Wang-Landau Monte Carlo simulation -...

Wang-Landau Monte Carlo simulation

Aleš Vítek IT4I, VP3

PART 1

Classical Monte Carlo Methods

Outline

1. Statistical thermodynamics, ensembles

2. Numerical evaluation of integrals, crude Monte Carlo (MC)

3. MC methods for statistical thermodynamics, Markov chains

4. Variants of MC methods

Statistical thermodynamics

microscopic description

(positions, momenta, interactions)




macroscopic description

(pressure, density, temperature)




macroscopic description

(pressure, density, temperature)

Statistical Thermodynamics


time averages Molecular Dynamics methods

0

0

1lim ( ( ), ( )) d

t

K Kt

B b b r t p t t


ensemble averages Monte Carlo methods

1

1

1

( , ) ( , ; , , ) d, ,

( , ; , , ) d

K K K K n

n

K K n

b r p r p A AB A A b

r p A A


Statistical (Gibbs) ensembles

microstate (m-state): positions and momenta

macrostate (m-state): macroscopic variables (A1, … )

1 m-state = many m-states Gibbs ensemble

probability distribution in system’s phase space, 1( , ; , , )K K nr p A A


Examples of statistical (Gibbs) ensembles

micro-canonical: N, V, E

canonical: N, V, T

grand-canonical: m, V, T

isobaric-isothermic: N, P, T

etc.


B

( , ; ), ; , , exp N K K

K K

H r p Vr p N V T

k T

Canonical Gibbs ensemble

2

1

( , ; ) ( ; )2

NK

N K K KK K

pH r p V W r V

M


1

1

1

( , ) ( , ; , , ) d, ,

( , ; , , ) d

K K K K n

n

K K n

b r p r p A AB A A b

r p A A

Statistical thermodynamics as a mathematical problem

3 3 3 3

1 13

1d d d ...d d

!hN NN

r p r pN


con con 1 1

1 kin

con 1 1

( ) ( ; , , ) d d, ,

( ; , , ) d d

K K n N

n

K n N

b r r A A r rB A A B

r A A r r

Statistical thermodynamics as a mathematical problem

Canonical ensemble

con

B

( ; ); , , exp N K

K

W r Vr N V T

k T

Numerical evaluation of integrals

1 1

( ) d ( ) ( )b n n

jj ja

b a b a b af x x f a j f x

n n n

1D – rectangular, trapezoid, Simpsonian, …


1 1

( ) d ( ) ( )b n n

jj ja

b a b a b af x x f a j f x

n n n

1D – rectangular, trapezoid, Simpsonian, …

1D – crude Monte Carlo

1

( ) d ( )b n

jja

b af x x f x

n

xj randomly distributed over [a,b]


0

sin dx xExample


Localized functions

( ) ( )db

a

f x x x


Localized functions

( ) ( )db

a

f x x x

Solution: Sampling from (x).


Localized functions

( ) ( )db

a

f x x x

Solution: Sampling from (x).

How? Markov chains.

Markov chains

(e) (e)

1 , , n

Simplified example – finite number of microstates

microstates: s1, …, sn

probability distribution: 1 = (s1), …, n = (sn)

equilibrium PD:

Markov chains


Markov chain: s(1) s(2) … s(N) …

s(K) is from {s1, …, sn}

transition probabilities: ik = ik (stochastic matrix)

1

0 1

1

ik

n

ikk

Markov chains

( 1) ( )

1

nN N

k i ikj


PD function evolution:

( 1) ( )N N

equilibrium PD function:

(e) (e)

1

n

k i ikj

(e) (e)

Markov chains


Theorem:

Let s(1), s(2), …, s(N), … be an infinite Markov chain of microstates generated using a

stochastic matrix ik . Then the generated microstates are distributed over the state

space of the system in congruence with the equilibrium PD function of the used stochastic matrix.

Remarks:

The Markov chain does not represent any physical time evolution.

From the statistical thermodynamics point of view, only the equilibrium PD function is physically relevant. (Not, e. g., the stochastic matrix.)

Markov chains

Construction of an appropriate stochastic matrix

(e) (e)

1 1

, 1n n

k j jk jkj k

Mathematically

2n linear algebraic equations for n2 variables

many (infinite number of) solutions

Markov chains

Construction of an appropriate stochastic matrix

(e) (e)

1 1

, 1n n

k j jk jkj k

Mathematically

2n linear algebraic equations for n2 variables

many (infinite number of) solutions

Detailed balance

(e) (e)

k kj j jk

Markov chains

Decomposition of the stochastic matrix

ik ik ikp a

pik … probability of proposing a particular transition

aik … probability of accepting the proposed transition

Metropolis solution [1]

(e)(e) (e)

(e

(e)

(e))mi min 1, ,n 1

ki ikp pk ki

k ki ki i ik ik ik

ik

kik

ii

p

pap a p a a

[1] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087.

Markov chains

Decomposition of the stochastic matrix

ik ik ikp a

pik … probability of proposing a particular transition

aik … probability of accepting the proposed transition

Metropolis solution [1]

(e)(e) (e)

(e

(e)

(e))mi min 1, ,n 1

ki ikp pk ki

k ki ki i ik ik ik

ik

kik

ii

p

pap a p a a

[1] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087.

Notice that the PD function need not be normalized!

