Chapter 10: MC Method Flashcards Preview

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Flashcards in Chapter 10: MC Method Deck (9):
1

MC Method
(Brief Outline)

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

2

Simple Sampling

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

3

Simple Sampling 
(Disadvantages)

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

4

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

5

Ising Model 
(Limitation)

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

6

Importance Sampling

Choose states according to Boltxmann weight using a Markhov vhain

7

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

8

Markhov Chains 
(Properties)

  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

9

Metropolis Algorithm

local-update scheme with one spin flip per move