COURSE OUTLINE

PHYS 498NSM	Non-Equilibrium Statistical Mechanics
Fall 2003

Course Outline

Introduction
- Deterministic differential equation (Newton's eq.)
- Stochastic differential equation (Langevin's eq.)
- Equation of motion for the probability distribution function
- Fokker-Planck equation (FPE)
- Smolukowski equation (SE)
- Examples from Molecular Dynamics Simulations
- Master equation (ME)
Probability Theory
- Random variables and probability density
- Mean value, moments, correlations, and covariances
- Characteristic function and cumulants
- Several random variables
- Conditional probability
- Correlation functions
- Time dependent random variables
- Chapman-Kolmogorov equation
- Wiener-Khintchine theorem
- Gaussian and Poisson distributions (central limit theorem)
- Limits of sequences of random variables
Stochastic differential equations
- Langevin equation
- Wiener process
- Ornstein-Uhlenbeck process
- Nonlinear Langevin equation
- Ito and Stratonovich calculus
- Change of variables
Markov Processes
- Markov processes
- Differential Chapman-Kolmogorov equation (DCKE)
- Jump process: The master equation
- Diffusion process: the FPE
- Deterministic process: Liouville's eq.
- Backward DCKE
- Examples of Markov processes: Wiener process; Random walk; Poisson process; Ornstein-Uhlenbeck process; Random telegraph noise
Focker-Plank equation
- Kramers-Moyal forward expansion
- Kramers-Moyal backward expansion
- Pawula theorem
- FPE for one variable
- FPE for N-variables
- Examples: 3D Brownian motion; 3D Brownian motion in external field; Brownian motion of two interacting particles in external field
- Change of variables; FPE in covariant form
Methods for solving the 1D FPE
- Boundary conditions
- Stationary solutions
- Eigenfunction method
- Einstein diffusion equation
- Smoluchowski diffusion equation
- Rates of diffusion controlled reactions
- Brownian oscillator
- Approximate methods
- Numerical methods
- The Brownian dynamics method
Methods for solving the N-dimensional FPE
- Boundary conditions
- Stationary solutions
- Detailed balance
- Eigenfunction method
- Other methods: Transformation of variables; Variational method; Numerical integration; Expansion into a complete set; WKB method
Linear response theory
- Response function
- Correlation functions
- Generalized susceptibility
- Fluctuation-dissipation theorem
- Examples: velocity autocorrelation function; echoes, hysteresis.
Correlation functions
- Observables connected to Brownian transport
- Examples of evaluating correlation functions
- Generalized moment expansion of correlation functions
- Examples of generalized moment expansion: mean first passage time approximation; barrier crossing rates; diffusion and reaction in finite domain; Moessbauer line shape function; ions diffusion through membranes; relaxation rates in MRI
Classical and quantum stochastic processes
- Two state system coupled to a classical heat bath
- Application to Brownian processes
- Two state system coupled to a quantum heat bath (spin-boson model)
- Application to electron transfer in proteins
- Three state system coupled to a quantum heat bath
- Application to tunneling

This document was last modified on 08/28/03 by Klaus Schulten