Chapter 10: MC Method Flashcards Preview

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Flashcards in Chapter 10: MC Method Deck (9):

MC Method
(Brief Outline)

  • Genreate random input for a sample domain
  • Execute scheme via detemrinistic description
  • Evaluate results to estimate certain target functions


Simple Sampling

  • each point in phase space visited with equal probability
    • random numbers drawn from uniform distrubution


Simple Sampling 

  • phase space of a complex system is not uniformly populated
    • simple sampling leads to many "unimportant" areas being visited


Ising Model

  • motivates need for Importance Sampling
  • spins on d-dimensional cubic lattice
  • Hamiltonian given by Heisenberg model
  • need partition function Z to get thermodynamic properties and free energy F
  • probabilty of each state given by Boltzmann Distribution


Ising Model 

  • Problem: direct enumeration of all states is impossible
  • Solution: generate N independent "representative" configurations using Importance Sampling to get estimator of A <A>N


Importance Sampling

Choose states according to Boltxmann weight using a Markhov vhain


Markhov Chains

  • sequence of events wherein each event is only dependent on the event directly preceding it
  • Wµν is transition probability of going from µ to ν


Markhov Chains 

  1. encoded ergodicity: all states reachable from any starting state given enough time
  2. transition probabilty out of µ must be unity
  3. equilibrium distrbution is a fixed point (i.e. once equilbirium is reached, system remains in equilibtrium
  4. principle of detail balance


Metropolis Algorithm

local-update scheme with one spin flip per move