparison of singular vectors and time-dependent normal modes (Floquet vectors). Singular vectors characterize the transient growth of linear disturbances in a specified inner product over a specified time interval and, as such, they complement and are related to Lyapunov vectors, which characterize the asymptotic growth of linear disturbances. The relationship between singular vectors and Floquet vectors (the analog of Lyapunov vectors for time-periodic systems) is analyzed in the context of a nonlinear baroclinic wave-mean oscillation. It is found that the singular vectors divide into two dynamical classes that are related to those of the Floquet vectors. Singular vectors in the “wave dynamical” class are found to asymptotically approach constant linear combinations of Floquet vectors. The most rapidly decaying singular vectors project strongly onto the most rapidly decaying Floquet vectors. In contrast, the leading singular vectors project strongly onto the leading adjoint Floquet vectors. Examination of trajectories that are near the basic cycle show that the leading Floquet vectors are geometrically tangent to the local attractor while the leading initial singular vectors point off the local attractor. A method for recovering the leading Floquet vectors from a small number of singular vectors is additionally demonstrated.

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