Markov Chain Monte Carlo
Outline
● Discrete Markov chains
● Example: The Ehrenfest Urn model ● The algorithm of Metropolis
● Example: The Ising model
Markov Chain Monte Carlo Discrete homogeneous Markov chains
Markov Chain Monte Carlo Discrete homogeneous Markov chains
Markov Chain Monte Carlo Discrete homogeneous Markov chains
Markov Chain Monte Carlo Discrete homogenous Markov chains
Markov Chain Monte Carlo Discrete homogenous Markov chains
Markov Chain Monte Carlo Discrete homogeneous Markov chains
Markov Chain Monte Carlo
Discrete homogeneous Markov chains Example: The Ehrenfest urn
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Markov Chain Monte Carlo
Discrete homogeneous Markov chains Example: The Ehrenfest urn
Markov Chain Monte Carlo
Discrete homogenous Markov chains Example: The Ehrenfest urn
Markov Chain Monte Carlo Discrete homogeneous Markov chains
Example: The Ehrenfest urn, Monte Carlo simulation
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Markov Chain Monte Carlo The algorithm of Metropolis
Markov Chain Monte Carlo The algorithm of Metropolis
Markov Chain Monte Carlo The algorithm of Metropolis
Markov Chain Monte Carlo
The algorithm of Metropolis Example: The Ising model
Markov Chain Monte Carlo
The algorithm of Metropolis Example: The Ising model
Markov Chain Monte Carlo The algorithm of Metropolis
Example: The Ising model (square lattice plaquette) 43
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Markov Chain Monte Carlo
The algorithm of Metropolis
Example: The Ising model (square lattice plaquette)
Markov Chain Monte Carlo The algorithm of Metropolis
Example: The Ising model (square lattice plaquette)
Entropy and specific heat as a function of inverse temperature. The peak in specific heat is a signature of a phase transition in the limit ∣V∣→∞.
Markov Chain Monte Carlo The algorithm of Metropolis
Example: The Ising model (square lattice plaquette)
Markov Chain Monte Carlo
The algorithm of Metropolis Example: The Ising model
Markov Chain Monte Carlo The algorithm of Metropolis
Example: The Ising model
At http://physics.ucsc.edu/~peter/ising/ising.html, it is possible to have a qualitative idea of the behavior of the square-lattice Ising model.
You can see videos on YouTube: http://www.youtube.com/watch?v=XtY5dLN5CGE (5′) including presentations on how to write a Monte Carlo algorithm: http://www.youtube.com/watch?v=gsMNqJee87E
Markov Chain Monte Carlo
The algorithm of Metropolis
Example: The Ising model (square lattice plaquette)
Markov Chain Monte Carlo
Markov Chain Monte Carlo
The end (for now)!