Lui Lam

Liquid Crystalline and Mesomorphic Polymers

With contributions by internationally renowned researchers, this book provides a comprehensive overview of recent developments in a new class of liquid-crystalline compounds. The book begins with discussions of the theoretical aspects of cholesteric phase formation in polymers. The next part discusses recent work on polymers with mesogenic side groups (their molecular architecture, phase behaviour, and dynamics in external fields, as well as their properties in mixtures with low-mass liquid crystals). The final group of chapters deals with mesomorphic systems that do not contain mesomorphic groups (for example, derivatives of polyorganophosphasenes, cellulose, and graphitizable carbons), as well as polymer-dispersed liquid-crystal films (which have important practical applications), and the new and exotic bowl-shaped liquid-crystal materials.

Introduction to nonlinear physics

Preface.- 1.1 A Quiet Revolution.- 1.2 Nonlinearity.- 1.3 Nonlinear Science.- 1.3.1 Fractals.- 1.3.2 Chaos.- 1.3.3 Pattern Formation.- 1.3.4 Solitons.- 1.3.5 Cellular Automata.- 1.3.6 Complex Systems.- 1.4 Remarks.- References.- Fractals and Multifractals.- Fractals and Diffusive Growth.- 2.1 Percolation.- 2.2 Diffusion-Limited Aggregation.- 2.3 Electrostatic Analogy.- 2.4 Physical Applications of DLA.- 2.4.1 Electrodeposition with Secondary Current Distribution.- 2.4.2 Diffusive Electrodeposition.- Problems.- References.- Multifractality.- 3.1 Definition of i(#)and/(a).- 3.2 Systematic Definition of x(q).- 3.3 The Two-Scale Cantor Set.- 3.3.1 Limiting Cases.- 3.3.2 Stirling Formula and/(a).- 3.4 Multifractal Correlations.- 3.4.1 Operator Product Expansion and Multifractality.- 3.4.2 Correlations of Iso-a Sets.- 3.5 Numerical Measurements of/(a).- 3.6 Ensemble Averaging and r(q).- Problems.- References.- Scaling Arguments and Diffusive Growth.- 4.1 The Information Dimension.- 4.2 The Turkevich-Scher Scaling Relation.- 4.3 The Electrostatic Scaling Relation.- 4.4 Scaling of Negative Moments.- 4.5 Conclusions.- Problems.- References.- Chaos and Randomness.- to Dynamical Systems.- 5.1 Introduction.- 5.2 Determinism Versus Random Processes.- 5.3 Scope of Part II.- 5.4 Deterministic Dynamical Systems and State Space.- 5.5 Classification.- 5.5.1 Properties of Dynamical Systems.- 5.5.2 A Brief Taxonomy of Dynamical Systems Models.- 5.5.3 The Relationship Between Maps and Flows.- 5.6 Dissipative Versus Conservative Dynamical Systems.- 5.7 Stability.- 5.7.1 Linearization.- 5.7.2 The Spectrum of Lyapunov Exponents.- 5.7.3 Invariant Sets.- 5.7.4 Attractors.- 5.7.5 Regular Attractors.- 5.7.6 Review of Stability.- 5.8 Bifurcations.- 5.9 Chaos.- 5.9.1 Binary Shift Map.- 5.9.2 Chaos in Flows.- 5.9.3 The Rossler Attractor.- 5.9.4 The Lorenz Attractor.- 5.9.5 Stable and Unstable Manifolds.- 5.10 Homoclinic Tangle.- 5.10.1 Chaos in Higher Dimensions.- 5.10.2 Bifurcations Between Chaotic Attractors.- Problems.- References.