Macromolecules in Solution and Brownian Relativity - 1st Edition - ISBN: 9780123739063, 9780080557984

Macromolecules in Solution and Brownian Relativity, Volume 15

1st Edition

Authors: Stefano Antonio Mezzasalma
Hardcover ISBN: 9780123739063
eBook ISBN: 9780080557984
Imprint: Academic Press
Published Date: 27th May 2008
Page Count: 248
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Table of Contents

1 Classical and Relativistic Mechanics 7 1.1 Historical Summary . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Analytical Mechanics . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 Lagrangian Mechanics and Hamilton's Principle . . 14 1.2.2 Hamiltonian Mechanics . . . . . . . . . . . . . . . . 18 1.2.3 Poisson's Brackets and Canonical Transformations . 19 1.2.4 Liouville's Theorem . . . . . . . . . . . . . . . . . . 21 1.3 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.1 Einstein's Postulates . . . . . . . . . . . . . . . . . . 22 1.3.2 Lorentz-Poincar_e Transformation . . . . . . . . . . . 23 1.3.3 Rules of Length Contraction and Time Dilation . . . 25 1.3.4 Classi_cation of Events . . . . . . . . . . . . . . . . 26 1.3.5 Notes on Tensor Analysis . . . . . . . . . . . . . . . 28 1.3.6 Covariant and Contravariant Vector Components . . 29 1.3.7 Tensor Formulation of Special Relativity . . . . . . . 31 1.3.8 Maxwell's Equations and Gauge Symmetry . . . . . 33 1.3.9 Lorentz-Poincar_e Invariance of Electrodynamics . . . 35 1.3.10 Doppler's E_ect . . . . . . . . . . . . . . . . . . . . 36 1.3.11 Criticism of the Einstein's Postulates . . . . . . . . . 37 1.4 Relativistic Mechanics . . . . . . . . . . . . . . . . . . . . . 40 1.4.1 Point Particle Dynamics . . . . . . . . . . . . . . . . 40 1.4.2 Energy and Momentum . . . . . . . . . . . . . . . . 41 1.4.3 Hamilton's Principle and Mechanics . . . . . . . . . 43 1.4.4 Experimental Con_rmations . . . . . . . . . . . . . . 44 1.4.5 Notes on General Field Theory and Noether's Theorem 45 1.5 General Relativity . . . . . . . . . . . . . . . . . . . . . . . 52 1.5.1 The Principle of Equivalence . . . . . . . . . . . . . 52 1.5.2 Curved and Accelerated Reference Frames . . . . . . 54 1.5.3 Curvatures, Geodesic Curves and Parallel Transport 55 1.5.4 Metric Tensor, A_ne Connection and Curvature Tensor 60 1.5.5 Tensor Densities . . . . . . . . . . . . . . . . . . . . 64 1.5.6 Covariant Di_erentiation and Principle of General Covariance . . . . . . . . . . . . . . . . . . . . . . . 65 1.5.7 Postulate of Geodesic Motion and Free Falling Frame 67 1.5.8 Extremal Proper Time . . . . . . . . . . . . . . . . . 69 1.5.9 Energy-Momentum Tensor and Conservation Law . 71 1.5.10 Einstein's Field Equations . . . . . . . . . . . . . . . 76 1.6 Particular Solutions and Reference Frames . . . . . . . . . . 82 1.6.1 Weak and Stationary Field Approximation . . . . . 82 1.6.2 Riemann's Normal and Harmonic Coordinates . . . 83 1.6.3 General Static Isotropic Metric . . . . . . . . . . . . 84 1.6.4 Geodesic Light Deflection and Parametrized Metric . 85 1.6.5 Schwarzschild's Metric . . . . . . . . . . . . . . . . . 88 1.6.6 Friedmann-Lemaîitre-Robertson-Walker Metric . . . 89 1.6.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . 92

