Here you can find the course page on the university database.
Please send the exercises to email@example.com with object “Exercise Ing 2018”.
Some notes on stochastic processes
Introduction to Monte Carlo methods
- Generation of sequences of uniformly distributed (pseudo)random numbers, using different simple congruent methods, illustrating their performance. Source code.
- Generation of exponentially distributed random numbers, using the inverse method. Source code.
- Generation of Gaussian random numbers, using the Box-Muller method. Source code.
Maxwell-Boltzmann distribution for ideal gases
- Calculation of the pressure from microscopic collisions with the wall (Ex. 3.4 Sethna). We run a simulation and compute the histogram of collisions in the right wall during an interval of time, as a function of the moment carried by the particles. Source code (need to be improved!).
Introduction to Complex Networks
- Some slides.
- Simulation: how to generate random Erdős–Rényi and small world networks (Ex. 3.7 Sethna).
- HOMEWORK (to submit by Thursday 12 April): Analyze the Marvel-Heros Social Network! If the network is too large, and you have problem in analyzing it, you can instead study the following brain network (see Exercise 2 in the homework); it is a .dat file describing the weighted adjacency matrix, so it is very easy to upload and start to work on it.
- Summary of the Mathematica program discussed in class (.cdf can be read also without having Mathematica: you can download CDF viewer for free in the Wolfram website) CDF Mathematica Notebook.
- HOMEWORK: exercise on fractals.
Monte Carlo simulation of equilibrium processes
- Summary and Exercise. Deadline: 1 June 2018.
- Simulation: Metropolis algorithm for the Ising model in a 2D-lattice. Source code.
- Simulation: Cluster algorithm for the Ising model in a 2D-lattice. Source code.
- Notes and Simulation: Fractal in the Ising Model. Source code.
Bibliography: W. Krauth, Cluster Monte Carlo algorithms.
Stochastic Interacting Particle Models and applications
- Review on the application of the Voter Model in Ecology.
- Voter Model code from Wolfram Demonstration Project.
- Notes and code on stochastic reactions formalism and Gillespie algorithm.
- Notes on how to solve diffusion Equation for the voter model.
Bibliography: D. Gillespie, Stochastic Simulation of Chemical Kinetics, Annual Review of Physical Chemistry.