PHYCS 498NSM
Non-Equilibrium Statistical Mechanics
Spring 1999
Instructor:
Professor
Klaus Schulten
Course Outline
Ising Spins on Regular Bodies
Analytical Theory
Monte Carlo Simulation
Application to Viral Infection, Fever and Vaccines
Classical Dynamics under the Influence of Stochastic Forces
Langevin Equation and Stochastic Differential Equations
Examples of Stochastic Processes
From Stochastic Differential Equations to Fokker-Planck Equations
Einstein and Smoluchowski Diffusion Equation
Free Diffusion
rms deviation
Green's Function
Smoluchowski equation
equilibrium and zero flux condition
fluctuation-dissipation theorem
final form of Smoluchowski equation
flux operator
Boundary Conditions
Examples:
freely diffusing particle in half-space with/without reactive wall
freely diffusing particle in bounded domain
Solution of the Smoluchowski Equation
Diffusion in Linear Potential
Diffusion in Harmonic Potential
solution through eigenfunction expansion
solution through transformation to time-dependent coordinate
Noise-induced limit cycles
Rates of Diffusion-Controlled Reactions
Equation for Relative Diffusion of Two Particles
Diffusion-Controlled Reactions under Stationary Conditions
Derivation of Bimolecular Reaction Rates
general case
free diffusion
diffusion in Coulomb field
diffusion in Debey-Hueckel potential
The Adjoint Smoluchowski Equation
Homogenous Time Property
Chapman-Kolmogorov Equation
Adjoint Smoluchowski Operator
Backward Smoluchowski Equation
Spectral Expansion of Propagator of Smoluchowski Equation
Variational Principle
Similarity to Selfadjoint Operator
Spectrum of Smoluchowski Operator
Left and Right Eigenfunctions of Smoluchowski Operator
Bi-orthogonality
Projection Operators
Expansion of Propagator
Asymptotic Behaviour
Observables Connected with Brownian Transport
Definition of Correlation Functions
Examples of Correlation Functions
particle number correlation function
reaction rate
reaction yield
photobleaching
Moessbauer line shape function
Spectral Expansion of Correlation Functions
Asynmptotic Behaviour
Evaluation of Particle Correlation Function Using the Backward Equation
Examples of Evaluated Correlation Functions
particle number for free diffusion and reaction in finite domain
photobleaching
Moessbauer line shape function for harmonically bound 57Fe
Generalized Moment Expansion of Correlation Functions
Laplace Transformed Correlation Function
Laurent Series of Correlation Functions
The Moments and Generalized Moments
Evaluation of Moments for 1-Dim. Systems
Recursive Evaluation
Solving the Inhomogeneous Smoluchowski Equation
Inverse Smoluchowski Operator Expressed Through Double Integral
Using Forward or Backward Operators
Double-Exponential Approximatons
Examples of Generalized Moment Expansion
Mean First Passage Time Approximation
Barrier Crossing Rates
Limit of High Barriers
Diffusion and Reaction in Finite Domain
pheromones diffusing in antennas of butterflies
single-exponential approximation
double-exponential approximation
Moessbauer Line Shape Function
Ions Diffusing Through a Membrane Pore
Relaxation Rates in Magnetiuc Resonance Imaging
The Master Equation
Discrete Systems
The Linear Rate Equation
conclusion from exitence of a single equilibrium
conclusion from particle conservation
conclusion from zero equilibrium flux condition
Left and Right Eigenvectors of Rate Operator
Spectrum of Rate Operator
Spectral Expansion of Propagator
Correlation Functions
Laplace Transfrom of Correlation Function and Laurent Series
Generalized Moment Expansion of Correlation Function
Birth-Death Processes
Examples
Linear Response Theory
Change of Correlation Function for Small Temporal Perturbation
Definition of Response Function
Fluctuation-Dissipation Theorem
Examples
Evaluation of Velocity Autocorrelation Function
Velocity Autocorrelation Function in Proteins
Temperature Quench Echoe
Dielectric Response in Photosynthetic Reaction Center
Theory of Echoes
Heuristics of Temperature and Dipole Echoes
Harmonic Theory of Echoes
Echoes in a Linear Harmonic Chain
Langevin Oscillator Theory of Echoes
Theory of Hysteresis
Dipole Fluctuations
Hysteresis in Linear Response Theory
Hysteresis in Multi-Stable Potentials
Mathematics of Financial Derivatives
Combined Classical/Quantum Mechanics
two oscillators
photodynamics of retinal
Two State Quantum System Coupled to Classical Bath
Kubo Line Shape Theory
Application to Brownian Processes
Examples: Brownian Oscillator, MRI Microscopy, Polymer Dynamics
Spin-Boson Model
Two-State System Coupled to Quantum Oscillator
Two-State System Coupled to Ensemble of Quantum Oscillators
Application to Electron Transfer
Path Integral Presentation
Three-State System Coupled to Ensemble of Quantum Oscillators
Application to Tunneling
This document was last modified on 02/05/00 by
Klaus Schulten