Statistical Mechanics: Part B: Time-Dependent Processes

ISBN-13: 9781461579083 / Angielski / Miękka / 2012 / 362 str.

Bruce J. Berne

Opis

The last decade has been marked by a rapid growth in statistical mechanics, especially in connection with the physics and chemistry of the fluid state. Our understanding in these areas has been considerably advanced and enriched by the discovery of new techniques and the sharpening of old techniques, ranging all the way from computer simulations to mode-mode coupling theories. Statistical mechanics brings together under one roof a broad spectrum of mathematical methods. The aim of these volumes is to provide a didactic treatment of those topics that are most useful for the study of problems of current interest to theoretical chemists. The emphasis throughout is on the techniques themselves and not on reviewing the enormous literature in statistical mechanics. Each author was charged with the following task. Given N pages, (a) pose the problem, (b) present those aspects of the particular technique that cJearly illustrate its internal workings, (c) apply the technique to the solUlion of several illustrative examples, and (d) write the chapter so that it will enable the reader to approach key citations to the literature intelligently. These volumes are designed for graduate students and research workers in statistical mechanics. Nevertheless, they should be useful in other areas as weIl.

Spis treści

1. Molecular Dynamics Techniques for Hard-Core Systems.- 1. Introduction.- 2. Statistical-Mechanical Background.- 2.1. Canonical-Ensemble Averages.- 2.2. Molecular Dynamics Averages.- 2.3. Periodic Boundary Conditions.- 3. Approach to Equilibrium.- 4. Equilibrium Thermodynamic Functions.- 4.1. Dynamical Pressure.- 4.2. Virial Theorem.- 4.3. Relationship of Dynamical and Canonical-Ensemble Pressures.- 4.4. Pressure from Collision Rate.- 5. Equilibrium Time Correlation Functions.- 5.1. Time Correlation Functions for Self-Diffusion.- 5.2. Computational Procedures and Results for D(2) (t).- 6. Nonequilibrium Molecular Dynamics.- 6.1. Numerical Details.- 6.2. Analysis of Data.- 7. Accuracy of Molecular Dynamics Trajectory.- 7.1. Constants of Motion.- 7.2. Growth of Trajectory Error.- 8. Programming Techniques.- 8.1. Simple Molecular Dynamics Program.- 8.2. Time Table Method.- 8.3. Cell Table Method.- 8.4. Position Updating.- References.- 2. Molecular Dynamics Methods: Continuous Potentials.- 1. Introduction.- 2. Integration Procedures for the Equations of Motion.- 3. Time-Saving Techniques.- 4. Potential Functions for Anisotropic Molecules.- 5. The Calculation of Time Correlation Functions and Static Properties.- 6. Systems with Long-Range Forces.- 7. Brownian Molecular Dynamics.- 8. Conclusion.- References.- 3. The Kinetic Theory of Gases.- 1. Introduction.- 2. The Boltzmann Equation.- 2.1. Boltzmann’s Derivation.- 2.2. The H-Theorem and the Approach to Equilibrium.- 2.3. Solutions of the Boltzmann Equation.- 3. The Kinetic Theory of Dense Gases.- 3.1. Introduction.- 3.2. The Enskog Theory and Its Modifications.- 3.3. The Liouville Equation and the BBGKY Hierarchy Equations.- 3.4. The Cluster Expansion Method and the Generalized Boltzmann Equation.- 4. Time Correlation Functions.- 4.1. Basic Theory.- 4.2. Computer Results for the Velocity Autocorrelation Function.- 4.3. Applications of the Kinetic Theory of Gases to the Time Correlation Functions.- 5. Discussion.- 5.1. Problems Connected with the Boltzmann Equation.- 5.2. Problems Connected with the Kinetic Theory of Dense Gases.- References.- 4. Renormalized Kinetic Theory of Dense Fluids.- 1. Introduction.- 2. Phase Space Time Correlation Functions.- 3. The Memory Function Formulation.- 4. Symmetrized Two-Particle Correlations with Renormalized Interactions.- 4.1. Formal Solution for G.- 4.2. Properties of Renormalized Interactions.- 4.3. Mode-Coupling Approximations.- 5. The Boltzmann-Enskog Theory of Thermal Fluctuations.- 6. The Velocity Autocorrelation Function of a Dense Fluid.- 6.1. The Enskog Approximation.- 6.2. Effects of Recollisions.- 7. Summary and Discussions.- Appendix A.- Appendix B.- Appendix C. Reduction of the Memory Function in the Boltzmann-Enskog Approximation.- References.- 5. Projection Operator Techniques in the Theory of Fluctuations.- 1. Introduction.- 2. Liouville Space.- 3. Relaxation Equations.- 4. Symmetry Properties.- 5. Navier-Stokes Equations.- 6. Conclusions.- References.- 6. Principles of Mode-Mode Coupling Theory.- 1. Introduction.- 2. Writing Mode-Mode Equations.- 3. Solving Mode-Mode Equations.- 4. Examples.- 4.1. Mutual Diffusion in a Binary Critical Mixture.- 4.2. Self-Diffusion.- 4.3. Reorientation.- 5. Conclusion.- References.- 7. Global Analysis of Nonlinear Chemical Kinetics.- 1. Introduction.- 1.1. Structural Stability of Kinetic Equations.- 1.2. Critical Points and Dynamics in the Linear Regime.- 1.3. Graphical Analysis of Stability Criteria.- 2. An Index Theorem for Chemical Kinetics.- 2.1. Chemical Dynamics on Manifolds.- 2.2. Limitations of the Index Theorem.- 3. Flow Box Analysis in Chemical Kinetics.- 3.1. Classification of Flows in Kinetic Systems.- 3.2. Computation of State Transition Diagrams from the Nonlinear Equations.- 3.3. Critical Points and Qualitative Dynamics from State Transition Diagrams.- 4. Some Limit Cycle Oscillations in Nonlinear Kinetic Equations.- 4.1. The Hopf Bifurcation.- 4.2. Global Analysis of Limit Cycle Oscillations.- 4.3. Cyclic Attractors and Limit Cycles in Higher Dimensions.- 5. Reaction and Diffusion in a Ring of Cells.- 5.1. Eigenvalues of Turing’s Equations.- 5.2. Global Analysis of a Nonlinear Cellular Oscillation.- 6. Discussion.- References.- Author Index.