2 Special Brownian Relativity 95 2.1 Brownian Motion and Difffsion (Notes) . . . . . . . . . . . 95 2.1.1 Historical Summary . . . . . . . . . . . . . . . . . . 95 2.1.2 Einstein's Approach and Bachelier-Wiener Process . 101 2.1.3 The Sutherland-Stokes-Einstein Equation . . . . . . 106 2.1.4 Notes on Hydrodynamics . . . . . . . . . . . . . . . 108 2.1.5 Smoluchowski's Formalism . . . . . . . . . . . . . . 111 2.1.6 Langevin's Equation and Spatio-Temporal Scales . . 114 2.1.7 Markov's Processes and Fokker-Planck Formalism . 118 2.1.8 Rotational Brownian Motion . . . . . . . . . . . . . 121 2.1.9 Notes on Fluctuation-Dissipation Theorem . . . . . 123 2.2 Postulates of Brownian Relativity . . . . . . . . . . . . . . . 129 2.2.1 Equivalence of Time-like and Shape-like Observers . 129 2.2.2 The Invariant Diffusive Interval . . . . . . . . . . . . 134 2.2.3 Random Walk and Rouse's Chain . . . . . . . . . . 136 2.2.4 Brownian Lorentz-Poincaré Transformations . . . . . 140 2.2.5 Fick's Diffusion Equation . . . . . . . . . . . . . . . 143 2.2.6 Fluctuations, Dissipation and Collisions . . . . . . . 146 2.3 Real Polymer in a Minkowskian Fluid . . . . . . . . . . . . 149 2.3.1 Intrinsic Viscosity as a Brownian-Lorentz Factor . . 149 2.3.2 Zimm's, Kirkwood's and Flory's Regimes . . . . . . 152 2.3.3 Characteristic Function for Diffusive Intervals . . . . 153 2.3.4 Real Polymer Size . . . . . . . . . . . . . . . . . . . 157

3 General Brownian Relativity 161 3.1 Geometric Approach to Polymers in Solution . . . . . . . . 161 3.1.1 Principle of Equivalence for Brownian Statistics and Geometry . . . . . . . . . . . . . . . . . . . . . . . . 161 3.1.2 Statistical Pseudo-Coordinates . . . . . . . . . . . . 163 3.1.3 Brownian Metric Tensor . . . . . . . . . . . . . . . . 164 3.1.4 Postulate of Geodesic Brownian Motion . . . . . . . 168 3.1.5 Brownian-Einstein Equations . . . . . . . . . . . . . 169 3.1.6 Energy-Momentum and Polymeric Stress Tensors . . 170 3.1.7 "Static and Isotropic" Polymer Solutions . . . . . . 172 3.1.8 Schwarzschild's Single Coil . . . . . . . . . . . . . . 173 3.1.9 Concentrated Polymer Solutions . . . . . . . . . . . 174 3.1.10 Weak and Stationary Limit . . . . . . . . . . . . . . 178 3.1.11 Macromolecular Continuity Equation . . . . . . . . . 179 3.1.12 Scaling and Polymer Volume Fraction . . . . . . . . 181 3.1.13 Fluctuating-Deflecting Entanglement Points . . . . . 184 3.1.14 Scaling Behavior in Semidilute Solutions . . . . . . . 187 3.2 Statistical Gauge Transformation . . . . . . . . . . . . . . . 191 3.2.1 Extended Heat-Diffusion Equation . . . . . . . . . . 191 3.2.2 Lagrangian Theory and Klein-Gordon Field . . . . . 194 3.3 Outlook and Notes . . . . . . . . . . . . . . . . . . . . . . . 198

4 The Covariant Scaling of Probability 201 4.1 Vineyard's Van Hove Distribution Function . . . . . . . . . 202 4.2 True Self-Avoiding Walk Polymer . . . . . . . . . . . . . . . 206 4.2.1 Self-Correlation Functions of Chain and Liquid Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.2.2 Ornstein-Uhlenbeck Spatial Process . . . . . . . . . 208 4.2.3 Percus-Yevick-Wertheim Scaling Hypothesis . . . . . 210 4.2.4 "True" Self-Avoidance from Molecular Correlations . 211 4.3 Turbulently Advected Passive Scalar . . . . . . . . . . . . . 217 4.3.1 Fluid Turbulence and Brownian Relativity . . . . . . 217 4.3.2 Passive Scalar Advection and Intermittency (Notes) 219 4.3.3 Polymer Topology and Turbulence Statistics . . . . 221 4.3.4 Brownian-Relativistic Anomalous Exponents . . . . 222 4.3.5 Passive Structure Exponents and (Star) Polymer Configurations . . . . . . . . . . . . . . . . . . . . . 224

