Understanding Time Evolution

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
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