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9 Pages

### mcmc3

Course: MCMC 06, Fall 2009
School: Fayetteville State...
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Word Count: 965

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MCMC Lecturenotes III Contents 1. The O(3) Model and the Heat Bath Algorithm 1 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....

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MCMC Lecturenotes III Contents 1. The O(3) Model and the Heat Bath Algorithm 1 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. 2 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 3 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 . 4 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 5 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 6 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 7 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 ...

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Fayetteville State University - MCMC - 06
Lecturenotes Statistics I Contents1. Uniform and General Distributions 2. Condence Intervals, Cumulative Distribution Function and Sorting1Uniform and General Distributions1 for 0 x &lt; 1; 0 elsewhere. The corresponding distribution function
Fayetteville State University - MCMC - 06
Lecturenotes MCMC IV Contents1. Multicanonical Ensemble 2. How to get the Weights? 3. Example Runs (2d Ising and Potts models) 4. Re-Weighting to the Canonical Ensemble 5. Energy and Specic Heat Calculation 6. Free Energy and Entropy Calculation 7
Fayetteville State University - MCMC - 06
Lecturenotes MCMC I Contents1. Statistical Physics and Potts Models 2. Sampling and Re-weighting 3. Importance Sampling and Markov Chain Monte Carlo 4. The Metropolis Algorithm1Statistical Physics and Potts ModelMC simulations of systems desc
Fayetteville State University - MCMC - 06
1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1M2q02d Ising 2d q=10 Potts
Fayetteville State University - MCMC - 06
35 30 25 int 20 15 10 5 0 0 50 100 t 150 200 250 L= 20 L= 40 L= 80 L=160 L= 10 L= 5
Fayetteville State University - MCMC - 06
20 18 16 14 12 int 10 8 6 4 2 0 50 100 t 150 200 250 L=40 1-hit Metropolis L=80 1-hit Metropolis L=40 2-hit Metropolis L=80 2-hit Metropolis L=40 heat bath L=80 heat bath
Fayetteville State University - MCMC - 06
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Ising 10-state Potts s
Fayetteville State University - MCMC - 06
-2 -3 -4 -5 f -6 -7 -8 -9 -10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ising 10 state Potts
Fayetteville State University - MCMC - 06
2500 2000 int 1500 1000 500 0 0 L=160 ts1 L= 80 ts1 L= 40 ts1 L= 20 ts1 1000 2000 3000 4000 5000 6000 7000 8000 t
Fayetteville State University - MCMC - 06
0 Energy per spin &lt;e0s&gt; Multicanonical data -0.5 &lt;e0s&gt;-1-1.5-2 0 0.1 0.2 0.3 0.4 0.5 0.6
Fayetteville State University - MCMC - 06
multicanonical1000Histograms500702 lattice0 1.0canonical-E/N1.5
Fayetteville State University - MCMC - 06
1.8 1.6 1.4 1.2 C/N 1 0.8 0.6 0.4 0.2 0 0 0.2Specific heat per spin0.4 0.60.81
Fayetteville State University - MCMC - 06
6000 5000 4000 3000 2000 1000 0 0 -0.5 e -1 -1.5 -2 H Random Sampling Weighted to =0.2 MC at =0.2 MC at =0.4
Fayetteville State University - MCMC - 06
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Fayetteville State University - MCMC - 06
50 40 int 30 20 10 0 0 50 100 t 150 200 250 L=20 random updating L=40 random updating L=20 systematic updating L=40 systematic updating
Fayetteville State University - MCMC - 06
1000 Histograms Multicanonical =0.711000.30.40.50.6 act0.70.80.91
Fayetteville State University - MCMC - 06
0 Random Start Ordered Start Exact-0.5e0s-1-1.5-2 0 50 100 Sweeps 150 200
Fayetteville State University - MCMC - 06
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Fayetteville State University - MCMC - 06
1 0.9 0.8 0.7 act 0.6 0.5 0.4 0.3 0.2 0.1 0 50 100 Sweeps 150 200 Disordered starts Metropolis 1-hit Metropolis 2-hit Heat Bath Ordered starts
Fayetteville State University - MCMC - 06
CLASSWORK 4 9/29/2006 =Consider a cubic lattice of size 100 x 100 x 100. For the site number is=553771 find the following: Coordinates of this site. Site numbersof th
