Quiz 3 - 04/13/03

Quiz 3 CHEM 430b/530b
Statistical Methods and Thermodynamics
04/13/03
 Metropolis Monte Carlo

(25 points) Item 1a: Describe the implementation of a Metropolis Monte Carlo algorithm to generate an ensemble of configurations with probability distribution $P(\xi)$.
(25 points) Item 1b: Prove that the Metropolis Monte Carlo algorithm described in item (1a) evolves any arbitrary distribution of configurations toward the equilibrium distribution $P(\xi)$.

Classical Fluids

(25 points) Item 2a: Explain how to compute the radial distribution function g(r) of liquid Ar, after having generated an ensemble of configurations of the system at thermal equilibrium.

(25 points) Item 2b: Derive an expression for the internal energy of liquid argon in terms of the radial distribution function g(r).
 

Solution:

Item 1a:

Read the description of the Metropolis Monte Carlo algorithm on page 74 of the lecture notes.

Item 1b:

Read the proof of the Metropolis Monte Carlo algorithm described on pages 75 and 76 of the lecture notes.

Item 2a:

The number of particles at a distance between r and r+dr from any particle in the fluid is

$$N(r)=\rho g(r) 4 \pi r^2 dr.$$ (420)

Therefore, the calculation of $g(r)$ involves averaging the number of particles at a distance between $r$ and $r+dr$ from any particle in the system and dividing that number by the element of volume $4 \pi r^2 dr$.

Item 2b:

See derivation of Eq. (408) on page 97 of the lecture notes.