Metropolis algorithm

… for evaluating configuration integrals of statistical thermodynamics (canonical ensemble)

B

B

( )

con 1

kin ( )

1

( )e d d

e d d

K

K

W r

k T

K N

W r

k T

N

b r r rB B

r r

and, consequently,

( ) ( )( ) ( )( )

B B B B

( ) (e)(e)

(e)e e e e

k ii i kiK

W WW WW r

k T k T k T k Tki

i


from the configuration generated in the preceding step, r(i), generate a new one using an algorithm obeying that the probability for old new is the same as for

new old (usually isotropic translation and/or rotation)

if W (new)<W (i) then r (i+1) = r (new)

if W (new)>W (i) then

r (i+1) = r (new) with probability

r (i+1) = r (i) with probability

repeat until “infinity” (convergence)

( new)B/

eiW k T

( new)B/

1 eiW k T


Then

B

B

( )

con 1 ( )

con( )1

1

( )e d d 1lim ( )

e d d

K

K

W r

k Tn

K N iKW r n

ik T

N

b r r rb r

nr r


Then

B

B

( )

con 1 ( )

con( )1

1

( )e d d 1lim ( )

e d d

K

K

W r

k Tn

K N iKW r n

ik T

N

b r r rb r

nr r

Canonical (or NVT or isochoric-isothermic)

Monte Carlo

A general sketch of an MC simulation

propose an initial configuration

employ the Metropolis algorithm to generate as many as possible further configurations

collect necessary data along the MC path to evaluate ensemble averages

calculate the averages as simple arithmetic means of the collected data

Troublesomes

Periodic boundary conditions - example

Literature

M. LEWERENZ, Monte Carlo Methods: Overview and Basics, in NIC Series Volume 10:

Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, NIC Juelich 2002 (http://www.fz-juelich.de/nic-series/volume10/volume10.html)

M. P. ALLEN, P. J. TILDESLEY, Computer Simulation of Liquids, Oxford University Press,

Oxford 1987.

D. P. LANDAU, K. BINDER, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, Cambridge 2000 and 2005.

http://www.fz-juelich.de/nic-series/volume10/volume10.html





Appendix

Simulated annealing [1] and basin hopping [2]

MC simulation at a high temperature

equilibrium structure for T = 0 K

T

0 K

sim

ula

ted

an

ne

ali

ng

gra

die

nt

optim

ization

ba

sin

ho

pp

ing

[1] S. Kirckpatrick, C. D. Gellat, M. P. Vecchi, Science 220 (1983) 671; [2] D. J. Wales, H. A. Scheraga, Science 285 (1999) 1368.

Example 2

Simulated annealing of small water clusters.

PART 2

Wang-Landau algorithm

Problem: low temperatures

Simulation at low temperatures – combination of integration and optimization

Molecular clusters, biomolecules:

Complicated energy landscapes

A lot of minims separated by large energy barriers

con con 1 1

1 kin

con 1 1

( ) ( ; , , ) d d, ,

( ; , , ) d d

K K n N

n

K n N

b r r A A r rB A A B

r A A r r

IF , then mean value of B can be expressed as a 1D integral where is classical density of states How to calculate ? Wang-Landau algorithm

con( ) ( )K Kb r b W r

Mean value of variable B

min

min

con con 1

1 kin

con 1

( ) ( ; , , ) ( )d, ,

( ; , , ) ( )d

nW

n

nW

b W W A A W WB A A B

W A A W W

( )W

1 1

1

1

, , ; ( , , ) ,

0

1

, ,

1 d d

( ) lim

1 d d

N N

N

N

r r W r r W W W

W

N

r r

r r

W

r r

( )W

Calculation of Ω Wang-Landau algorithm – basic idea

- Completaly random walk in configuration space => energy

histogram h(W)~Ω(W), but low energy configurations will not be

visited

- Metropolis algorithm: configurations are sampled by ρ(Ei(r), =>

energy histogram h(W)~Ω(W) x ρ(W(r))

- If we use ρ(Wi(r)) = 1/(Ω(W)), then energy histogram will be

constant function

Wang-Landau algorithm – calculation of density of states

- Start of simulation: Ω(W) is unknown, we start with constant Ω(W) = 1

and from random initial configuration r

- Following configuration r2 created by small random change of foregoing

configuration r1 is accepted with probability

- Visiting of energy Wi: change of Ω(Wi) → Ω(Wi) x k0,where modification

factor k0 >1.

- e. g., k0 =2, big k → big statistical errors, small k → long simulation

- After reaching of flat energy histogram h(W), modification factor is

changed e.g. by rule , energy histogram is deleted (density of

states isn‘t deleted!) and we start a new simulation

- Computations is finished, when k reaches predeterminated value (e. g.

1,000 001)

min 1, 1,

new new old

newold old

W r W

WW r

1j jk k

Wang-Landau algorithm – accuracy of results depends on:

- Final value of modification factor kfinal

- Softness of deals of energy

- Criterion of flatness of histogram

- Complexity of simulated system

- Details of implemented algorithm

Wangův-Landau algorithm: parallelization

- MPI, each core has one own walker, all cores share actual density

of states and actual energy histogram

W-L algorithm-example:

THE JOURNAL OF CHEMICAL PHYSICS 134, 024109 (2012)

Thank you for your attention

Wang-Landau Monte Carlo simulation -...

Documents

Transcript of Wang-Landau Monte Carlo simulation -...