PHYCS 498NSM Non-Equilibrium Statistical Mechanics
Spring 1999
Instructor: Professor Klaus Schulten

Course Outline
  1. Ising Spins on Regular Bodies
    1. Analytical Theory
    2. Monte Carlo Simulation
    3. Application to Viral Infection, Fever and Vaccines

  2. Classical Dynamics under the Influence of Stochastic Forces
    1. Langevin Equation and Stochastic Differential Equations
    2. Examples of Stochastic Processes
    3. From Stochastic Differential Equations to Fokker-Planck Equations

  3. Einstein and Smoluchowski Diffusion Equation
    1. Free Diffusion
      • rms deviation
      • Green's Function
    2. Smoluchowski equation
      • equilibrium and zero flux condition
      • fluctuation-dissipation theorem
      • final form of Smoluchowski equation
      • flux operator
    3. Boundary Conditions
    4. Examples:
      • freely diffusing particle in half-space with/without reactive wall
      • freely diffusing particle in bounded domain

  4. Solution of the Smoluchowski Equation
    1. Diffusion in Linear Potential
    2. Diffusion in Harmonic Potential
      • solution through eigenfunction expansion
      • solution through transformation to time-dependent coordinate

  5. Noise-induced limit cycles
  6. Rates of Diffusion-Controlled Reactions
    1. Equation for Relative Diffusion of Two Particles
    2. Diffusion-Controlled Reactions under Stationary Conditions
    3. Derivation of Bimolecular Reaction Rates
      • general case
      • free diffusion
      • diffusion in Coulomb field
      • diffusion in Debey-Hueckel potential

  7. The Adjoint Smoluchowski Equation
    1. Homogenous Time Property
    2. Chapman-Kolmogorov Equation
    3. Adjoint Smoluchowski Operator
    4. Backward Smoluchowski Equation

  8. Spectral Expansion of Propagator of Smoluchowski Equation
    1. Variational Principle
    2. Similarity to Selfadjoint Operator
    3. Spectrum of Smoluchowski Operator
    4. Left and Right Eigenfunctions of Smoluchowski Operator
    5. Bi-orthogonality
    6. Projection Operators
    7. Expansion of Propagator
    8. Asymptotic Behaviour

  9. Observables Connected with Brownian Transport
    1. Definition of Correlation Functions
    2. Examples of Correlation Functions
      • particle number correlation function
      • reaction rate
      • reaction yield
      • photobleaching
      • Moessbauer line shape function
    3. Spectral Expansion of Correlation Functions
    4. Asynmptotic Behaviour
    5. Evaluation of Particle Correlation Function Using the Backward Equation
    6. Examples of Evaluated Correlation Functions
      • particle number for free diffusion and reaction in finite domain
      • photobleaching
      • Moessbauer line shape function for harmonically bound 57Fe

  10. Generalized Moment Expansion of Correlation Functions
    1. Laplace Transformed Correlation Function
    2. Laurent Series of Correlation Functions
    3. The Moments and Generalized Moments
    4. 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
    5. Double-Exponential Approximatons

  11. Examples of Generalized Moment Expansion
    1. Mean First Passage Time Approximation
    2. Barrier Crossing Rates
    3. Limit of High Barriers
    4. Diffusion and Reaction in Finite Domain
      • pheromones diffusing in antennas of butterflies
      • single-exponential approximation
      • double-exponential approximation
    5. Moessbauer Line Shape Function
    6. Ions Diffusing Through a Membrane Pore
    7. Relaxation Rates in Magnetiuc Resonance Imaging

  12. The Master Equation
    1. Discrete Systems
    2. The Linear Rate Equation
      • conclusion from exitence of a single equilibrium
      • conclusion from particle conservation
      • conclusion from zero equilibrium flux condition
    3. Left and Right Eigenvectors of Rate Operator
    4. Spectrum of Rate Operator
    5. Spectral Expansion of Propagator
    6. Correlation Functions
    7. Laplace Transfrom of Correlation Function and Laurent Series
    8. Generalized Moment Expansion of Correlation Function
    9. Birth-Death Processes
    10. Examples

  13. Linear Response Theory
    1. Change of Correlation Function for Small Temporal Perturbation
    2. Definition of Response Function
    3. Fluctuation-Dissipation Theorem
    4. Examples
      • Evaluation of Velocity Autocorrelation Function
      • Velocity Autocorrelation Function in Proteins
      • Temperature Quench Echoe
      • Dielectric Response in Photosynthetic Reaction Center

  14. Theory of Echoes
    1. Heuristics of Temperature and Dipole Echoes
    2. Harmonic Theory of Echoes
    3. Echoes in a Linear Harmonic Chain
    4. Langevin Oscillator Theory of Echoes

  15. Theory of Hysteresis
    1. Dipole Fluctuations
    2. Hysteresis in Linear Response Theory
    3. Hysteresis in Multi-Stable Potentials

  16. Mathematics of Financial Derivatives
  17. Combined Classical/Quantum Mechanics
    1. two oscillators
    2. photodynamics of retinal

  18. Two State Quantum System Coupled to Classical Bath
    1. Kubo Line Shape Theory
    2. Application to Brownian Processes
    3. Examples: Brownian Oscillator, MRI Microscopy, Polymer Dynamics

  19. Spin-Boson Model
    1. Two-State System Coupled to Quantum Oscillator
    2. Two-State System Coupled to Ensemble of Quantum Oscillators
    3. Application to Electron Transfer

  20. Path Integral Presentation
    1. Three-State System Coupled to Ensemble of Quantum Oscillators
    2. Application to Tunneling

This document was last modified on 02/05/00 by Klaus Schulten