Understanding Time Evolution

"Understanding Time Evolution first considers that the evolution of quantum operators is canonical with the total Hamiltonian, and that the generator of the temporal evolution of the classical variables is the mean value of this Hamiltonian, evaluated with a purely quantum Density Matrix. The authors introduce the general MaxEnt Density Matrix for systems where quantum and classical degrees of freedom interact. This methodology can describe the interaction between microscopic and macroscopic degrees of freedom. Next, the objectivity of the mathematical description of electric charge transport is explored. It is shown that the description of electric charge transport using fractional order derivatives is non-objective. Similarly, the closing study explores the mathematical description of mechanical movement"-- Provided by publisher UNDERSTANDING TIMEEVOLUTION UNDERSTANDING TIMEEVOLUTION CONTENTS PREFACE Chapter 1SEMIQUANTUM TIME EVOLUTION:CLASSICAL LIMIT, DISSIPATION ANDQUANTUM MEASUREMENT Abstract 1. INTRODUCTION 2. THE MODEL 3. MATTER–FIELD INTERACTION AND CLASSICALLIMIT 3.1. Numerical Results 3.2. Using Information Quantifiers 4. DISSIPATION: A TWO-LEVEL MODEL COUPLEDTO A CLASSICAL ELECTROMAGNETIC FIELD 4.1. Fixed Points (FP) 4.2. Attractors 5. THE MEASUREMENT PROBLEM 5.1. An Example of Measurement’s Procedure CONCLUSION ACKNOWLEDGMENTS REFERENCES Chapter 2SEMIQUANTUM TIME EVOLUTION II:DENSITY MATRICES Abstract 1. INTRODUCTION 2. THE SEMIQUANTUM MODEL 3. BASIC PROPERTIES OFA MAXENT DENSITY MATRIX 4. MAXENT DENSITY MATRIX FOR THESEMIQUANTUM PROBLEM 5. CLASSICAL LIMIT: HAMILTONIANAND PREVIOUS RESULTS 6. DENSITY MATRIX AND CLASSICAL LIMIT 7. RESULTS 8. DENSITY MATRIX AND A DISSIPATIVE SYSTEM 9. GENERALIZED BLOCH EQUATIONS 9.1. The Fixed Points 9.2. Expectation Values 9.3. Stability Considerations 10. LINEARIZATION PROCEDURE ACKNOWLEDGMENTS CONCLUSION REFERENCES Chapter 3OBJECTIVE AND NONOBJECTIVEMATHEMATICAL DESCRIPTION OFTHE ELECTRIC CHARGE TRANSPORT ABSTRACT INTRODUCTION DESCRIPTION OF THE ELECTRON TRANSPORTIN ELECTRIC CIRCUIT WITH INTEGER ORDERDERIVATIVES IS OBJECTIVE VON-SCHWEIDLER DESCRIPTION OF THE ELECTRONTRANSPORT IN ELECTRIC CIRCUIT USING INTEGERORDER DERIVATIVE IS OBJECTIVE CAPUTO, RIEMANN-LIOUVILLE FRACTIONAL ORDERDERIVATIVE, DEFINED WITH INTEGRALREPRESENTATION ON FINITE INTERVAL AND GENERALLIOUVILLE-CAPUTO, GENERAL RIEMANN-LIOUVILLE NONBJECTIVE DESCRIPTION OF THE ELECTRON TRANSPORTIN ELECTRIC CIRCUIT USING CAPUTO FRACTIONALORDER DERIVATIVE DEFINED WITH FORMULA (11) ORRIEMANN-LIOUVILLE FRACTIONAL ORDER DERIVATIVEDEFINED WITH FORMULA (12) OBJECTIVE DESCRIPTION OF THE ELECTRON TRANSPORTIN ELECTRIC CIRCUIT USING GENERALLIOUVILLE–CAPUTO FRACTIONAL ORDER DERIVATIVEDEFINED WITH FORMULA (13) OR GENERAL ROEMANNLIOUVILLEFRACTIONAL ORDER DERIVATIVES DEFINEDWITH FORMULA (14) OBJECTIVE DESCRIPTION OF THE ION TRANSPORT ACROSSA PASSIVE MEMBRANE OF A BIOLOGICAL NEURONUSING INTEGER ORDER DERIVATIVE