Nonlinear Phenomena at Phase Transitions and Instabilities

This NATO Advanced Study Institute, held in Geilo between March 29th and April 9th 1981, was the sixth in a series devoted to the subject of phase transitions and instabilities. The present institute was intended to provide a forum for discussion of the importance of nonlinear phenomena associated with instabilities in systems as seemingly disparate as ferroelectrics and rotating buckets of oil. Ten years ago, at the first Geilo school, the report of a central peak in the fluctuation spectrum of SrTi0 close to its 3 106 K structural phase transition demonstrated that the simple soft-mode theory of such transitions was incomplete. The missing ingredient was the essential nonlinearity of the system. Parti­ cipants at this year's Geilo school heard assessments of a decade of experimental and theoretical effort which has been expended to elucidate the nature of this nonlinearity. The importance of order­ ed clusters and the walls which bound them was stressed in this con­ text. A specific type of wall, the soliton, was discussed by a number of speakers. New experimental results which purport to demonstrate the existence of solitons in a one-dimensional ferromagnet were presented. A detailed discussion was given of the role of solitons in transport phenomena in driven multistable systems, typified by a sine-Gordon chain.

Anharmonic Properties near Structural Phase Transitions.- Intention and Summary.- I. From early Experiments to the New View.- a) Classical and Nonclassical Behaviour.- b) Central-Peak Research.- c) Computer Simulation, and Anharmonic Theory.- d) The New View.- II. Recent Consolidating Results.- a) Anharmonicity at TC in SrTiO3.- b) Displacive-Order-Disorder Crossover in Ferroelectric Oxides.- c) Evidence for Precursor Order in RbCaF3 and KMnF3.- d) Dynamic Components in SrTiO3.- e) Critical Dyamics in KMnF3.- III. Incommensurate Systems.- The Theory of Structural Phase Transitions: Universality and Quasi-Elastic Scattering Phenomena.- Preamble.- 1. Background.- 1.1 A Simple Model and the Basic Observables.- 1.2 General Theory of Scaling Susceptibilities.- 2. Novel Universal Quantities.- 2.1 Adiabatic and Isothermal Susceptibilities.- 2.2 The Onset of Superlattice Scattering.- 3. Universal Patterns of Short Range Order.- 3.1 Introduction.- 3.2 General Theory of the Block Coordinate p.d.f.- 3.3 One Dimension: Exact Results.- 3.4 Two and Three Dimensions: Approximate Results.- 3.5 Discussion.- 3.6 Summary and Prospects.- Dynamic Correlations in the Ordered Phase of Perovskites.- I. Introduction.- II. Model.- III. Dynamic Correlations below TC.- IV. Results and Conclusions.- Phason Light Scattering in BaMnF4.- Nonlinear Excitations in Some Anharmonic Lattice Models.- 1) Introduction.- 2) Nonlinear Excitations in a model Ferroelectric gl Static Solutions 33 Stationary Solutions.- Stability of Static Periodons.- The Periodon-Phonon Coupling.- 3) Some Remarks on Toda-Lattice.- Solitons in The One-Dimensional Planar Ferromagnet CsNiF3.- 1. Abstract.- 2. Introduction.- 3. Experiment.- 4. Data evalutation.- On The Possibility to Create Nonthermal Solitons in a One-Dimensional Magnetic Sine Gordon System.- 1. Introduction.- 2. The Model.- 3. Results.- 4. Discussion.- Transport and Fluctuations in Linear Arrays of Multistable Systems.- I. Classification of Multistable Systems.- II. Open Systems.- Thermal Instabilities in Electrical Conductors.- Chains of Phase Couples and Externally Synchronized Oscillators.- III. Transport in The Driven Sine-Gordon Chain.- Propagation Velocity of Driven Kinks.- The Steady State Density of Kinks.- IV. Fluctuations in The Sine-Gordon Chain.- The Kink Counting Approach.- The Hydrodynamic Approach.- Universality.- Appendix: The Simplicity of Transport in the Smoluchowski Equation.- Non-Linear Thermal Convection.- 1. Introduction.- 2. The Basic Equations and Boundary Conditions.- 3. The Linear Theory.- a. The Horizontally Unbounded Layer.- b. The Effect of Lateral Walls.- 4. Non-Linear Convection.- 4.1 Perturbation Approach.- 4.2 Numerical Approaches.- 4.3 Comparison with Experiments.- 4.4 The mean field theory.- 5. Convection in a Small Prandtl Number Fluid.- Nematic Instability Induced by An Elliptical Shear.- I. Experimental Apparatus.- II. Instability Mechanism.- III. Modification of The Convective Structure on Increasing the Control Parameter N.- IV. The Time Evolution of The Convective Structure at Fixed N.- Instabilities and Fluctuations.