5 Shape Mechanics 231 5.1 Brownian Simultaneity and Uncertainty . . . . . . . . . . . 232 5.2 Geometrical Lorentz-Poincaré Symmetry . . . . . . . . . . . 234 5.3 The Static Uncertainty Relation . . . . . . . . . . . . . . . 236 4 CONTENTS 5.3.1 For Geometry . . . . . . . . . . . . . . . . . . . . . . 236 5.3.2 For Topology . . . . . . . . . . . . . . . . . . . . . . 238 5.3.3 For Matter and Position . . . . . . . . . . . . . . . . 239 5.4 Materiality and Geometry of Energy . . . . . . . . . . . . . 240 5.4.1 Energy Surface as a "Material Shell" . . . . . . . . . 240 5.4.2 Horocyclic Energy Projection . . . . . . . . . . . . . 242 5.4.3 A Geometrical System of Units . . . . . . . . . . . . 243 5.5 n-Molecular Systems and Pairwise Potential . . . . . . . . . 245 5.5.1 Total Curvature Equation . . . . . . . . . . . . . . . 245 5.5.2 Liquid Density Correlation Functions . . . . . . . . 248 5.6 The Shape-Mechanical Issue . . . . . . . . . . . . . . . . . . 252 5.6.1 Figures as Slits in Young's Experiment . . . . . . . . 253 5.6.2 The Geometrical Wave Function . . . . . . . . . . . . . . . . . 254 5.6.3 Geometrical Wave Equation (in Harmonic Form) . . 255 5.6.4 N-mer Conformations versus n-mer Configurations in Polysaccharide Molecules . . . . . . . . . . . . . . 259 5.7 Outlook and Notes . . . . . . . . . . . . . . . . . . . . . . . 266


Description

1 Classical and Relativistic Mechanics 7 1.1 Historical Summary . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Analytical Mechanics . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 Lagrangian Mechanics and Hamilton's Principle . . 14 1.2.2 Hamiltonian Mechanics . . . . . . . . . . . . . . . . 18 1.2.3 Poisson's Brackets and Canonical Transformations . 19 1.2.4 Liouville's Theorem . . . . . . . . . . . . . . . . . . 21 1.3 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.1 Einstein's Postulates . . . . . . . . . . . . . . . . . . 22 1.3.2 Lorentz-Poincar_e Transformation . . . . . . . . . . . 23 1.3.3 Rules of Length Contraction and Time Dilation . . . 25 1.3.4 Classi_cation of Events . . . . . . . . . . . . . . . . 26 1.3.5 Notes on Tensor Analysis . . . . . . . . . . . . . . . 28 1.3.6 Covariant and Contravariant Vector Components . . 29 1.3.7 Tensor Formulation of Special Relativity . . . . . . . 31 1.3.8 Maxwell's Equations and Gauge Symmetry . . . . . 33 1.3.9 Lorentz-Poincar_e Invariance of Electrodynamics . . . 35 1.3.10 Doppler's E_ect . . . . . . . . . . . . . . . . . . . . 36 1.3.11 Criticism of the Einstein's Postulates . . . . . . . . . 37 1.4 Relativistic Mechanics . . . . . . . . . . . . . . . . . . . . . 40 1.4.1 Point Particle Dynamics . . . . . . . . . . . . . . . . 40 1.4.2 Energy and Momentum . . . . . . . . . . . . . . . . 41 1.4.3 Hamilton's Principle and Mechanics . . . . . . . . . 43 1.4.4 Experimental Con_rmations . . . . . . . . . . . . . . 44 1.4.5 Notes on General Field Theory and Noether's Theorem 45 1.5 General Relativity . . . . . . . . . . . . . . . . . . . . . . . 52 1.5.1 The Principle of Equivalence . . . . . . . . . . . . . 52 1.5.2 Curved and Accelerated Reference Frames . . . . . . 54 1.5.3 Curvatures, Geodesic Curves and Parallel Transport 55 1.5.4 Metric Tensor, A_ne Connection and Curvature Tensor 60 1.5.5 Tensor Densities . . . . . . . . . . . . . . . . . . . . 64 1.5.6 Covariant Di_erentiation and Principle of General Covariance . . . . . . . . . . . . . . . . . . . . . . . 65 1.5.7 Postulate of Geodesic Motion and Free Falling Frame 67 1.5.8 Extremal Proper Time . . . . . . . . . . . . . . . . . 69 1.5.9 Energy-Momentum Tensor and Conservation Law . 71 1.5.10 Einstein's Field Equations . . . . . . . . . . . . . . . 76 1.6 Particular Solutions and Reference Frames . . . . . . . . . . 82 1.6.1 Weak and Stationary Field Approximation . . . . . 82 1.6.2 Riemann's Normal and Harmonic Coordinates . . . 83 1.6.3 General Static Isotropic Metric . . . . . . . . . . . . 84 1.6.4 Geodesic Light Deflection and Parametrized Metric . 85 1.6.5 Schwarzschild's Metric . . . . . . . . . . . . . . . . . 88 1.6.6 Friedmann-Lemaîitre-Robertson-Walker Metric . . . 89 1.6.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . 92