- Probability, Random Processes, and the Statistical Description of Dynamics.- 6.1 Nondeterminism in Dynamics.- 6.2 Measure and Probability.- 6.2.1 Estimating a Density Function from Data.- 6.3 Nondeterministic Dynamics.- 6.4 Averaging.- 6.4.1 Stationarity.- 6.4.2 Time Averages and Ensemble Averages.- 6.4.3 Mixing.- 6.5 Characterization of Distributions.- 6.5.1 Moments.- 6.5.2 Entropy and Information.- 6.6 Fractals, Dimension, and the Uncertainty Exponent.- 6.6.1 Pointwise Dimension.- 6.6.2 Information Dimension.- 6.6.3 Fractal Dimension.- 6.6.4 Generalized Dimensions.- 6.6.5 Estimating Dimension from Data.- 6.6.6 Embedding Dimension.- 6.6.7 Fat Fractals.- 6.6.8 Lyapunov Dimension.- 6.6.9 Metric Entropy.- 6.6.10 Pesins Identity.- 6.7 Dimensions, Lyapunov Exponents, and Metric Entropy in the Presence of Noise.- Problems.- References.- Modeling Chaotic Systems.- 7.1 Chaos and Prediction.- 7.2 State Space Reconstruction.- 7.2.1 Derivative Coordinates.- 7.2.2 Delay Coordinates.- 7.2.3 Broomhead and King Coordinates.- 7.2.4 Reconstruction as Optimal Encoding.- 7.3 Modeling Chaotic Dynamics.- 7.3.1 Choosing an Appropriate Model.- 7.3.2 Order of Approximation.- 7.3.3 Scaling of Errors.- 7.4 System Characterization.- 7.5 Noise Reduction.- 7.5.1 Shadowing.- 7.5.2 Optimal Solution of Shadowing Problem with Euclidean Norm.- 7.5.3 Numerical Results.- 7.5.4 Statistical Noise Reduction.- 7.5.5 Limits to Noise Reduction.- Problems.- References.- Pattern Formation and Disorderly Growth.- Phenomenology of Growth.- 8.1 Aggregation: Patterns and Fractals Far from Equilibrium.- 8.2 Natural Systems.- 8.2.1 Ballistic Growth.- 8.2.2 Diffusion-Limited Growth.- 8.2.3 Growth of Colloids and Aerosols.- Problems.- References.- Models and Applications.- 9.1 Ballistic Growth.- 9.1.1 Simulations and Scaling.- 9.1.2 Continuum Models.- 9.2 Diffusion-Limited Growth.- 9.2.1 Simulations and Scaling.- 9.2.2 The Mullins-Sekerka Instability.- 9.2.3 Orderly and Disorderly Growth.- 9.2.4 Electrochemical Deposition: A Case Study.- 9.3 Cluster-Cluster Aggregation.- Appendix: A DLA Program.- Problems.- References.- Solitons.- Models and Applications.- 10.1 Introduction.- 10.2 Origin and History of Solitons.- 10.3 Integrability and Conservation Laws.- 10.4 Soliton Equations and their Solutions.- 10.4.1 Korteweg-de Vries Equation.- 10.4.2 Nonlinear Schrodinger Equation.- 10.4.3 Sine-Gordon Equation.- 10.4.4 Kadomtsev-Petviashvili Equation.- 10.5 Methods of Solution.- 10.5.1 Inverse Scattering Method.- 10.5.2 Backlund Transformation.- 10.5.3 Hirota Method.- 10.5.4 Numerical Method.- 10.6 Physical Soliton Systems.- 10.6.1 Shallow Water Waves.- 10.6.2 Dislocations in Crystals.- 10.6.3 Self-Focusing of Light.- 10.7 Conclusions.- Problems.- References.- Nonintegrable Systems.- 11.1 Introduction.- 11.2 Nonintegrable Soliton Equations with Hamiltonian Structures.- 11.2.1 The fl4 Equation.- 11.2.2 Double Sine-Gordon Equation.- 11.3 Nonlinear Evolution Equations.- 11.3.1 Fisher Equation.- 11.3.2 The Damped 0* Equation.