1 modern monte carlo methods: (2) histogram reweighting (3) transition matrix monte carlo jian-sheng...
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Modern Monte Carlo Modern Monte Carlo Methods: Methods:
(2) Histogram Reweighting(2) Histogram Reweighting(3) Transition Matrix Monte Carlo(3) Transition Matrix Monte Carlo
Jian-Sheng WangJian-Sheng WangNational University of SingaporeNational University of Singapore
Modern Monte Carlo Modern Monte Carlo Methods: Methods:
(2) Histogram Reweighting(2) Histogram Reweighting(3) Transition Matrix Monte Carlo(3) Transition Matrix Monte Carlo
Jian-Sheng WangJian-Sheng WangNational University of SingaporeNational University of Singapore
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Outline• Histogram reweighting• Transition matrix Monte Carlo• Binary-tree summation Monte
Carlo
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Methods for Computing Density of States
• Reweighting methods (Salsburg et al, 1959, Ferrenberg-Swendsen, 1988)
• Multi-canonical simulation (Berg et al, 1992)
• Broad Histogram (de Oliveira et al, 1996)• TMMC and flat-histogram (Wang,
Swendsen, et al, 1999)• F.Wang-Landau method (2001)
Micheletti, Laio, and Parrinello, Phys Rev Lett, Apr (2004)
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Density of States• The density of states n(E) is the
count of the number of (microscopic) states with energy E, assuming discrete energy levels.
( )
( ) 1E X E
n E
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Partition Function in n(E)
• We can express partition function in terms of density of states:
( ) ( )
( )
( )
E X E X
X E E X E
E
E
Z e e
n E e
Thus, if n(E) is calculated, we effectively solved the statistical-mechanics problem.
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4. Reweighting 4. Reweighting Methods Methods
4. Reweighting 4. Reweighting Methods Methods
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Ferrenberg-Swendsen Histogram Reweighting• Do a canonical ensemble simulation at
temperature T=1/(kBβ), and collect energy histogram, i.e., the counts of occurrence of energy E.
• Thus, density of states can be determined up to a constant:
( ) ( ) EH E n E e
( ) ( ) En E H E e
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Calculate Moments of Energy
• From the density of states, we can calculate moments of energy at any other temperature,
( ' )
( ' )'
( )
( )
n E
n EE
E
E H E eE
H E e
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Reweighting Result
Result from a single simulation of 2D Ising model at Tc, extrapolated to other temperatures by reweighting
From Ferrenberg and Swendsen, Phys Rev Lett 61 (1988) 2635.
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Range of Validity of n(E)
Red curve marked FS is from Ferrenberg-Swendsen method
Relative error of density of states |nMC/nexact-1| from Ferrenberg-Swendsen method and transition matrix Monte Carlo, 3232 Ising at Tc.
From J S Wang and R H Swendsen, J Stat Phys 106 (2002) 245.
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Multiple Histogram Method
• Conduct several simulations at different temperatures Ti
• How to combine histogram results Hi(E) properly at different temperatures?
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Minimize error at each E
• We do a weighted average from M simulations
• The optimal weight is
( ) ( ) ( ) iEii i i i
i i i
Zn E wn E w H E e
N
( ) /iEi i i iw H E N e Z
Where the proportionality constant is fixed by normalization Σwi = 1, and Zi= ΣE n(E) exp(-βiE)
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Multiple Histogram Example
Multiple histogram calculation of the specific heat of the 3D three-state anti-ferromagnetic Potts model, using a cluster algorithm
From J S Wang, R H Swendsen, and R Kotecký, Phys Rev Lett 63 (1989) 109.
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5. Transition Matrix 5. Transition Matrix Monte Carlo (TMMC) Monte Carlo (TMMC) 5. Transition Matrix 5. Transition Matrix
Monte Carlo (TMMC) Monte Carlo (TMMC)
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Transition Matrix (in energy)
• We define transition matrix
which has the propertyh(E) T(E->E ’) = h(E ’) T(E ’->E)
( ) , ( ' ) '
( ' ) '
1( -> ') ( -> ')
( )
( -> ')
E X E E X E
EE X E
T E E W X Xn E
W X X
h(E) = n(E) e-E/(kT) is energy distribution or exact energy histogram.
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Transition Matrix Monte Carlo
• Compute T(E->E ’) with any valid MC algorithms that have micro-canonical property
• Obtain h(E), or equivalently n(E) from energy detailed balance equationSee J.-S. Wang and R. H. Swendsen, J Stat Phys 106 (2002) 245.
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Example for Ising Model
• Using single-spin-flip dynamics, the transition matrix W in spin configuration space is
0, if and ' diff er more than 1 spin
1( -> ') min(1, ), diff er by exactly 1 spin
( -> ), diagonal term by normalization
EW eNW
N = Ld is the number of sites.
