PHYS 498NSM Non-Equilibrium Statistical Mechanics Fall 2003

Course Outline

1. 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)

2. 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

3. Stochastic differential equations

• Langevin equation
• Wiener process
• Ornstein-Uhlenbeck process
• Nonlinear Langevin equation
• Ito and Stratonovich calculus
• Change of variables

4. 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

5. 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

6. 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

7. 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

8. Linear response theory

• Response function
• Correlation functions
• Generalized susceptibility
• Fluctuation-dissipation theorem
• Examples: velocity autocorrelation function; echoes, hysteresis.

9. 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

10. 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