Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Monte Carlo Methods in Statistical Mechanics
Mario G. Del Pópolo
Atomistic Simulation CentreSchool of Mathematics and Physics
Queen’s University BelfastBelfast
Mario G. Del Pópolo Statistical Mechanics 1 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 2 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Integrals as averages
Standard numerical quadrature:
I =
Z b
af (x)dx ≈ δx
NXi=1
f (xi)
Random sampling with distribution functionρ(x):
I =
Z b
af (x)dx =
Z b
a
„f (x)
ρ(x)
«ρ(x)dx
≈ 1M
MXi=1
f (xi)
ρ(xi)
=
fif (xi)
ρ(xi)
flρ
Mario G. Del Pópolo Statistical Mechanics 3 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Uniform sampling
Use a random variable X uniformly distributed on [a, b]. Then
ρ(x) =1
b − a
and
I =
Z b
af (x)dx = (b − a)
Z b
af (x)ρ(x)dx = 〈f (x)〉ρ
with variance σ2I =
˙I2¸
− 〈I〉2 given by:
σ2I = (b − a)2
Df (x)2
Eρ− 〈f (x)〉2
ρ
Using N random numbers, x1, · · · , xN , the integral and its variance areestimated by:
iN ≈ b − aN
NXj=1
f (xj) and s2I =
1N
0@ (b − a)2
N
NXj=1
f 2(xj)− i2N
1AMario G. Del Pópolo Statistical Mechanics 4 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Statistical error
The standard error in iN is given by:
Standard error:
σiN =σI√N
Decreasing the error by one order of magnitude impliesincreasing the sample size, N, in two orders of magnitude
Mario G. Del Pópolo Statistical Mechanics 5 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 6 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Importance sampling
In the most general case:
I =
Z b
af (x)dx =
Z b
a
„f (x)
ρ(x)
«ρ(x)dx =
fif (xi)
ρ(xi)
flρ
the variance of I is:
σ2I =
Z b
a
f 2(x)
ρ(x)dx − I2
and the corresponding estimators are:
iN ≈ 1N
NXj=1
f (xj)
ρ(xj)and s2
I =1N
0@ 1N
NXj=1
f 2(xj)
ρ(xj)− i2
N
1A
Mario G. Del Pópolo Statistical Mechanics 7 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Importance sampling
In the importance sampling method ρ(x) is chosen to be largewhere f (x) is large and small where f (x) is smallA significant random sample is concentrated in the region wheref (x) is large, and contributes more to the integral, instead ofbeing distributed uniformly over the whole interval [a, b]
Under such conditions the following inequality is fulfilled:∫ b
a
f 2(x)
ρ(x)dx < (b − a)
∫ b
af 2(x)dx
The use of ρ(x) reduces the variance, σ2I , with respect to uniform
sampling and leads to a lower standard error: σiN = σI√N
Mario G. Del Pópolo Statistical Mechanics 8 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Quadrature vs. random samplingImportance sampling
Importance vs. uniform sampling
f (x) =
rσ
πexp (−σx2) and ρ(x) =
1π
11 + x2
Mario G. Del Pópolo Statistical Mechanics 9 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 10 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Calculating ensemble averages
In classical statistical mechanics the ensemble average of B(rN , pN)is calculated as:
〈B〉e =
∫ ∫B(rN , pN)f [N]
0 (rN , pN)drNdpN
with, for example,
f [N]0 (rN , pN) =
1h3NN!
exp (−βH)
QN,V ,Tor f0(rN , pN ; N) =
exp (−β(H− Nµ))
Ξµ,V ,T
How to evaluate 〈B〉e numerically ?
6N-dimensional integral → quadrature and uniform samplingunfeasible
Mario G. Del Pópolo Statistical Mechanics 11 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Calculating ensemble averages
Solution → Importance sampling: generate random configurationsaccording to a distribution W (rN) so:
〈B〉ce =
RB(rN)f c
0 (rN)drNRf c0 (rN)drN
=
RB(rN) ρ(rN )
ρ(rN )f c0 (rN)drNR ρ(rN )
ρ(rN )f c0 (rN)drN
where we have focused on the configurational contribution to 〈B〉e. Clearly:
〈B〉ce =
〈B/ρ〉ρ〈1/ρ〉ρ
where 〈 〉ρ signifies averages over the distribution:
W (rN) = ρ(rN)× f c0 (rN)
Mario G. Del Pópolo Statistical Mechanics 12 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Ensemble averages
Challenge: What is the most convenient form of ρ(rN) ?
