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