Metropolis Algorithm

Simple Metropolis algo based on Markow chain theory

  • Probability - \(P_i(t)\)
  • Transition probabilty
    • \(w(i\rightarrow j)\) (stochastic matrix or transition probability)
    • \(\Sigma_j w_{ij} = 1\) Largest Eigenvalue \(\lambda_{max} = 1\)
    • Example (of stochastic matrix:): \(w = [1/3 1 2/3][0 0 1/3][2/3 0 0]\)
    • \(\Sigma_i P_i(t) = 1\)
  • Markow Chain:
    • \(P_i(t+\epsilon) = \Sigma_j w(j \rightarrow i) P_j(t)\)
    • Know/Have a model:
      • \(P_i(t + \epsilon) \rightarrow P_T(\vec{k}_j \alpha) = \frac{|\psi_T(\vec{k}_j \alpha)|^2}{\int d\vec{k} |\psi_T(\vec{k}_j \alpha)|^2}\)
    • Metropolis test: we accept new move if \(\omega \in [0,1]\), \(\omega \leq \frac{P_T(\vec{k'};\vec{a})}{P_T(\vec{k};\vec{a})}\)
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