__Instructor:__

`Rob Coalson`

`coalson@pitt.edu
`

`Eberly`` 321, (412)624-8261`

` `

**Lecture Notes**

**1)
****Notes, Jan. 11, 2009:
**Problem of Random Flights, Markoff's method, with applications to polymer extension and swelling; the Central Limit Theorem; the Diffusion Equation.

**2)
****Notes, Jan. 13, 2009: **Langevin Eq.; free particle velocity distribution, Einstein relation between friction and random force; position distribution for free particle: reduction to Diffusion Equation in high friction limit.

**3)
****Notes, Jan. 15, 2009: **Langevin Eq. for particle in a constant force field; stochastic Langevin dynamics algorithm for motion in an arbitrary force field and its reduction in the high-friction limit.

**4)
****Notes, Jan. 20: **Velocity auto-correlation function for a particle undergoing Langevin dynamics; correlation functions of the random force.

**5)
****Notes, Jan. 22, 2009: **Connection of stochastic ordinary differential equations (ODEs) to deterministic particle differential equations (PDEs): equivalence of Langevin Eq. and phase space Fokker Planck Eq.; equivalence of high-friction Langevin Eq. and Smoluchowski Eq.

**6)
****Notes, Jan. 27, 2009: **Thermally Activated Barrier Crossing: Transition State Theory, Kramer's Theory in the high and low friction limits.

**7)
****Notes, Feb. 5**Transformation of the Smoluchowski Eq. to Schrodinger form.

**8)
****Notes, Feb. 10, 2009: **Generalized Langevin Eq. (GLE): Phenomenology and Fluctuation-Dissipation Theorems.

**9)
****Notes, Feb. 17, 2009: **Derivation of the Generalized Langevin Eq. from a microscopic Hamiltonian (a system coordinate coupled blinearly to a bath of harmonic oscillators).

**10)
****Notes,
Feb. 25, 2009: **1-D Drift-Diffusion Equations (steady state of the Smoluchowski Eq.): quadrature formulae for the steady state concentration profile and current; application to ion permeation through channel proteins (GHK theory).

**11)
****Notes,
Mar. 5, 2009: **Computing mean first passage times in high-friction Brownian motion via the Adjoint Equation: reduction to quadrature for 1D motion; PDE formulation for multi-dimensional motion.

**12)
****Notes,
Mar. 24, 2009: **Fermi's Golden Rule for state-to-state quantum mechanical transition rates: derivation using time-dependent perturbation theory; requirement of a dense manifold of accepting states; counterexample: quantum dynamics of an isolated two-level system.

**13)
****Notes,
Mar. 26, 2009: **Decay of a Doorway State: Exact vs. Golden Rule quantum dynamics.

**14)
****Notes,
Apr. 9, 2009: **Spectroscopy and the Golden Rule. Time-dependent perturbation theory with a monochromatic driving field: transition probabilities and cross sections for 1-photon absorption and emission; time-independent (sum over states) vs. time-dependent (correlation function) formulations; Application to vibronic absorption spectra: Franck-Condon factors, time-correlation function formulation for pure initial states and finite temperature systems.

**15)
****Notes,
July. 20, 2009: **Simple Quantum Relaxation Theory: Population Relaxation of a System Coupled to a Bath. (Golden Rule for System Transitions; Detailed Balance; Pauli Master Equations; Semi-classical Evaluation of Heisenberg Correlation Functions.)

**Problem Sets**

**1)
****Problem Set 1,
Solution Key**

**2)
****Problem Set 2,
Solution Key**

**3)
****Problem Set 3,
Solution Key**

**Guest Lectures**

**1)
****MC_feb09. **Dr. Mary Cheng: Brownian Dynamics Simulation of Ion Permeation through Protein Channels.

**2)
****IK_apr09. **Dr. Igor Kurnikov: Non-adiabatic Electron Transfer in Chemistry and Biology.

**3)
****KW_apr09. **Dr. Kim Wong: Approaches for Calculating Non-adiabatic, Energy Transfer, and Chemical Reaction Rates.

**Student Presentations**

**1)
****JX_apr09. **Jiawei Xu: Linear Response Theory and Its Applications.

**2)
****AS_apr09. **Andrey Sharapov: Debye-Falkenhagen Theory and Its Computer Simulation.

**3)
****GZ_apr09. **Guozhen Zhang: Poisson-Nernst-Planck Theory Approach to the Calculation of Ion Transport through Protein Channels.

**4)
****FY_apr09. **Fangyong Yan: Graphical Rule-Based Modeling of Signal Transduction Systems.