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

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 here (3MB).

For convenience, the lecture notes are also provided as individual chapters, and can be downloaded by clicking on the chapter title in the table of contents below.

Despite careful editing, the notes still contain many typos and missing (or faulty) cross-references. Bringing these to my attention will be greatly appreciated!

1  Introduction

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 ź =  ]-ź, ź[
4.2.2  Diffusion in a Half-Space Wź = [0, ź[
4.3  Diffusion in a One-Dimensional Harmonic Potential

5.  The Brownian Dynamics Method Applied
5.1  Diffusion in a Linear Potential
5.2  Diffusion in a Harmonic Potential
5.3  Harmonic Potential with a Reactive Center
5.4  Free Diffusion in a Finite Domain
5.5  Hysteresis in a Harmonic Potential
5.6  Hysteresis in a Bistable Potential

6.  Noise-Induced Limit Cycles
6.1  The Bonhoeffer-van der Pol Equations
6.2  Analysis
6.2.1  Derivation of Canonical Model
6.2.2  Linear Analysis of Canonical Model
6.2.3  Hopf Bifurcation Analysis
6.2.4  Systems of Coupled Bonhoeffer-van der Pol Neurons
6.3  Alternative Neuron Models
6.3.1  Standard Oscillators
6.3.2  Active Rotators
6.3.3  Integrate-and-Fire Neurons
6.3.4  Conclusions

7.  Adjoint Smoluchowski Equation
7.1  The Adjoint Smoluchowski Equation
7.2  Correlation Functions

8.  Rates of Diffusion-Controlled Reactions
8.1  Relative Diffusion of two Free Particles
8.2  Diffusion-Controlled Reactions under Stationary Conditions
8.2.1  Examples

9.  Ohmic Resistance and Irreversible Work

10.  Smoluchowski Equation for Potentials: Extremum Principle and Spectral Expansion
10.1  Minimum Principle for the Smoluchowski Equation
10.2  Similarity to Self-Adjoint Operator
10.3  Eigenfunctions and Eigenvalues of the Smoluchowski Operator
10.4  Brownian Oscillator

11.  The Brownian Oscillator
11.1  One-Dimensional Diffusion in a Harmonic Potential

12.  Fokker-Planck Equation in x and v for Harmonic Oscillator

13.  Velocity Replacement Echoes

14.  Rate Equations for Discrete Processes

15.  Generalized Moment Expansion

16.  Curve Crossing in a Protein: Coupling of the Elementary Quantum Process to Motions of the Protein
16.1  Introduction
16.2  The Generic Model: Two-State Quantum System Coupled to an Oscillator
16.3  Two-State System Coupled to a Classical Medium
16.4  Two State System Coupled to a Stochastic Medium
16.5  Two State System Coupled to a Single Quantum Mechanical Oscillator
16.6  Two State System Coupled to a Multi-Modal Bath of Quantum Mechanical Oscillators
16.7  From the Energy Gap Correlation Function DR(t)] to the Spectral Density J(w )
16.8  Evaluating the Transfer Rate
A.  Numerical Evaluation of the Line Shape Function

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