PHYS 498NSM Non-Equilibrium Statistical Mechanics
Fall 2003

Lecture Notes


Most of the material covered in the course is presented (in a slightly different order) in the following lecture notes, available in printing quality PDF format. Despite careful editing, the notes still contain many typos and missing (or faulty) cross-references. Bringing these to my attention will be greatly appreciated! As the course progresses, additional course material will be added to this page.


2.  Dynamics under the Influence of Stochastic Forces
    2.1  Newton's Equation and Langevin's Equation
    2.2  Stochastic Differential Equations
    2.3  How to Describe Noise
    2.4  Ito calculus
    2.5  Fokker-Planck Equations
    2.6  Stratonovich Calculus
    2.7  Appendix: Normal Distribution Approximation
        2.7.1  Stirling's Formula
        2.7.2  Binomial Distribution

3.  Einstein Diffusion Equation
    3.1  Derivation and Boundary Conditions
    3.2  Free Diffusion in One-dimensional Half-Space
    3.3  Fluorescence Microphotolysis
    3.4  Free Diffusion around a Spherical Object
    3.5  Free Diffusion in a Finite Domain
    3.6  Rotational Diffusion

4.  Smoluchowski Diffusion Equation
    4.1  Derivation of the Smoluchoswki Diffusion Equation for Potential Fields
    4.2  One-Dimensional Diffuson in a Linear Potential
        4.2.1  Diffusion in an infinite space W =  ]-, [
        4.2.2  Diffusion in a Half-Space W = [0, [
    4.3  Diffusion in a One-Dimensional Harmonic Potential

5.  Random Numbers
    5.1  Randomness
    5.2  Random Number Generators
        5.2.1  Homogeneous Distribution
        5.2.2  Gaussian Distribution
    5.3  Monte Carlo integration

6.  Brownian Dynamics
    6.1  Discretization of Time
    6.2  Monte Carlo Integration of Stochastic Processes
    6.3  Ito Calculus and Brownian Dynamics
    6.4  Free Diffusion
    6.5  Reflective Boundary Conditions

7.  The Brownian Dynamics Method Applied
    7.1  Diffusion in a Linear Potential
    7.2  Diffusion in a Harmonic Potential
    7.3  Harmonic Potential with a Reactive Center
    7.4  Free Diffusion in a Finite Domain
    7.5  Hysteresis in a Harmonic Potential
    7.6  Hysteresis in a Bistable Potential

9.  Adjoint Smoluchowski Equation
    9.1  The Adjoint Smoluchowski Equation
    9.2  Correlation Functions

10.  Rates of Diffusion-Controlled Reactions
    10.1  Relative Diffusion of two Free Particles
    10.2  Diffusion-Controlled Reactions under Stationary Conditions
        10.2.1  Examples

12.  Smoluchowski Equation for Potentials: Extremum Principle and Spectral Expansion
    12.1  Minimum Principle for the Smoluchowski Equation
    12.2  Similarity to Self-Adjoint Operator
    12.3  Eigenfunctions and Eigenvalues of the Smoluchowski Operator
    12.4  Brownian Oscillator

13.  The Brownian Oscillator
    13.1  One-Dimensional Diffusion in a Harmonic Potential
 

18.  Curve Crossing in a Protein: Coupling of the Elementary Quantum Process to Motions of the Protein
    18.1  Introduction
    18.2  The Generic Model: Two-State Quantum System Coupled to an Oscillator
    18.3  Two-State System Coupled to a Classical Medium
    18.4  Two State System Coupled to a Stochastic Medium
    18.5  Two State System Coupled to a Single Quantum Mechanical Oscillator
    18.6  Two State System Coupled to a Multi-Modal Bath of Quantum Mechanical Oscillators
    18.7  From the Energy Gap Correlation Function DE[R(t)] to the Spectral Density J(w)
    18.8  Evaluating the Transfer Rate
 

   Bibliography