- 11.3.3 The Damped Driven Sine-Gordon Equation.- 11.4 A Method of Constructing Soliton Equations.- 11.5 Formation of Solitons.- 11.6 Perturbations.- 11.7 Soliton Statistical Mechanics.- 11.7.1 The ^System.- 11.7.2 The Sine-Gordon System.- 11.8 Solitons in Condensed Matter.- 11.8.1 Liquid Crystals.- 11.8.2 Polyacetylene.- 11.8.3 Optical Fibers.- 11.8.4 Magnetic Systems.- 11.9 Conclusions.- Problems.- References.- Special Topics.- Cellular Automata and Discrete Physics.- 12.1 Introduction.- 12.1.1 A Weil-Known Example: Life.- 12.1.2 Cellular Automata.- 12.1.3 The Information Mechanics Group.- 12.2 Physical Modeling.- 12.2.1 CA Quasiparticles.- 12.2.2 Physical Properties from CA Simulations.- 12.2.3 Diffusion.- 12.2.4 Soundwaves.- 12.2.5 Optics.- 12.2.6 Chemical Reactions.- 12.3 Hardware.- 12.4 Current Sources of Literature.- 12.5 An Outstanding Problem in CA Simulations.- Problems.- References.- Visualization Techniques for Cellular Dynamata.- 13.1 Historical Introduction.- 13.2 Cellular Dynamata.- 13.2.1 Dynamical Schemes.- 13.2.2 Complex Dynamical Systems.- 13.2.3 CD Definitions.- 13.2.4 CD States.- 13.2.5 CD Simulation.- 13.2.6 CD Visualization.- 13.3 An Example of Zeemans Method.- 13.3.1 Zeemans Heart Model: Standard Cell.- 13.3.2 Zeemans Heart Model: Physical Space.- 13.3.3 Zeemans Heart Model: Beating.- 13.4 The Graph Method.- 13.4.1 The Biased Logistic Scheme.- 13.4.2 The Logistic/Diffusion Lattice.- 13.4.3 The Global State Graph.- 13.5 The Isochron Coloring Method.- 13.5.1 Isochrons of a Periodic Attractor.- 13.5.2 Coloring Strategies.- 13.6 Conclusions.- References.- From Laminar Flow to Turbulence.- 14.1 Preamble and Basic Ideas.- 14.1.1 What Is Turbulence?.- 14.2 From Laminar Flow to Nonlinear Equilibration.- 14.2.1 A Linear Analysis: The Kelvin-Helmholz Instability.- 14.2.2 A Weakly Nonlinear Analysis: Landaus Equation.- 14.3 From Nonlinear Equilibration to Weak Turbulence.- 14.3.1 The Quasi-Periodic Sequence.- 14.3.2 The Period Doubling Sequence.- 14.3.3 The Intermittent Sequence.- 14.3.4 Fluid Relevance and Experimental Evidence.- 14.4 Strong Turbulence.- 14.4.1 Scaling Arguments for Inertial Ranges.- 14.4.2 Predictability of Strong Turbulence.- 14.4.3 Renormalizing the Diffusivity.- 14.5 Remarks.- References.- Active Walks: Pattern Formation, Self-Organization, and Complex Systems.- 15.1 Introduction.- 15.2 Basic Concepts.- 15.3 Continuum Description.- 15.4 Computer Models.- 15.4.1 A Single Walker.- 15.4.2 Branching.- 15.4.3 Multiwalkers and Updating Rules.- 15.4.4 Track Patterns.- 15.5 Three Applications.- 15.5.1 Dielectric Breakdown in a Thin Layer of Liquid.- 15.5.2 Ion Transport in Glasses.- 15.5.3 Ant Trails in Food Collection.- 15.6 Intrinsic Abnormal Growth.- 15.7 Landscapes and Rough Surfaces.- 15.7.1 Groove States.- 15.7.2 Localization-Delocalization Transition.- 15.7.3 Scaling Properties.- 15.8 Fuzzy Walks.- 15.9 Related Developments and Open Problems.- 15.10 Conclusions.- References.- Appendix: Historical Remarks on Chaos.- Contributors.

