LANGEVIN DYNAMICS IN ATOMIC, POLYMERIC, AND COLLOIDAL SYSTEMS

By DAVID MARTIN

CONTENTS BACKGROUND, MECHANICS AND STATISTICAL MECHANICS BACKGROUND THE MECHANICAL DESCRIPTION OF A SYSTEM OF PARTICLES PHASE SPACE AND EQUATIONS OF MOTION IN EQUILIBRIUM IN A NON-ISOLATED SYSTEM NEWTON’S EQUATIONS IN OPERATOR FORM THE LIOUVILLE EQUATION LIOUVILLE EQUATION IN AN ISOLATED SYSTEM EXPRESSIONS FOR EQUILIBRIUM THERMODYNAMIC AND LINEAR TRANSPORT PROPERTIES LIOUVILLE EQUATION IN A NON-ISOLATED SYSTEM NON-EQUILIBRIUM DISTRIBUTION FUNCTION AND CORRELATION FUNCTIONS OTHER APPROACHES TO NON-EQUILIBRIUM PROJECTION OPERATORS SUMMARY CONCLUSIONS REFERENCES THE EQUATION OF MOTION FOR A TYPICAL PARTICLE AT EQUILIBRIUM: THE MORI–ZWANZIG APPROACH THE PROJECTION OPERATOR THE GENERALISED LANGEVIN EQUATION THE GENERALISED LANGEVIN EQUATION IN TERMS OF THE VELOCITY EQUATION OF MOTION FOR THE VELOCITY AUTOCORRELATION FUNCTION THE LANGEVIN EQUATION DERIVED FROM THE MORI APPROACH: THE BROWNIAN LIMIT GENERALISATION TO ANY SET OF DYNAMICAL VARIABLES MEMORY FUNCTIONS DERIVATION OF EXPRESSIONS FOR LINEAR TRANSPORT COEFFICIENTS CORRELATION FUNCTION EXPRESSION FOR THE COEFFICIENT OF NEWTONIAN VISCOSITY APPROXIMATE METHODS TO CALCULATE CORRELATION FUNCTIONS AND MORI-ZWANZIG MEMORY FUNCTIONS TAYLOR SERIES EXPANSION SPECTRA MORI’S CONTINUED FRACTION METHOD USE OF INFORMATION THEORY PERTURBATION THEORIES MODE COUPLING THEORY MACROSCOPIC HYDRODYNAMIC THEORY MEMORY FUNCTIONS CALCULATED BY THE MOLECULAR-DYNAMICS METHOD THE GENERALISED LANGEVIN EQUATION IN NON-EQUILIBRIUM DERIVATION OF GENERALISED LANGEVIN EQUATION IN NON-EQUILIBRIUM LANGEVIN EQUATION FOR A SINGLE BROWNIAN PARTICLE IN A SHEARING FLUID THE LANGEVIN EQUATION AND THE BROWNIAN LIMIT A DILUTE SUSPENSION – ONE LARGE PARTICLE IN A BACKGROUND EXACT EQUATIONS OF MOTION FOR A(T) LANGEVIN EQUATION FOR A(T) LANGEVIN EQUATION FOR VELOCITY 5 2 MANY-BODY LANGEVIN EQUATION MANY-BODY LANGEVIN EQUATION FOR A(T) MANY-BODY LANGEVIN EQUATION FOR VELOCITY LANGEVIN EQUATION FOR THE VELOCITY AND THE FORM OF THE FRICTION COEFFICIENTS GENERALISATION TO NON-EQUILIBRIUM THE FOKKER-PLANCK EQUATION AND THE DIFFUSIVE LIMIT APPROACH TO THE BROWNIAN LIMIT AND LIMITATIONS A BASIC LIMITATION OF THE LE AND FP EQUATIONS THE FRICTION COEFFICIENT SELF-DIFFUSION COEFFICIENT (DS) THE INTERMEDIATE SCATTERING FUNCTION F(Q,T) SYSTEMS IN A SHEAR FIELD LANGEVIN AND GENERALISED LANGEVIN DYNAMICS EXTENSIONS OF THE GLE TO COLLECTIONS OF PARTICLES NUMERICAL SOLUTION OF THE LANGEVIN EQUATION GAUSSIAN RANDOM VARIABLES A BD ALGORITHM TO FIRST-ORDER IN T A SECOND FIRST-ORDER BD ALGORITHM A THIRD FIRST-ORDER BD ALGORITHM THE BD ALGORITHM IN THE DIFFUSIVE LIMIT HIGHER-ORDER BD SCHEMES FOR THE LANGEVIN EQUATION GENERALISED LANGEVIN EQUATION THE METHOD OF BERKOWITZ, MORGAN AND MCCAMMON THE METHOD OF ERMAK AND BUCKHOLZ THE METHOD OF CICCOTTI AND RYCKAERT OTHER METHODS OF SOLVING THE GLE SYSTEMS IN AN EXTERNAL FIELD BOUNDARY CONDITIONS IN SIMULATIONS PBC IN EQUILIBRIUM PBC IN A SHEAR FIELD PBC IN ELONGATIONAL FLOW BROWNIAN DYNAMICS FUNDAMENTALS CALCULATION OF HYDRODYNAMIC INTERACTIONS ALTERNATIVE APPROACHES TO TREAT HYDRODYNAMIC INTERACTIONS THE LATTICE BOLTZMANN APPROACH DISSIPATIVE PARTICLE DYNAMICS BROWNIAN DYNAMICS ALGORITHMS THE ALGORITHM OF ERMAK AND MCCAMMON APPROXIMATE BD SCHEMES BROWNIAN DYNAMICS IN A SHEAR FIELD LIMITATIONS OF THE BD METHOD ALTERNATIVES TO BD SIMULATIONS LATTICE BOLTZMANN APPROACH POLYMER DYNAMICS TOXVAERD APPROACH DIRECT USE OF BROWNIAN DYNAMICS RIGID SYSTEMS THEORIES BASED ON DISTRIBUTION FUNCTIONS, MASTER EQUATIONS AND STOCHASTIC EQUATIONS FOKKER-PLANCK EQUATION THE DIFFUSIVE LIMIT AND THE SMOLUCHOWSKI EQUATION SOLUTION OF THE N-BODY SMOLUCHOWSKI EQUATION POSITION-ONLY LANGEVIN EQUATION QUANTUM MONTE CARLO METHOD MASTER EQUATIONS THE IDENTIFICATION OF ELEMENTARY PROCESSES KINETIC MC AND MASTER EQUATIONS KMC PROCEDURE WITH CONTINUUM SOLIDS 10 AN OVERVIEW