NONOBJECTIVE DESCRIPTION OF THE ION TRANSPORTACROSS A PASSIVE MEMBRANE OF A BIOLOGICALNEURON CELL USING CAPUTO FRACTIONAL ORDERDERIVATIVE DEFINED WITH FORMULA (11) ORRIEMANN-LIOUVILLE FRACTIONAL ORDER DERIVATIVEDEFINED WITH FORMULA (12) OBJECTIVE DESCRIPTION OF THE ION TRANSPORT ACROSSA PASSIVE MEMBRANE OF A BIOLOGICAL NEURONUSING GENERAL LIOUVILLE-CAPUTO FRACTIONALORDER DERIVATIVE DEFINED WITH FORMULA (13) ORGENERAL RIEMANN-LIOUVILLE FRACTIONAL ORDERDERIVATIVE DEFINED WITH FORMULA (14) HODGKIN-HUXLEY DESCRIPTION OF THE ION TRANSPORTACROSS THE MEMBRANE OF THE SQUID GIANT AXONWITH INTEGER ORDER DERIVATIVE IS OBJECTIVE HODGKIN-HUXLEY TYPE DESCRIPTION OF THE IONTRANSPORT ACROSS THE MEMBRANE OF THE SQUIDGIANT AXON USING CAPUTO OR RIEMANN-LIOUVILLEFRACTIONAL ORDER DERIVATIVES DEFINED WITHINTEGRAL REPRESENTATION ON A FINITE INYERVAL,IS NONOBJECTIVE HODGKIN-HUXLEY TYPE DESCRIPTION OF THE IONTRANSPORT ACROSS THE MEMBRANE OF THE SQUIDGIANT AXON USING GENERAL LIOUVILLE-CAPUTOOR GENERAL RIEMANN-LIOUVILLE FRACTIONALORDER DERIVATIVE, DEFINED WITH INTEGRALREPRESENTATION ON INFINITE INTERVAL,IS OBJECTIVE MORRIS-LECAR DESCRIPTION OF THE ION TRANSPORTACROSS THE MEMBRANE OF THE BARNACLE GIANTMUSCLE FIBER WITH INTEGER ORDER DERIVATIVEIS OBJECTIVE MORRIS-LECAR TYPE DESCRIPTION OF THE IONTRANSPORT ACROSS THE MEMBRANE OF THEBARNACLE GIANT MUSCLE FIBER USING CAPUTO ORRIEMANN-LIOUVILLE FRACTIONAL ORDERDERIVATIVE, DEFINED ON FINITE INTERVAL,IS NONOBJECTIVE MORRIS-LECAR TYPE DESCRIPTION OF THE IONTRANSPORT ACROSS THE MEMBRANE OF THEBARNACLE GIANT MUSCLE FIBER USING GENERALLIOUVILLE-CAPUTO OR GENERAL RIEMANN-LIOUVILLEFRACTIONAL ORDER DERIVATIVES, DEFINED ONINFINITE INTERVAL, IS OBJECTIVE OBJECTIVE DESCRIPTION OF THE ION TRANSPORT ALONGDENDRITE AND AXON HAVING PASSIVE MEMBRANE INCLASSICAL CABEL THEORY (AXON MODEL) USINGINTEGER ORDER DERIVATIVES DESCRIPTION OF THE ION TRANSPORT IN THE FRAMEWORKOF FRACTIONAL ORDER NERVE AXON MODEL USINGCAPUTO OR RIEMANN-LIOUVILLE FRACTIONAL ORDERPARTIAL DERIVATIVE, DEFINED WITH INTEGRALREPRESENTATION ON A FINITE INTERVAL,IS NONOBJECTIVE DESCRIPTION OF THE ION TRANSPORT IN THE FRAMEWORKOF FRACTIONAL ORDER NERVE AXON MODEL, WHICHUSES GENERAL LIOUVILLE-CAPUTO OR GENERALRIEMANN-LIOUVILLE FRACTIONAL ORDER PARTIALDERIVATIVES, DEFINED WITH INTEGRALREPRESENTATION ON INFINITE INTERVAL,IS OBJECTIVE DESCRIPTION OF THE ION TRANSPORT ACROSS THE ACTIVEAXON MEMBRANE AND ALONG THE AXON, USINGINTEGER ORDER PARTIAL DERIVATIVES(ACTIVE NERVE AXON MODEL) IS OBJECTIVE DESCRIPTION OF THE ION TRANSPORT ACROSS THE ACTIVEAXON MEMBRANE AND ALONG THE AXON, USINGCAPUTO FRACTIONAL ORDER PARTIAL DERIVATIVEOR RIEMANN-LIOUVILLE FRACTIONAL ORDER PARTIALDERIVATIVE, DEFINED WITH INTEGRALREPRESENTATION ON A FINITE INTERVAL(FRACTIONAL ORDER ACTIVE NERVE AXON MODEL)IS NONOBJECTIVE DESCRIPTION OF THE ION TRANSPORT IN THE FRAMEWORKOF THE ACTIVE FRACTIONAL ORDER NERVE AXONMODEL, WHICH USES GENERAL LIOUVILLE-CAPUTO ORGENERAL RIEMANN-LIOUVILLE FRACTIONAL ORDERPARTIAL