Statistical Mechanics of Complex Systems

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

Please send the exercises to suweis@pd.infn.it 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.

Fractals

  • 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

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