2 Special Brownian Relativity 95 2.1 Brownian Motion and Difffsion (Notes) . . . . . . . . . . . 95 2.1.1 Historical Summary . . . . . . . . . . . . . . . . . . 95 2.1.2 Einstein's Approach and Bachelier-Wiener Process . 101 2.1.3 The Sutherland-Stokes-Einstein Equation . . . . . . 106 2.1.4 Notes on Hydrodynamics . . . . . . . . . . . . . . . 108 2.1.5 Smoluchowski's Formalism . . . . . . . . . . . . . . 111 2.1.6 Langevin's Equation and Spatio-Temporal Scales . . 114 2.1.7 Markov's Processes and Fokker-Planck Formalism . 118 2.1.8 Rotational Brownian Motion . . . . . . . . . . . . . 121 2.1.9 Notes on Fluctuation-Dissipation Theorem . . . . . 123 2.2 Postulates of Brownian Relativity . . . . . . . . . . . . . . . 129 2.2.1 Equivalence of Time-like and Shape-like Observers . 129 2.2.2 The Invariant Diffusive Interval . . . . . . . . . . . . 134 2.2.3 Random Walk and Rouse's Chain . . . . . . . . . . 136 2.2.4 Brownian Lorentz-Poincaré Transformations . . . . . 140 2.2.5 Fick's Diffusion Equation . . . . . . . . . . . . . . . 143 2.2.6 Fluctuations, Dissipation and Collisions . . . . . . . 146 2.3 Real Polymer in a Minkowskian Fluid . . . . . . . . . . . . 149 2.3.1 Intrinsic Viscosity as a Brownian-Lorentz Factor . . 149 2.3.2 Zimm's, Kirkwood's and Flory's Regimes . . . . . . 152 2.3.3 Characteristic Function for Diffusive Intervals . . . . 153 2.3.4 Real Polymer Size . . . . . . . . . . . . . . . . . . . 157

3 General Brownian Relativity 161 3.1 Geometric Approach to Polymers in Solution . . . . . . . . 161 3.1.1 Principle of Equivalence for Brownian Statistics and Geometry . . . . . . . . . . . . . . . . . . . . . . . . 161 3.1.2 Statistical Pseudo-Coordinates . . . . . . . . . . . . 163 3.1.3 Brownian Metric Tensor . . . . . . . . . . . . . . . . 164 3.1.4 Postulate of Geodesic Brownian Motion . . . . . . . 168 3.1.5 Brownian-Einstein Equations . . . . . . . . . . . . . 169 3.1.6 Energy-Momentum and Polymeric Stress Tensors . . 170 3.1.7 "Static and Isotropic" Polymer Solutions . . . . . . 172 3.1.8 Schwarzschild's Single Coil . . . . . . . . . . . . . . 173 3.1.9 Concentrated Polymer Solutions . . . . . . . . . . . 174 3.1.10 Weak and Stationary Limit . . . . . . . . . . . . . . 178 3.1.11 Macromolecular Continuity Equation . . . . . . . . . 179 3.1.12 Scaling and Polymer Volume Fraction . . . . . . . . 181 3.1.13 Fluctuating-Deflecting Entanglement Points . . . . . 184 3.1.14 Scaling Behavior in Semidilute Solutions . . . . . . . 187 3.2 Statistical Gauge Transformation . . . . . . . . . . . . . . . 191 3.2.1 Extended Heat-Diffusion Equation . . . . . . . . . . 191 3.2.2 Lagrangian Theory and Klein-Gordon Field . . . . . 194 3.3 Outlook and Notes . . . . . . . . . . . . . . . . . . . . . . . 198

