Fokker-Planck Equation
Tuesday, March 26, 2013
2:01 PM
Reading: Gardiner Secs. 3.4-3.6
We now will employ the PDE (Fokker-Planck) approach to the Langevin model for Brownian motion. We
prepare by stating the following general result, whose derivation is g
Langevin Equation Model for Brownian Motion
Tuesday, February 26, 2013
2:01 PM
Reading: Gardiner Sec. 1.2
Homework 2 due Tuesday, March 19.
The Brownian motion model we have been working with so far is phenomenological;
based on properties of successive o
Method of Moments
Friday, March 08, 2013
2:37 PM
Homework 2 due Tuesday, March 19.
Sloppy method of moments:
Consider the Langevin equation written in terms of delta-correlated force.
Try to compute the statistics of the velocity covariance at a single mo
Physical Interpretation of Langevin Equation Solution Statistics
Tuesday, March 19, 2013
1:53 PM
What about the statistics of the position of a Brownian particle under the
Langevin equation?
Method of moments approach was the second homework problem on th
Velocity Autocorrelation in Langevin Equation
Tuesday, March 05, 2013
1:59 PM
Reading: Gardiner Secs. 4.1-4.3, 4.4.4
Homework 2 due Tuesday, March 19.
We now will use the formal stochastic calculus rule for computing the autocorrelation
function of the ve
Dimensional Analysis for Klein-Kramers Equation
Tuesday, April 02, 2013
2:08 PM
Last time, we argued by a hand-waving procedure that the
following stochastic differential equations describing the
dynamics of a particle subject to an external force:
Might
Analytical Uses of the Fokker-Planck Equation
Friday, March 29, 2013
1:58 PM
When one is working with systems of stochastic differential equations that are not exactly
solvable, then what can one do?
Numerical simulation of trajectories and numerical eva
Relation of Langevin and Wiener Process Model
Friday, March 22, 2013
2:20 PM
From last time we computed that:
Presumably this should be negligible if the random kicks are supposed to be
independent over different time intervals. To see this, we need to re
Stochastic Calculus
Friday, March 01, 2013
2:03 PM
Reading: Gardiner Sec. 4.2
Homework 2 will be posted tonight, due Tuesday, March 19.
From last time, we argued that the thermal force should have the
following properties:
Zero mean
Covariance proportio
Reconciliation of PDE and Trajectory Perspectives
Friday, February 22, 2013
2:02 PM
We recall that from the trajectory-based (Lagrangian) perspective, the continuum
limit description was:
From the probability distribution perspective (Eulerian),
the conti
Probability and Random Variables
Tuesday, January 29, 2013
1:18 PM
References:
Sections from my lectures notes with Majda on Turbulent Diffusion
Minier and Peirano, Sec. 2
Gardiner, Secs. 2.1-2.5, 2.8
Office hours:
Wednesdays 11 AM - 12 PM (this class
Brownian motion
Friday, January 25, 2013
2:03 PM
References: K, "Brownian Motion"
Gardiner, Sec. 1.2
Nelson, Dynamical Theories of Brownian Motion Secs. 1 -4
Just for historical purposes
Brownian motion is the irregular trajectories seen in particles at
Books for course
Tuesday, January 22, 2013
3:09 PM
MCM practice at 5:15 PM in Walker 5113.
Office hours: Between 01/23 and 5 Pm on 01/24: please fill
out Office hours survey on the class website.
Two online websites for finding used books:
bookfinder.com
Important Facts about Random Variables
Friday, February 01, 2013
2:01 PM
Homework 1 hopefully posted over the weekend.
Two more fundamental concepts about random variables.
Often it is useful to summarize a random variable not in complete detail (as would
Brownian Motion Increments
Tuesday, February 05, 2013
2:05 PM
Homework 1 posted, due Friday, February 22.
We argued last time that the increment of the position of a Brownian particle between two
successive experimental observations can be expressed as a
Kolmogorov/Smoluchowski equation approach to Brownian motion
Tuesday, February 12, 2013
1:53 PM
Readings: Gardiner, Secs. 1.2, 3.8.1, 3.8.2
Einstein
Homework 1 due February 22.
Conditional probability for random variables.
Let's transfer our results for c
Extension to Multiple Dimensions
Friday, February 08, 2013
2:03 PM
No office hours today.
To start, let's see why the following simple SDE:
Is a good continuum limit description of our discrete-time model for Brownian
motion. Here W(t) is to be understood
Interlude on Numerical Simulation of Random Variables
Friday, February 15, 2013
2:00 PM
Homework 1 due Friday, February 22 at 2 PM.
No class on Tuesday, February 19.
Reading: Kloeden and Platen, Sec. 1.3
Simulating models with random variables is often ca