- 1. Role of Fluctuations.- 2. Description of Random Processes.- 3. Phase Transitions.- 4. The Laser Instability.- 5. The Transition to Chaos in Optics.- 6. Instability Transients.- 7. Fluctuations in Hydrodynamic Instabilities.- Steady States, Limit Cycles and The Onset of Turbulence. A Few Model Calculations and Exercises.- 1. Introduction: From Steady States to Chaos in The Time Evolution of Non-Linear Systems.- 1.1 Background.- 1.2 Landau Picture of The Onset of (Quasiperiodic Gaussian) Turbulence.- 1.3 Lorenz Model, Discrete Maps and Related Matters.- 2. Steady States, Limit Cycles and Phase Transition picture.- 2.1 Steady States and Their Stability.- 2.2 Phase Transition Picture: An Illustration of Phase Coexistence. Tricritical points, Triple Points, Etc. in Convective Instability.- 2.3 Limit Cycles and Their Stability.- 3. Strange Attractors.- 3.1 Fractal Dimension: An Illustration.- 3.2 Lyapunov Characteristic Exponents.- 3.3 Exercise: Lorenz Model.- 3.4 Exercise: Two-Component Lorenz Model.- 4. Discrete Maps: Feigenbaum’s Cascade and Pomeau’s Intermittencies.- 4.1 Period-Doubling Cascades.- 4.2 Pomeau’s Intermittencies.- The Physical Mechanism of Oscillatory and Finite Amplitude Instabilities in Systems with Competing Effects.- I. Introduction.- II. Oscillatory Instability.- III. Finite Amplitude Instability.- IV. Concluding Remarks.- Investigation of Fluctuations and Oscillatory States in Rayleigh-Benard Systems by Neutron Scattering.- 1. Introduction.- 2. Observations of Oscillatory States in a Homeotropic Nematic Sample.- A Rayleigh-Bénard Experiment: Helium in a Small Box.- I. The Experiment.- II. Basics of Convection: Busse Theory.- III. The Wavenumber Selection: Effect of Side Walls and The Dynamics of a 3 Rolls to 2 rolls Transition.- IV. The Oscillatory Instability.- V. Two Oscillators: Entrainment and Locking.- VI. Routes to Turbulent Convection.- Period Doubling Bifurcation Route to Chaos.- Space-Time Symmetry in Doubly Periodic Circular Couette Flow.- 1. Introduction.- 2. Review of Experimental Results.- 3. Theory.- The Structure and Dynamics of Non-Stationary Taylor-Vortex Flow.- Pattern Formation During Crystal Growth: Theory.- 1. Introduction.- 2. Dendrites.- 3. Directional Solidification.- 4. Eutectics.- Electrothermal Instabilities at Magnetic Critical Points.- Abstract.- 1. Electronic Transport Coefficients at TC.- 2. Differential Negative Resistivity Conditions near TC.- 3. Continuously Forced Ballast Resistor.- Instabilities During Crystal Growth: Experiments.- 1. Introduction.- 2. Normal Growth and Lateral Growth.- The Nonfaceted — Faceted Transition.- The Solid-Vapor Interface.- The Solid-Solution Interface.- The Solid-Melt Interface.- 3. Sharp Interface-Diffuse Interface.- Light Scattering at the Solid Liquid Interface.- Quasi Eleastic Light Scattering Technique.- Light Scattering at the Ice-Water Interface.- Intensity Hysteresis and Dynamics.- Transient Phenomena.- Light Scattering at the Solid-Liquid Interfaces of D2O and Salol.- 4. Growth Into Supercooled Melt.- Growth of Dendrites.- 5. The Dynamics of Freezing and Melting.- Periodic Fluctuations at The Solid Liquid Interface of Salol.- 1. Introduction.- 2. Experimental.- 3. Observations.- 4. Oscillatory Scattering.- 5. Diffusive Scattering.- On The Dynamics of Epitaxial Phase Transformations.- 1. Surface Molecular Dynamics (SMD).- 2. Surface-Liquid Systems.- The Pinning of a Domain Wall by Weakened Bonds in Two Dimensions.- 1. Introduction.- 2. Solution of The Model.- 3. Discussion.- Melting in Two Dimensions.- I. Introduction.- Theoretical Background.- II. Dislocation Mediated Melting.- Dislocation Mediated Melting at a Smooth Substrate.- Periodic Substrate Potentials.- III. Fluid Phases.- Experiments and Computer Simulations.- IV. Doslocation Theory of Liquid Crystal Films.- Molecular Dynamics Study of 2-D Melting: Long Range Potentials.- 1. Introduction.- 2. Molecular Dynamics (MD).- 3. Results.- a) Coulomb System.- b) Dipolar System.- Experimental Studies of Two Dimensional Melting.- 1. Introduction.- 2. Crytalline Order and Melting in 2D.- 3. Exoerimental 2D Phase Diagrams.- 4. Diffraction Lineshapes in the 2D Solid Phase.- 5. Incommensurate Melting.- 6. Commensurate Melting.- 7. Conclusion.- An Unusual Polymorphism in the 2D Melting: The Smectic F and I Phases.- 1. Structure of the SG Phase.- 2. Structure of the 2d SF and SI Phase.- 3. Analysis of the 2d Order.- 4. The SI and SF Phases and the 2d Melting Theory.- Participants.