Topics to be covered
(the list of topics and their order
is preliminary for the quantum part)
Elements of probability theory
- definitions of probability
- binomial, Gaussian, and Poisson distributions
- characteristic functions and sums of random variables, Central
Limit Theorem,
- random walks
Random number generators
- randomness in Nature vs codes
- congruential rndm(), overflow problems, period, shuffled rndm()
- transformation method
- generating arbitrary distributions; rejection and hybrid methods
- universal integrals evaluator
Classical Monte Carlo. Metropolis Algorithm for
the
Ising model
- Why using random sampling make sense at all; body volume problem
- ergodicity, balance Eq., detailed balance Eq., Metropolis
algorithm
- single flip algorithm for the Ising model; heat bath update
- convergence, errorbars, blocking, bootstrap and jackknife
methods
- autocorrelation function and thermalization time
- quantities to calculate, estimators
- correlation functions, fast Fourier transform
- reweighting, histogram, and multiple histogram method
Other classical models
- continuous spin model, XY- and O(N)-models
- q-state Potts model
- glasses
- first-order transitions
- conserved order-parameter Ising model and the classical
lattice gas
- polymers
- kinetic equations
From "simple" to "art": more efficient and
elaborate methods
- What a better data structure can do?
- Swendsen-Wang, Wollf and Niedermayer cluster
algorithms
- invaded cluster algorithm
- worm algorithm (WA)
- high-temperature expansions for various models and WA
- WA for polymers
- entropic sampling; Wang-Landau approach
- simulated tempering, parallel tempering
Quantum Monte Carlo
- quantum-to-classical maping; Feynman representation of
Quantum
mechanics
using path-integrals in discrete and continuous spaces;
quantum
configuration space
- sign problem
- diagrammatic expansions, Feynman diagrams
More qunatum methods
(if we have time)
- path-integral MC in continuous space
- diagrammatic MC; how it works for the interacting lattice
systems
- Worm algorithm for quantum models
- MC estimators in quantum systems
- Stochastic series expansion
- Variational MC
- Diffusion/projection MC