Volume 15: Macromolecules in Solution and Brownian Relativity, 1st Edition

Key Features

Cross-disciplinarity
Novelty
Potentiality

Description

This book illustrates the recent picture of statistical physics of polymers and polymer solutions that emerges from some paradigms of contemporary science joint together. Among its principal aims are discussing the consequences of a novel self-diffusion theory, which benefits from an extension towards relativistic-like principles, and the generalization of usual concepts met in polymer science in terms of geometry alone. The monograph gives the whole fundamentals necessary to handle the view proposed, which is set in the final chapters. All the formers see about to provide the reader with a comprehensive treatation of the necessary fundamentals of classical, relativistic, quantum and statistical mechanics. Among the most important mechanical theories ever developed, a chapter on the Brownian movement and another on macromolecules prepare the ground that is specific to face universality and scaling behaviors in polymer solutions. The scope of the book is therefore two-fold: On the one hand, it wishes to involve the readers and scholars into a new research on polymer physics and chemistry. On the other, to get close chemical physicists and physical chemists to disciplines which, traditionally, are far from their direct fields of interest.

Readership

Theoretical scientists
Applied scientists
Computationalists

Macromolecules in Solution and Brownian Relativity, 1st Edition

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