ρ(rN)must be similar to Bc(rN)f c0 (rN)
Ideal choice → ρ(rN) = f c0 (rN)
The problem has been rephrased: How to generate a series ofrandom configurations so that each state occurs with probabilityρ(rN) = f c
0 (rN) ?Generate a Markov chain of sates, Γn ≡ (rN
n ), with a limitingdistribution f c
0 (rN)
Mario G. Del Pópolo Statistical Mechanics 13 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 14 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Markov chains
Markov chain: sequence of random configurations (states or trials)satisfying the following two conditions:
1 The outcome of each trial belongs to a finite set of outcomesΓ1, Γ2, · · · , Γm, called the state space
2 The outcome of each trial depends only on the outcome of theimmediately preceding one
Conditional probability for sequence of steps:
Pr (j, t + 1|i, t ; kt−1, t − 1; · · · ; k0, 0)
Statement 2 implies:
Pr (j, t + 1|i, t ; kt−1, t − 1; · · · ; k0, 0) = Pr (j, t + 1|i, t) Markov process
Pr (j, t + 1|i, t) = Πij are the elements of a transition matrix Π linking states Γi
and Γj
Mario G. Del Pópolo Statistical Mechanics 15 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Markov chainsBe ρi(t) the probability of being in state i at time t . Then:
ρj(t) =X
i
ρi(t − 1)Πij
or using the row vector ρ(t) = (ρ1(t), ρ2(t), · · · ) and then transition matrix:
ρ(t) = ρ(t − 1)Π
The solution can be written in terms of the left eigenvalues (λi ) and lefteigenvectors (Φi ) of Π:
ρ(t) =X
i
Φiλti and ΦiΠ = λiΦi
Π is an stochastic matrix soP
j Πij = 1 ∀ i . It can be proofed that:1 λi ≤ 1 ∀ i2 There is at least one eigenvalue equal to unity, let us say λ1 = 13 If the Markov chain is irreducible, there is only one eigenvalue equal to
unity
Mario G. Del Pópolo Statistical Mechanics 16 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Markov chainsAccording to the previous considerations:
limt→∞
ρ(t) = limt→∞
Xi
Φiλti = Φ1
Φ1 is the unique limiting distribution of the Markov chainThe stationary distribution satisfies:
Φ1Π = Φ1 or ρ0Π = ρ0
In statistical mechanics:
ρ0: vector with elements ρ0(Γn) (Γn = position in phase space)
Need to determine the elements of Π satisfying:
Πi,j ≥ 0 ∀ i ;X
j
Πi,j = 1 ∀ i andX
i
ρiΠij = ρj
Πi,j must not depend on the normalisation constant (partition function) ofρ0
Mario G. Del Pópolo Statistical Mechanics 17 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 18 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Ensemble averagesMarkov chainsDetailed balance
Condition of detailed balance
A useful trick in searching for a solution of the previous equations isto replace:∑
i
ρiΠij = ρj → ρiΠij = ρjΠji Detailed balance
Summing over all states i :∑i
ρiΠij =∑
i
ρjΠji = ρj
∑i
Πji = ρj
In the practice:
Need to generate a sequence of configurations (states, Γn)according to the specified equilibrium distribution ρ0(Γ)
Use ρiΠij = ρjΠji and∑
j Πi,j = 1 to build the transition matrixelements in terms of ρ0
Mario G. Del Pópolo Statistical Mechanics 19 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 20 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Metropolis algorithm
This is an asymmetrical solution to the previous problem:
Metropolis method
Πij = αij for ρj ≥ ρi and i 6= jΠij = αij(ρj/ρi) for ρj < ρi and i 6= j
Πii = 1−∑
j 6=i Πij
α is a symmetrical matrix often called the underlying matrix ofthe Markov chainSince we use (ρj/ρi) we circumvent the problem of calculatingthe normalisation factor (partition function)
Mario G. Del Pópolo Statistical Mechanics 21 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Barker algorithm
Symmetrical solution:
Barker method
Πij = αijρj/(ρi + ρj) for i 6= j
Πii = 1−∑j 6=i
Πij
In both the Metropolis and the Barker method the Markov chain willbe irreducible provided ρi > 0 ∀ i and the underlying symmetricMarkov chain is irreducible.
Mario G. Del Pópolo Statistical Mechanics 22 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Ensemble averages
The calculation of ensemble average of B(rN , pN) :
〈B〉e =
∫ ∫B(rN , pN)f [N]
0 (rN , pN)drNdpN
= 〈B〉ide +
∫B(rN)ρ0(rN)(rN)drN
is achieved by averaging over M successive states of the Markovchain. The average converges to the desired value as M →∞.
〈B〉M =1M
M∑t=1
B(rNt ) =
∑rN∈Γ
B(rN)ρ0(rN)+O(M−1/2) ≡ 〈B〉e+O(M−1/2)
Non-ergodicity can be a serious problem
Mario G. Del Pópolo Statistical Mechanics 23 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 24 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Sampling the canonical ensemble
Aim: Generate a series of particleconfigurations distributed according to:
ρ0(rN) =exp (−βVN )
Zwith
Z =
ZdrN exp (−βVN )
In order to implement the Metropolisalgorithm we need to specify α, whichsatisfies:
αij = αji
For particle n at position rn, αij is defined as:
αij = 1/NR rin ∈ R
αij = 0 rin /∈ R
Mario G. Del Pópolo Statistical Mechanics 25 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Sampling the canonical ensemble
Metropolis algorithm:
Πij = αij for ρj ≥ ρi and i 6= j
Πij = αij (ρj/ρi ) for ρj ≥ ρi and i < j
Πii = 1−Xj 6=i
Πij for i = j
with
ρj
ρi=
exp (−βVN(rNj ))
exp (−βVN(rNi ))
= exp (−βδV jiN)
A randomly chosen particle is movedaccording to α→ Trial move
If δV jiN < 0 then ρj ≥ ρi and the new
configuration is accepted
If δV jiN > 0 the new configuration is accepted
with probability exp (−βδV jiN)
Mario G. Del Pópolo Statistical Mechanics 26 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Organisation of a simulation
Ising H = −JX<ij>
si sj −NX
i=1
Hsi
Read input
?
