General information 
Course unit name: Monte Carlo Methods
Course unit code: 574646
Academic year: 20212022
Coordinator: Bruno Julia Diaz
Department: Department of Quantum Physics and Astrophysics
Credits: 3
Single program: S
Estimated learning time 
Total number of hours 75 
Facetoface and/or online activities 
26 
 Lecture 
Facetoface and online 
20 

 Lecture with practical component 
Facetoface and online 
6 
Supervised project 
15 
Independent learning 
34 
Competences to be gained during study 
Capacity to develop Monte Carlo algorithms to solve quantum mechanical problems

Learning objectives 
Referring to knowledge
Referring to abilities, skills
To gain an experience of preparing a Final project which includes development of the code, analysis of the results, preparation of a poster or an online presentation 
Teaching blocks 
1.
Introduction
* 1.1 Random processes. Discrete and continuous random variables.
1.2 Probability distribution function. Its moments.
1.3 Normal distribution. BoxMuller random generator.
1.4 Central Limit Theorem.
2.
Evaluation of integrals
* 2.1 Crude Monte Carlo method (hitormiss)
2.2 Monte Carlo method with importance sampling
2.3 Estimation of statistical variance
3.
Random walks and Metropolis algorithm
* 3.1 Diffusion equation. Brownian motion.
3.2 Discrete and continuous random walks.
3.3. Random walk solution to Laplace equation
3.4 Importance sampling. Detailed balance condition.
3.5 Metropolis algorithm. Generation of random numbers according to simple laws of the probability distribution.
4.
Classical Monte Carlo method for manybody systems.
* 4.1 Classical Monte Carlo method. MaxwellBoltzmann distribution.
4.2 Observables in classical systems. Energy. Density profile. Pair distribution function. Static structure factor.
4.3 Thermal phase transitions
4.4 Simulated annealing method
5.
Variational Monte Carlo methods for bosons and fermions.
* 5.1 Observables in quantum systems. Hamiltonian. Energy. Onebody density matrix. Momentum distribution.
5.2 Two observables for the kinetic energy.
5.3 Variational principle. Upper bound to the groundstate energy.
5.4 Variational Monte Carlo method for bosons.
5.5 Jastrow and Nosanow trial wave functions. Gassolid quantum phase transition.
5.6 Variational Monte Carlo method for fermions. Slater determinants. Fixed node approximation.
6. Diffusion Monte Carlo method.
* 6.1 Imaginarytime projection method. Exact estimator for the groundstate energy.
6.2 Diffusion Monte Carlo method for bosons. Drift Force. Local energy. Branching.
7.
Path Integral Monte Carlo method
* 7.1 Path Integral formalism
7.2 Path Integral Monte Carlo method
7.3 Observables.
Teaching methods and general organization 

Official assessment of learning outcomes 

Reading and study resources 
Consulteu la disponibilitat a CERCABIB
Book
Malvin H. Kalos, Paula A. Whitlock “Monte Carlo Methods”
Kurt Binder “Monte Carlo Simulation in Statistical Physics”
Rafael Guardiola “Monte Carlo methods in quantum manybody theories”
Article