Chapter 10: MC Method

Question 1

MC Method

(Brief Outline)

Answer 1

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

Question 2

Simple Sampling

Answer 2

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

Question 3

Simple Sampling

(Disadvantages)

Answer 3

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

Question 4

Ising Model

Answer 4

• 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

Question 5

Ising Model

(Limitation)

Answer 5

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

Question 6

Importance Sampling

Answer 6

Choose states according to Boltxmann weight using a Markhov vhain

Question 7

Markhov Chains

Answer 7

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

Question 8

Markhov Chains

(Properties)

Answer 8

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

Question 9

Metropolis Algorithm

Answer 9

local-update scheme with one spin flip per move