DERIVATIVES, DEFINED WITH INTEGRALREPRESENTATION ON INFINITE INTERVAL,IS OBJECTIVE DESCRIPTION OF THE ION TRANSPORT IN A NEURALNETWORK USING FRACTIONAL ORDER DERIVATIVES,HAVING INTEGRAL REPRESENTATION ON FINITEINTERVAL, IS NONOBJECTIVE DESCRIPTION OF THE ION TRANSPORT IN A NEURALNETWORK USING FRACTIONAL ORDER DERIVATIVES,HAVING INTEGRAL REPRESENTATION ON INFINITEINTERVAL, IS OBJECTIVE DESCRIPTION OF THE ION TRANSPORT IN A NEURALNETWORK USING INTEGER ORDER DERIVATIVEIS OBJECTIVE CONCLUSION REFERENCES Chapter 4OBJECTIVE AND NONOBJECTIVEMATHEMATICAL DESCRIPTION OF THEMECHANICAL MOVEMENT OF A MATERIALPOINT, DUE TO THE USE OF DIFFERENT TYPEOF FRACTIONAL ORDER DERIVATIVES ABSTRACT INTRODUCTION SOME EXAMPLES OF OBJECTIVE AND NONOBJECTIVEMATHEMATICAL DESCRIPTIONS Objective Description of the Movement of a Material Particle Caputo, Riemann-Liouville Fractional Order Derivatives Definedwith Integral Representation on Finite Interval General Liouville-Caputo, General Riemann-Liouville FractionalOrder Derivatives Defined with Integral Representation on InfiniteInterval Objective Description of the Material Point Velocity Using FirstOrder Derivative Objective Description of the Material Point Velocity Using GeneralCaputo-Liouville Fractional Order Derivative Defined withFormula (8) Objective Description of the Material Point Velocity Using GeneralRiemann-Liouville Fractional Order Derivative Defined withFormula (9) Nonobjective Description of the Material Point Velocity UsingCaputo Fractional Order Derivative Defined with Formula (6) Nonobjective Description of the Material Point Velocity UsingRiemann-Liouville Fractional Order Derivative Definedwith Formula (7) Objective Description of the Material Point Acceleration withSecond Order Derivatives Objective Description of the Material Point Acceleration UsingGeneral Liouville-Caputo Fractional Order Derivative Definedwith Formula (8) Objective Description of the Material Point Acceleration UsingGeneral Riemann-Liouville Fractional Order Derivative Definedwith Formula (9) Nonobjective Description of the Material Point Acceleration UsingCaputo Fractional Order Derivative Defined with Formula (6) Nonobjective Description of the Material Point Acceleration UsingRiemann-Liouville Fractional Order Derivative Defined withFormula (7) Objective Description of the Material Point Dynamics Using Firstand Second Order Derivative Objective Description of the Material Point Dynamics UsingGeneral Liouville-Caputo Fractional Order Derivative Definedwith Formula (8) Objective Description of the Material Point Dynamics UsingGeneral Riemann-Liouville Fractional Order Derivative Definedwith Formula (9) Nonobjective Description of the Material Point Dynamics UsingCaputo Fractional Order Derivative Defined with Formula (6) Nonobjective Description of the Material Point Dynamics UsingRiemann-Liouville Fractional Order Derivative Defined withFormula (7) CONCLUSION CONFLICT OF INTEREST REFERENCES INDEX Blank Page