Statistical Mechanics of Complex Systems

Here you can find the course page on the university database.

Please send the exercises to with object “Exercise Ing 2018”.

Some notes on stochastic processes

Here you can find notes on stochastic processes. Here you can find notes on persistence times of a birth-death process.

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.
  • Exercises.

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.

Examples of exams of previous years

Laboratory of Interdisciplinary Physics