# Homeworks - PHYS 334 - Fall 2015

## Homework schedule

Homework 1 (16-09-2015) - Demonstrate by induction that the mean of a random walk is identically null and its variance proportional to the time step.

Homework 2 (30-09-2015) - Simulate of Langevin equation (free diffusion) with and without mass.

Show that the solution with mass converges to the solution without mass.

Simulate a Brownian particle in a harmonic potential.

Show that the equipartition of energy holds independently of m (mass) and k (trap stiffness).

Homework 3 (2-10-2015) - Study how the potential and position distribution of a Brownian particle depends on its parameters.

Show numerically how the Boltzmann distribution is satisfied.

Homework 4 (7-10-2015) - Reproduce the results in Figs. 3 and 4 of the rope article.

Also, show numerically that the motion of a Brownian particle at a fixed temperature

(fluctuation-dissipation relations!) converges towards the iso-thermal integral (as mass goes to 0)

and that a stochastic equation with state-dependent diffusion (but constant friction)

converges towards the Stratonovich integral (as the correlation time of the noise goes to 0).

Homeworks 5 (21-10-2015) - Exercises 3.1, 3.2, 3.3 and 3.8.

Homeworks 6 (4-11-2015) - Study the entropy change in a volume 2V with 2N particles when a barrier is introduced with and without Gibbs factor.

Homework 7 (13-11-2015) - Simulate the Ising model and explore its behavior as a function of temperature and magnetic field.

Homework 8 (20-11-2015) - Demonstrate strong dependence on initial condition using molecular dynamics.