Fayetteville State University - MCMC - 06
lectureRMC: RMC continuation of MUCA lecture -&gt; Seminars/y2005/FSU_PhysChem.
Fayetteville State University - MCMC - 06
CLASSWORK 3 9/19/2006 =1. Create your own library, `MyLib', of Fortran 77 routines. Let the first entry be a Fortran functions ggau_df.f and/or ggau_qdf.f, which
Fayetteville State University - MCMC - 05
Write in you answers for the following simulations and turn this worksheet in. Use your personal seed for all simulations. Send all (asked for) plots in ONE e-mail. - a0303_06: CPU time for the run: Energy em per spin with error bar:
Fayetteville State University - MCMC - 07
Continuous Systems: Heisenberg Spin ModelWe give an example of a model with a continuous energy function. The 2d version of the model is known as -model and of interest in eld theory because of its analogies with 4d Yang-Mills theory. In statistical
Fayetteville State University - MCMC - 07
Lecturenotes Statistics I Contents1. Uniform and General Distributions 2. Condence Intervals, Cumulative Distribution Function and Sorting1Uniform and General DistributionsUniform distribution (probability density): u(x) = 1 for 0 x &lt; 1; 0 e
Fayetteville State University - MCMC - 07
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Fayetteville State University - MCMC - 07
Bayesian StatisticsKolmogorov Axioms and Conditional Probabilities We denote events by A, B, C, . . . , and use the following notation: 1. A B = A and B, the event that A and B both occur. 2. Ac = not A, the event that A does not occur. 3. E, the e
Fayetteville State University - MCMC - 07
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Fayetteville State University - MCMC - 07
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Parallel ComputingAfter briey discussing the often neglected, but in praxis frequently encountered, issue of trivially parallel computing, we turn to parallel computing with information exchange. Our illustration is the replica exchange method, als
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Fayetteville State University - MCMC - 07
1000 Histograms Multicanonical =0.711000.30.40.50.6 act0.70.80.91
Fayetteville State University - MCMC - 07
20 18 16 14 12 int 10 8 6 4 2 0 50 100 t 150 200 250 L=40 1-hit Metropolis L=80 1-hit Metropolis L=40 2-hit Metropolis L=80 2-hit Metropolis L=40 heat bath L=80 heat bath
Fayetteville State University - MCMC - 07
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Fayetteville State University - MCMC - 07
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14 12 10 int 8 6 4 2 0 10 20 30 t SW L=160 SW L= 80 SW L= 40 SW L= 20 W L=160 W L= 80 W L= 40 W L= 20 40 50 60
Fayetteville State University - MCMC - 07
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PHY5846C: INTRODUCTION TO EXPERIMENTAL TECHNIQUES1. The luminosity of a colliding beam machine is de ned by N= ; where N interactions occur per second for cross section . For a machine with n1; n2 particles per bunch in the two beams with nB bunches
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Fayetteville State University - PHY - 3802
percent, since the ratio5/540 reduces to 0.0093 (rounded off) in decimal form.Significant DigitsThe accuracy of a measurement is often described in t e r m s of the number of significant digits used in expressing it. If the digits of a number resu
Fayetteville State University - PHY - 3802
Fayetteville State University - PHY - 3802
Fayetteville State University - PHY - 3802
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Fayetteville State University - PHY - 3802
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Fayetteville State University - PHY - 3802
IM.1. Franck-Hertz Experiment1. Purpose:Perform the historic Franck-Hertz experiment to demonstrate the existence of discrete energy levels in mercury, and to determine the minimum kinetic energy needed by an electron in order to collide inelastica
Fayetteville State University - PHY - 3802
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Fayetteville State University - PHY - 3802
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