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Transition Matrix for Ising model
where <N (σ,E ’-E )>E is micro-canonical average of number of ways that the system goes to a state with energy E ’, given the current energy is E.
( ' )1( -> ') ( , ' ) min(1, )E E
ET E E N E E e
N
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The Ising Model
- +
+
+
+
++
+
++
++
+
++
+
+-
---
-- -
- --
- ----
---- Total energy is
E(σ) = - J ∑<ij> σi σj
sum over nearest neighbors, σ = ±1
NE) is the number of sites, such that flip spin costs energy E.
σ = {σ1, σ2, …, σi, … }
E=0
E=-8J
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Broad Histogram Equation (Oliveira)
n(E)<N(σ,E ’-E)>E = n(E ’)<N(σ’,E-E ’)>E ’
• This equation is used to determine density of states as well as to construct a “flat-histogram” algorithm
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Flat Histogram Algorithm
1. Pick a site at random2. Flip the spin with probability
3. Where E is current and E ’ is new energy
4. Accumulate statistics for <N(σ,E ’-E)>E
'( ' , ' ) ( )
min 1, min 1,( , ' ) ( ' )
E
E
N E E n EN E E n E
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HistogramsHistograms for 2D Ising 32x32 with 107 Monte Carlo steps. Insert is a blow-up of the flat-histogram.
From J-S Wang and L W Lee, Computer Phys Comm 127 (2000) 131.
Flat-histogram
Canonical
Broad histogram
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2D Ising ResultSpecific heat of a 256x256 Ising model, using flat-histogram/multi-canonical method. Insert shows relative error. 3 x 107 Monte Carlo sweeps are used.
From J-S Wang, “Monte Carlo and Quasi-Monte Carlo Methods 2000,” K-T Fang et al, eds.
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5. Binary Tree 5. Binary Tree Summation Monte Summation Monte
Carlo Carlo
5. Binary Tree 5. Binary Tree Summation Monte Summation Monte
Carlo Carlo
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Newman-Ziff Method for Percolation
Start with an empty lattice, compute Q(Γ0)
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Newman-Ziff MethodRandomly occupy a bond, compute Q(Γ1)
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Newman-Ziff MethodRandomly occupy an unoccupied bond, compute Q(Γ2)
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Newman-Ziff MethodAnd so on and compute Q(Γb) with b number of bonds
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Newman-Ziff MethodUntil all bonds are occupied, compute Q(ΓM)
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Newman-Ziff Method• Any quantity as a function of p is
computed as (for percolation, q = 1)
• Each sweep takes time of O(N)
0
!(1 )
!( )!
Mb M b
bb
MQ p p Q
b M b
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Binary Tree Summation• Work in the Fortuin-Kasteleyn
representation, P(Γ) pb(1-p)M-bqNc
• Putting bonds of β-type only (i.e. always merge two clusters into one)
• The steps that do no merge cluster are not explicitly simulated
• Compute weights w(b,i)
See J.-S. Wang, O. Kozan, and R. Swendsen, `Computer Simulation Studies in Condensed Matter Physics XV', p.189, Eds. D. P. Landau, et al (Springer-Verlag, Heidelberg, 2002).
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BTS algorithm1. Start with an empty lattice, n0=0,
n1=M, i=0, compute Q(0)2. Pick a type-β bond at random,
merge the clusters A and B3. n0 n0+ nAB – 1, n1 n1-nAB, i i+14. Compute Q(i), goto 2 if i < N-15. Compute weight w(b,i)Where M is total number of bonds, N is number of sites, n0 is number of γ-type bonds and n1 is number of β-type bonds. nAB is number of unoccupied bonds connecting clusters A and B.
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Compute Weight• w(0,i) = δi,0
• w(b+1,i) = w(b,i) (n0(i)-b+i) +
w(b,i-1)n1(i-1)/q
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Simulated and Re-constructed
Configurations
111 2 N-1 N Mb (bonds)0
1
N-1
22
i (merge sequence)
n1/qn0
simulated path
reconstructed path
fully occupied lattice
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Statistical Average at Fixed p
11
0
1
0
0
( , ) ( )
( , )
(1 )
N
b bi
N
bi
Mb M b
b bb
Q W w b i Q i
W w b i
Q p p c Q
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cb play the rule of density of states
• We compute cb from
0 1 1
0 1
/ ( 1)( )
bb
b
n n q b cn n M b c
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Comparison
Relative error for density of states (or cb for BTS) after 106 Monte Carlo steps.
Note: |n(E)/nex(E)-1| |S(E) – Sex(E)| ] where S(E) = ln n(E).
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Some features of BTS• Independent sample in each sweep• Any real values of p can be used
(including negative p)• It is not an importance sampling
method (similar to Sequential MC)• Each sweep takes O(N2)
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Summary• Cluster algorithms are best at Tc
• TMMC produces n(E) and uses more information from the samples
• BTS is an interesting variation