monte carlo methods in statistical...

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



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





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





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




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





Outline




4 Biasing





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





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





Importance vs. uniform sampling

f (x) =

rσ

πexp (−σx2) and ρ(x) =

1π

11 + x2




Ensemble averagesMarkov chainsDetailed balance

Outline




4 Biasing





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





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)





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)





Outline




4 Biasing





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





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





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





Outline




4 Biasing





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




Metropolis methodCanonical simulationsOther ensembles

Outline




4 Biasing





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)





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.





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





Outline




4 Biasing





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





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)





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

-





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





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





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





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





Outline




4 Biasing





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)))]




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.




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.


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