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Problems: Quantum Fields on the Lattice

Prof. Dr. Andreas Wipf WiSe 2019/20

MSc. Julian Lenz

Sheet 4

9 Detailed Balance

A statistical system has two states ω = 1, 2 with equilibrium probabilities P

ω

. Construct the most general form of a stochastic matrix W (ω, ω

0

) such that detailed balance is fulfilled, i.e.

P (ω)W (ω, ω

0

) = P (ω

0

)W (ω

0

, ω ) (1) for ω, ω

0

∈ {1, 2} . What is the optimal choice for W such that

W

eq

= lim

n→∞

W

n

(2)

is approached fastest?

10 Two-Dimensional Ising Model

Write a program for simulation of the two-dimensional Ising model via 1. the Metropolis algorithm

2. the Wolff algorithm.

Assume J = 1 such that the critical temperature is approximately T

c

≈ 2.269. Implement the following observables:

1. magnetization 2. susceptibility

3. autocorrelation time of the magnetization.

Which differences do you find between the algorithms? Which differences do you find between cold and

hot start?

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