Initialise
?Markovchain
?��
��Accumulate
averages
?
���
@@@
���
@@@
End ofrun ?
N Y - Write outputvalues
-
Mario G. Del Pópolo Statistical Mechanics 27 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Boundary conditions
Periodic boundary conditions:
Avoid surface effectsPeriodicity introducescorrelations
System size N:
Finite size effects depend oncorrelation length and range ofinteractionsDependence on cell symmetryand shape
Configurational energy:
The Ewald method (trulyperiodic b.c.)Truncation of the intermolecularforces: Minimum image pluscutoff
L
x
Mario G. Del Pópolo Statistical Mechanics 28 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Assessment of the results
The main advantage of Monte Carlo methods is the great flexibility inthe choice of the stochastic matrix Π
When designing a new MC algorithm or running a MC simulations one must:
Ensure the algorithm samples the desired ensemble distribution →detailed balance condition
Ensure every state can eventually be reached from any other ( Markovchain must be irreducible or ergodic)
Test accuracy of random number generator
Standard checks on simulation results:
Steady-state distribution must be reached (discard initial relaxation)Same distribution must be reached starting from different initialconditionsEstimate statistical uncertainties and correlation timesFinite size effects
Mario G. Del Pópolo Statistical Mechanics 29 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Quality of random number generator
Set of X-Y coordinatesproduced with a bad randomnumber generator
Coordinates produced with agood random number generator
Figure taken from Binder & Landau
Mario G. Del Pópolo Statistical Mechanics 30 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Time scales
Evolution of the internal energy,U, and magentisation, M, in theIsing model in the absence ofmagentic field. Note:
Initial relaxation. The twoquantities evolve withdifferent characteristic timescalesIntermediate times. Seriesare stationary and showequilibrium fluctuationsLonger time scale. Globalspin inversion.
Figure taken from Binder & Landau
Mario G. Del Pópolo Statistical Mechanics 31 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Outline
1 Multidimensional integralsQuadrature vs. random samplingImportance sampling
2 Multidimensional integrals in statistical mechanicsEnsemble averagesMarkov chainsDetailed balance
3 Symmetric and asymmetric algorithmsMetropolis methodCanonical simulationsOther ensembles
4 Biasing
Mario G. Del Pópolo Statistical Mechanics 32 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Metropolis methodCanonical simulationsOther ensembles
Simulations in other ensembles
Isobaric-isothermal ensemble (N, P, T ) → allows fluctuations in thevolume
Grand canonical ensemble (µ, V , T ) → allows fluctuations in the numberof particles
etc, etc. · · ·
Example: In a Metropolis grand canonical simulation:
Particle displacements are accepted with probability:min [1, exp (−βδVij)]
Particles are destroyed with probability:min [1, exp (−βδVij + ln (N/zV ))], where z = exp (βµ)/∆3
Particles are created with probability:min [1, exp (−βδVij + ln (zV/(N + 1)))]
Mario G. Del Pópolo Statistical Mechanics 33 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Comment on biasing and detailed balance
Condition of detailed balance:
ρ0(o)× α(o → n)× Pacc(o → n) = ρ0(n)× α(n → o)× Pacc(n → o)
Pacc is the probability that the trial move(o → n) will be accepted. For acanonical simulation it follows that:
Pacc(o → n)
Pacc(n → o)= exp (−βV)
α(n → o)
α(o → n)
Using Metropolis solution, the acceptance rule for a trial MC move is:
Pacc(o → n) = min„
1, exp (−βV)α(n → o)
α(o → n)
«By biasing the probability to generate a trial conformation, α, one couldmake the term on right hand side very close to one. In that case almost
every trial move will be accepted.
Mario G. Del Pópolo Statistical Mechanics 34 / 35
Multidimensional integralsMultidimensional integrals in statistical mechanics
Symmetric and asymmetric algorithmsBiasing
Bibliography
"Computer Simulations of Liquids" , by M. P. Allen, D. J. Tildesley.Oxford University Press, 1987" A guide to Monte Carlo simulations in Statistical Physics" , byD. P. Landau and K. Binder. Cambridge University Press, 2005"Modern Theoretical Chemistry", Volume 5, part A. Edited by B.Berne, Plenum Press, 1977."The Monte Carlo Methods in the Physical Sciences ", Edited byJ. E. Gubernatis, AIP Conference Proceedings, vol 690, 2003." Monte Carlo Methods in Chemical Physics ", Edited by D.Ferguson, J. I. Siepmann and D. G. Truhlar. Advances inChemical Physics, vol 105, Wiley, 1999.
Mario G. Del Pópolo Statistical Mechanics 35 / 35
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