MCMC Lecturenotes III Contents
1. The O(3) Model and the Heat Bath Algorithm
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The O(3) Model and the Heat Bath Algorithm
We give an example of a model with a continuous energy function. The 2d version of the model is of interest to eld theorists because of its analogies with the four-dimensional Yang-Mills theory. In statistical physics the d-dimensional model is known as the Heisenberg ferromagnet. Expectation values are calculated with respect to the partition function Z =
i
dsi eE({si}) .
(1)
si,1 The spins si = si,2 are normalized to (si)2 = 1 si,3
(2)
+1 2 1 and the measure dsi is dened by dsi = d cos(i) di , (3) 4 1 0 where the polar (i) and azimuth (i) angles dene the spin si on the unit sphere.
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The energy is E =
ij
(1 sisj ) ,
(4)
where the sum goes over the nearest neighbor sites of the lattice. We would like to update a single spin s. The sum of its 2d neighbors is S = s1 + s2 + . . . + s2d1 + s2d . Hence, the contribution of spin s to the action is 2d sS. We propose a new spin s with the measure (3) by drawing two uniformly distributed random numbers r [0, ) for the azimuth angle and cos(r ) = xr [1, +1) for the cosine of the polar angle. This denes the probability function f (s , s) of the Metropolis process, which
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accepts the proposed spin s with probability w(s s ) = 1 e(SsSs
)
for Ss > Ss, for Ss < Ss.
If sites are chosen with the uniform probability distribution 1/N per site, where N is the total number of spins, it is obvious that the procedure fullls detailed balance. It is noteworthy that the procedure remains valid when the spins are chosen in the systematic order 1, . . . , N , then 1, . . . , N again, and so on. Balance still holds, whereas detailed balance is violated (exercise). The heath bath algorithm Repeating the Metropolis algorithm again and again for the same spin s leads to the equilibrium distribution of this spin, which reads P (s ; S) = const e Ss with P (s ; S) ds = 1 .
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One would prefer to choose s directly with the probability W (s s ) = P (s ; S) = const e s
S
,
as s is then immediately Boltzmann distributed with respect to its neighbor spins. The algorithm, which creates this distribution, is called the heat bath algorithm. Implementation of this algorithm becomes feasible when the energy function is suciently simple to allow for an explicit calculation of the probability P (s S). ; This is an easy task for the O(3) -model. Let = angle(s , S), x = cos() and S = |S| . For S = 0 a new spin s is simply obtained by random sampling. We assume in the following S > 0. The Boltzmann weight becomes exp(xS) and the normalization
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constant follows from
+1
dx exS =
1
2 sinh(S) . S
Therefore, the desired probability is S P (s ; S) = exS =: f (x) 2 sinh(S) and the method of the rst lecture can be used to generate events with the probability density f (x). With S exp(+xS) exp(S) xS y = F (x) = dx f (x ) = dx e = 2 sinh(S) 2 sinh(S) 1 1
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x
x
a uniformly distributed random number y r [0, 1) translates into 1 x = cos = ln [ exp(+S) y r exp(+S) + y r exp(S)] . S
r r
(5)
Finally, one has to give s a direction in the plane orthogonal to S. This is done by choosing a random angle r uniformly distributed in the range 0 r < 2. Then, xr = cos r and r completely determine s with respect to S. Before storing s in the computer memory, we have to calculate coordinates of s with respect to a Cartesian coordinate system, which is globally used for all spins of the lattice. This is achieved by a linear transformation. We dene S3 cos = , sin = S 1 cos2 , S1 S2 cos = and sin = . S sin S sin
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Unit vectors of a coordinate frame K , with z in the direction of S and y in the x y plane, are then dened by sin cos cos sin cos z = sin sin , x = cos sin and y = cos . 0 sin ...