PHYCS 498NSM | Non-Equilibrium Statistical Mechanics |
Spring 1999 | |
Instructor: | Professor Klaus Schulten |
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
¥ = ]-¥,
¥[
4.2.2 Diffusion in
a Half-Space W¥ = [0, ¥[
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