4 The Covariant Scaling of Probability 201 4.1 Vineyard's Van Hove Distribution Function . . . . . . . . . 202 4.2 True Self-Avoiding Walk Polymer . . . . . . . . . . . . . . . 206 4.2.1 Self-Correlation Functions of Chain and Liquid Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.2.2 Ornstein-Uhlenbeck Spatial Process . . . . . . . . . 208 4.2.3 Percus-Yevick-Wertheim Scaling Hypothesis . . . . . 210 4.2.4 "True" Self-Avoidance from Molecular Correlations . 211 4.3 Turbulently Advected Passive Scalar . . . . . . . . . . . . . 217 4.3.1 Fluid Turbulence and Brownian Relativity . . . . . . 217 4.3.2 Passive Scalar Advection and Intermittency (Notes) 219 4.3.3 Polymer Topology and Turbulence Statistics . . . . 221 4.3.4 Brownian-Relativistic Anomalous Exponents . . . . 222 4.3.5 Passive Structure Exponents and (Star) Polymer Configurations . . . . . . . . . . . . . . . . . . . . . 224

5 Shape Mechanics 231 5.1 Brownian Simultaneity and Uncertainty . . . . . . . . . . . 232 5.2 Geometrical Lorentz-Poincaré Symmetry . . . . . . . . . . . 234 5.3 The Static Uncertainty Relation . . . . . . . . . . . . . . . 236 4 CONTENTS 5.3.1 For Geometry . . . . . . . . . . . . . . . . . . . . . . 236 5.3.2 For Topology . . . . . . . . . . . . . . . . . . . . . . 238 5.3.3 For Matter and Position . . . . . . . . . . . . . . . . 239 5.4 Materiality and Geometry of Energy . . . . . . . . . . . . . 240 5.4.1 Energy Surface as a "Material Shell" . . . . . . . . . 240 5.4.2 Horocyclic Energy Projection . . . . . . . . . . . . . 242 5.4.3 A Geometrical System of Units . . . . . . . . . . . . 243 5.5 n-Molecular Systems and Pairwise Potential . . . . . . . . . 245 5.5.1 Total Curvature Equation . . . . . . . . . . . . . . . 245 5.5.2 Liquid Density Correlation Functions . . . . . . . . 248 5.6 The Shape-Mechanical Issue . . . . . . . . . . . . . . . . . . 252 5.6.1 Figures as Slits in Young's Experiment . . . . . . . . 253 5.6.2 The Geometrical Wave Function . . . . . . . . . . . . . . . . . 254 5.6.3 Geometrical Wave Equation (in Harmonic Form) . . 255 5.6.4 N-mer Conformations versus n-mer Configurations in Polysaccharide Molecules . . . . . . . . . . . . . . 259 5.7 Outlook and Notes . . . . . . . . . . . . . . . . . . . . . . . 266

Key Features

Cross-disciplinarity Novelty Potentiality

Readership

Theoretical scientists Applied scientists Computationalists


Details

No. of pages:
248
Language:
English
Copyright:
© Academic Press 2008
Published:
Imprint:
Academic Press
eBook ISBN:
9780080557984
Hardcover ISBN:
9780123739063

About the Authors

Stefano Antonio Mezzasalma Author

Affiliations and Expertise

Department of Chemical and Pharmaceutical Sciences, University of Trieste, Trieste, Italy