Computational Physics: Advanced Monte Carlo Methods
First semester 2016-17
This course provide an Introduction to Monte Carlo techniques in
Equilibrium and Non-equilibrium Systems.
The program is:
Monte Carlo computation of Integrals, Importance sampling, Markov Chains
and Detailed Balance, The Metropolis Algorithm, Critical slowing down,
Cluster algorithms, Monte Carlo in continuous time, Kawasaki algorithm:
local and non-local, Coupled chemical reactions: the Gillespie algorithm.
Lecture notes (pdf)
One of the first papers on Monte Carlo methods N. Metropolis and
S. Ulam, The Monte
Carlo method, JSTOR 44, 335 (1949).
M.E.J. Newman and G.T. Barkema, "Monte Carlo Methods in Statistical
Physics" (Oxford University Press). (This is an excellent and very
detailed book about Monte Carlo simulations in classical Statistical
Physics, where the emphasis is on lattice models).
D. Frenkel and B. Smit, "Understanding molecular simulations", Academic Press (2002).
(A very detailed and good book, containing also some material on Molecular Dynamics simulations).
B.A. Berg, Markov-Chain
Monte Carlo Methods for Simulations of Biomolecules.
An online course on Monte
Carlo simulation is provided by P. Coddington, Monte
Carlo simulations in Statistical Physics
D. T. Gillespie, Exact Stochastic
Simulation of Coupled Chemical Reactions, J. Phys. Chem. 81, 2340 (1977)
Applet fot the Buffon needle
Demonstration of Buffon's needle
Estimating pi with the hit-and-miss method
Calculation of pi using rain
Applet for Ising model simulations (I) -
This applet gives the possibility of choosing between different algorithms (Single spin flip,
Applet for Ising model simulations (II) -
This applet is very fast, but only restricted to single spin flip. It shows also magnetization
and energy plots produced while the simulation runs
Simple Codes in Octave/Matlab
Octave is a language intended for numerical computations, which is
essentially compatible with Matlab, but it is free. You can download it from this website. Here below some examples of simple codes in Octave. They
should also work in Matlab, although I haven't tested them.
Uniform Random Numbers with a gap
Estimating the area of a circle of radius 1 using the hit-and-miss method
A simple example of Markov process via matrix products and direct Monte Carlo simulation
How to select efficiently from q states according to a given probability distribution (q large)
(to run these codes in Octave copy them on a file, say file.m,
and they type source("file.m") from the octave input line).
The exam consists in the discussion of the assignments, with some
questions about the theory. All relevant output of your exercises should
be collected in a single pdf file containing the plots produced by
your simulations. Each figure should contain a caption describing
(i) what you plot (eg. x-y axes with their units, MC steps or, better, use
sweeps), (ii) the parameters used (e.g.T=0.8, q=10 Potts model) and (iii)
the type of algorithm used (eg. "Local Kawasaki", "Heat Bath"...). Send
this file with your codes by e-mail a few days before the exam. The
codes should also contain some comments so that the different parts can
For the exam all assignments up to 4.8 (included) should be solved.
In addition choose one of the additional assignments from the list below.
Wolff algorithm for the Ising model
The Wolff algorithm is a cluster algorithm for the Ising model which
does not suffer from the critical slowing down as the single spin
flip Metropolis algorithm. In this assignment the student uses the
Wolff algorithm to compute the autocorrelation time τ for the
two dimensional Ising model at the critical point T=Tc. Show that tau
diverges as a power law as a function of the lattice size L: τ ~
L^z and determine the value of the dynamical exponent z. Use finite size
scaling ansatz to analyze the behavior of the magnetic susceptibility in
the vicinity of the critical point and determine the exponents γ
and ν. To solve this assignment check the book by Newman and Barkema
where details are discussed.
The Kawasaki dynamics and continuous time algorithm
The Kawasaki dynamics consists in swapping the positions of two opposite
spins of an Ising model using Metropolis acceptance rule. In this way the
total magnetization of the system remains conserved. There are different
ways to implement the Kawasaki algorithm. One can use for instance a local
or a non-local algorithm. In this assignment the student implements both
algorithms as discussed in Problems 4.10 and 4.11 of the lecture notes.
Facultative: The student may also try to implement a continuous time
algorithm, as discussed in the book by Newman and Barkema.
The heat bath algorithm for the Potts model
The Metropolis algorithm is very popular and simple to implement. However
there are situations in which it does not work very well and there are
more suitable Monte Carlo algorithms. This is the case of the Potts model,
which is a generalization of the Ising model to more states per site.
In this assignment the student compares the performances of the Metropolis
and Heat Bath algorithms for the q-state Potts model (Problem 4.9 of the
Hard spheres fluid
In the assignment about hard spheres the student learns to use the
Monte Carlo algorithm for a continuum system (Problem 4.12 of the
lecture notes). The scope of the problem is the calculation of the
pair correlation function g(r). Faculatitve: from this quantity one can
compute the pressure of the system using the virial theorem and compare
the results with those of the virial expansion. Another possibility is
to try some cluster algorithms which are quite easy to implement.
Coupled chemical reactions and the Gillespie algorithm
The Gillespie algorithm is a Monte Carlo algorithm to simulate systems of
coupled chemical reactions. The system is assumed to be well-mixed so that
no spatial effects are taken into account. In this assignment the students
solve the last three problems of the lecture notes (Birth-annihilation
process, Lotka-Volterra Model, Brusselator).
Kosterlits-Thouless transition in the XY model
The two dimensional xy model is a model of interacting spins, which
differently from the Ising case, can point to any direction on the xy
plane. The peculiarity of this system is that there can be no spontaneous
symmetry breaking, as it has been rigorously proved. Therefore there
is no low temperature ferromagnetic phase. Nonetheless there is a phase
transition. The nature of this phase transition was elucidated by Kosterlitz
and Thouless who got the Nobel prize for Physics in 2016 for this (and other)
work. It is a topological phase transition involving the unbinding of
vortices and anti-vortices.