An Introduction to Monte Carlo Methods in Statistical Physics.

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Shift Register (Tausworthe ( Tausworthe) ) Algorithm A new random number is produced by combining two different existing numbers from a pre-existing table: X n = X n − p . XOR . X n − q where p and q must be properly chosen. The .XOR. operator is the bitwise exclusive-OR operator. The best choices of the pairs (p,q) are given by primitive Mersenne trinomials: + q x p x +1
Examples of pairs which satisfy this condition are: p = 250 q = 103 ← popular, known as R250 p = 1279 q = 216, 418 p = 9689 q = 84, 471, 1836, 2444, 4187 Large p and q yield better sequences. The “table” must be properly initialized. Use a good congruential generator to generate a different random number for each bit in turn. Lagged Fibonacci Generators Replace the exclusive-or by another operator, e.g., X n = X n − p * X n − q
Page 1 and 2: Workshop on Markov Chain Monte Carl
Page 3 and 4: Goal of these lectures ⇒ to provi
Page 5 and 6: What problems can we solve with Mon
Page 7 and 8: What strategy should we follow? •
Page 9 and 10: Simulation NATURE Experiment Theory
Page 11 and 12: Probability Distributions and the C
Page 13 and 14: Relate this to σ 2 = - < A> 2 2 2
Page 15 and 16: The sequence of states {X t } is a
Page 17: Congruential Method A fixed multipl
Page 21 and 22: Example: Plot points using consecut
Page 23 and 24: Non-uniform Distributions of Random
Page 25 and 26: Using Monte Carlo for Numerical Int
Page 27 and 28: Using Monte Carlo for Numerical Int
Page 29 and 30: Numerical Integration by Monte Carl
Page 31 and 32: Intelligent methods for Monte Carlo
Page 33 and 34: Probabilistic interpretation: Consi
Page 35 and 36: Monte Carlo and Stat. Mech.- An Int
Page 37 and 38: The Percolation problem The Percola
Page 39 and 40: Probability that the system is in s
Page 41 and 42: Single Spin-Flip Monte Carlo Method
Page 43 and 44: Single spin-flip sampling for the I
Page 45 and 46: Some Practical Advice 1. In the ver
Page 47 and 48: Historical Background (from The Mat
Page 49 and 50: Boundary Conditions Boundary Condit
Page 55 and 56: Correlation times Correlation times
Page 57 and 58: Finite Sampling Time Effects Finite
Page 59 and 60: A case study: L×L L Ising square l
Page 61 and 62: Finite Sampling/Finite Lattice Size
Page 63 and 64: • The spins may then be integrate
Page 65 and 66: At high T, clusters are small. At l
Page 67 and 68: “Improved estimators” It may be
Page 69 and 70:
• Generate a random number 0 < rn
Page 71 and 72:
Wolff embedding trick and cluster-f
Page 73 and 74:
Improvements in Performance (Ising
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Test for the 2-dim Ising model
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Reformulate the problem ⇒ an effe
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H eff How to get : Find where it is
Page 81 and 82:
Phase Switch Monte Carlo Phase Swit
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A New Approach to Monte Carlo Simul
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Metropolis Recipe: Metropolis Recip
Page 87 and 88:
A Quite Different Approach A Quite
Page 89 and 90:
Density of States for the 2-dim 2 I
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Density of States: Large 2-dim 2 Is
Page 93 and 94:
Specific Heat of the 2-dim 2 Ising
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Wang-Landau Landau vs Metropolis Sa
Page 97 and 98:
q=10 Potts Model: Determine T c
Page 99 and 100:
What About a System with Complex
Page 101 and 102:
Distribution of States: 3d EA Spin
Page 103 and 104:
3d EA Spin Glass Model 3d EA Spin G
Page 105 and 106:
Triangular Ising Model with Two-Bod
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Critical Endpoint: Singularity in t
Page 109 and 110:
Overview: Wang-Landau sampling The
Page 111 and 112:
Summary and Overview Summary and Ov
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An Introduction to Monte Carlo Methods in Statistical Physics.

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