Active walker models for complex systems

Abstract Many complex systems can be described in a unified way by the simple model of active walker(s). Examples of these complex systems are river formation, movement of heat-seeking missiles, ant swarms, worm movements, polymer reptation, retinal neurons and vessels, electrodeposit and dielectric breakdown patterns, percolation in soft materials, ion transport in glasses, rough surfaces, biological evolution, population dynamics, etc. In an active walker model (AWM), the walker changes the landscape as it walks and its steps are influenced by the changing landscape. In this paper, developments of the AWM are summarized and new results are presented. In particular, the three important areas of application of the AWM-- track patterns, intrinsic abnormal growth, and rough surfaces--are discussed.

ACTIVE WALKS: THE FIRST TWELVE YEARS (PART II)

Active walk is a paradigm for self-organization and pattern formation in simple and complex systems, originated by Lam in 1992. In an active walk, the walker (an agent) changes the deformable landscape as it walks and is influenced by the changed landscape in choosing its next step. Active walk models have been applied successfully to various biological, chemical and physical systems from the natural sciences, and to economics and many other systems from the social sciences. More recently, it has been used to model human history. In this review, the history, basic concepts, formulation, theories, applications, new developments and open problems of active walk are summarized and discussed. New experimental, theoretical and computer modeling results are included.

Solitons in liquid crystals: Recent developments

Abstract Liquid crystal is a state of matter intermediate between isotropic liquid and anisotropic crystal. The mechanical and optical properties of liquid crystals are highly nonlinear. Consequently, they are naturally soliton-bearing media. After a brief general introduction, five topics in recent developments on solitons in liquid crystals are presented, namely (i) optical solitons, (ii) solitons in nematics under a rotating magnetic field, (iii) solitons in electroconvective nematics, (iv) incommensurate solitons in smectic A, and (v) the soliton model for the chevron structure in ferroelectric smectic C∗ and in smectic A.

Growth pattern formation in copper electrodeposition: Experiments and computational modelling

Abstract Experimental results on electrodeposit patterns of copper sulphate solutions in linear cells are presented. Varying the control parameters of the experiment: concentration, cell thickness and voltage between electrodes, many different morphologies are found. A generalized Biased Random Walk computational model in 2D and 3D is introduced for the simulation of the physical experiments that approximately takes into account the control parameters of the physical experiments. The results of the computational model compare well with some of the experimental results.

Dielectric breakdown patterns in thin layers of oils

Abstract Experimental dielectric breakdown patterns are generated in thin cells of oils. Both mineral oil and olive oil are used. The cell is 6 μm thick and consists of two glass plates with conductive coating on the inner surfaces. A dc voltage across the cell is either increased gradually to above the threshold or applied with a fixed value above the threshold. Different types of patterns due to chemical reactions occurring at the inner surfaces are left behind after the dielectric breakdown process takes place. These patterns consist of filaments with wiggling or winding, depending on the experimental conditions. Time variations of the voltage and current are recorded during and after the discharge process. Some of these patterns are compared with those generated by the active walker model. The physical processes behind these patterns are discussed.

Neural network for classification of active walker patterns

Abstract A radial-basis-function neural network is constructed to classify patterns generated by the boundary probabilistic active walker model. The patterns generated by this model fall into five classes referred to as blob, jellyfish, diamond, lollipop, and needle. Some of these exhibit abnormal growth behavior such as irreproducible or transformational growth. For a test sample of 300 patterns, our neural network classifier cannot distinguish jellyfish from diamond, but can recognize blob, lollipop, needle, and jellyfish/diamond as four different classes with a success rate of about 99%. The failures occur within some parts of the sensitive zone, i.e., the zone in a parameter space in which more than one type of pattern are generated from different computer runs, due to the differing sequence of random numbers employed in each run. In developing this neural network classifier a number of issues are addressed, including the large dimension of the input space, the rotational variation within the classes, the intricacies of the patterns, and the definition of the metric.

Bilinear effect in complex systems

The distribution of the lifetime of Chinese dynasties (as well as that of the British Isles and Japan) in a linear Zipf plot is found to consist of two straight lines intersecting at a transition point. This two-section piecewise-linear distribution is different from the power law or the stretched exponent distribution, and is called the Bilinear Effect for short. With assumptions mimicking the organization of ancient Chinese regimes, a 3-layer network model is constructed. Numerical results of this model show the bilinear effect, providing a plausible explanation of the historical data. The bilinear effect in two other social systems is presented, indicating that such a piecewise-linear effect is widespread in social systems.