5.72 Problem set #6
J. Cao
(The problems can be solved by following the lecture notes.)
I. The self-intermediate scattering function (, ) = [()(0)] can be expressed
in even moments
2 = (1)
(0,2)
( , 0).
4
2
2
2
1) Show 0 = 1, 2 = 0 , and 4 = 30 + 0 2 ,
5.72 Problem Set #5 (Spring 2012)
Cao
I. Density fluctuations
1) Derive the equilibrium value of density fluctuations
1
() = | |2 = 1 + 0 ()
2) ( = 0) = 0 , where
1
=
is the isothermal compressibility.
3) Show 2 Re (, = ) = (, ) where (, ) = (, ) is
5.72 Problem set #4 (spring 2012)
Cao
I. Harmonic Oscillator
2
1) For a harmonic oscillator, show that () = ()(0) satisfies + 0 = 0.
2) Solve for () and its Fourier transform () = () .
3) The forced oscillator obeys the equation of motion
2
+ 0 = () .
Problem set # 3 for 5.72 (spring 2012)
Cao
I. A Gaussian variable with = 0 and 2 = is described by the probability
function
2
1
2
() =
.
2
1) Calculate () = () .
2) Show that 2 =
(2)!
2 !
.
II. A Brownian particle in the potential of a constant force f
Problem Set #2 for 5.72 (spring 2012)
I. A diffusive oscillator satisfies the Fokker-Planck equation
2
= 2 + (),
with = 2 .
1) Find the equilibrium distribution.
2) Show = 0 .
3) Assume that takes the form of
(0 )2
1
2 ()
(0 , , ) =
.
2()
Find () expli
Problem set #1 for 5.72 (spring 2012)
Cao
I. Consider a random walk on a linear three-site model. and are the probabilities of
moving right and left respectively. ( + = 1)
1
2
3
1) Write the transition matrix .
2) Find the stationary distribution and show
Chapter 4
Calculations of Time Correlation
Functions
In the last chapter, we explored the low-frequency, long-wavelength behavior of a system that is
disturbed from equilibrium. In the rst section of this chapter, we study the opposite limit, and
describe
Chapter 3
Hydrodynamics and Light Scattering
Hydrodynamics describes the low-frequency, long-wavelength behavior of a system that is disturbed
from equilibrium. When a system is disturbed from equilibrium, some quantities and parameters
decay very quickly
Chapter 2
Non-equilibrium Thermodynamics
2.1
Response, Relaxation and Correlation
At the beginning of the 21st century, the thermodynamics of systems far from equilibrium remains
poorly understood. However, it turns out that many nonequilibrium phenomena
Chapter 1
Stochastic Processes and Brownian
Motion
Equilibrium thermodynamics and statistical mechanics are widely considered to be core subject
matter for any practicing chemist [1]. There are plenty of reasons for this:
A great many chemical phenomena
Final Projects for 5.72
Jianshu Cao
May 29, 2012
Select any two of the projects. You are advised but not required to
choose projects relevant to your research. You are advised but not
required to choose one from project 13 on reaction rate theory and
on