Monte Carlo Methods

H. Rieger, Saarland University, Saarbrücken, Germany

Summerschool on Computational Statistical Physics, 4.-11.8.2010

NCCU Taipei, Taiwan

Monte Carlo Methods =Stochastic evaluation of physical quantities using random numbers

Example: Pebble game on the beach of Monaco = computation of using random events

Kid‘s game: Adult‘s game:

„Direct sampling“ „Markov chain sampling“

[after W. Krauth, Statistical Mechanics, Algorithms and Computation, Oxford Univ. Press]

Direct sampling

easy

„hard spheres in 2d“ – hard!

No direct sampling algorithm for hard spheres

(NOT random sequantial adsorption!)

What to do, when a stone from the lady lands here?

1) Simply move on2) Climb over the fence, and continue

until, by accident, she will reenterthe heliport

3) ….?

Move a pebble in a 3x3 lattice probabilistically such that each site a is visited with the same probability p(a) = 1/9

w(ab) = ¼ b a

w(aa) = 0 w(aa) = 1/4 w(aa) = 1/2

Markov chain sampling

Detailled Balance

ab

c

Together with

This yields:

The following condition for p(a) must hold:

This condition is fulfilled when

etc.

This is called „detailled balance condition“ – and leads here to w(ab,c)=1/4, w(aa)=1/2

More or less large piles close to the boundary due to „rejections“

Adult‘s pebbel game – solution: if a stone lands outside the heliport, stay there put a stone on top of the present stone and continue, i.e. reject move outside the square!

Rejections

Master equation

Markov chain described by „Master equation“ (t = time step)

The time independent probability distribution p(a) is a solution of this equation,

if the transition probabilities w(ab) etc. fulfill

For a given p(a) one can, for instance, choose

Which is the „Metropolis“ rule

detailed balance:

Monte Carlo for Thermodynamic Equilibrium

a are configurations of a many particle system,E(a) the energy of configuration a.

Thermodynamic equilibrium at temperature T is then described by theBoltzman distribution

is the normalization,called partition function

Thus the Metropolis rule for setting up a Markov chain leading toThe Boltzmann distribution is

= 1/kBT inverse temperature

Hard spheres

a = (r1,r2, … , rN) - configurations = coordinates of all N spheres in d dimension

All spheres have radius , they are not allowed to overlap, otherwise all configurations have the same energy (no interactions):

H(a) = if there is a pair (i,j) with |ri-rj|<2 H(a) = 0 otherwise

Define w(ab) in the following way: In configuration a choose randomly a particle i and displace it by a random vector - this constitutes configuration b.

w(ab) = 1 if b is allowed (no overlaps), w(ab) = 0 (reject) if displaced particle overlaps with some other particle

i.e.:

Hard spheres (2)

Tagged particle Tagged particle

Iteration: t t

Soft spheres / interacting particles

a = {(r1,p1),(r2,p2),…,(rN,pN)} - configurations = coordinates and momenta of all N particles in d dimension

Partition function

Example: LJ (Lennard-Jones)Energy:

L = box size

Peforming themomentum integral(Gaussian) Left with the configuration integral I

MC simulation for soft spheres: Metropolis

if

otherwise

Choose randomly particle i, its position is ri Define new position by ri‘=ri+, a random displacement vector, [-,]3

All othe rparticle remain fixed.

Acceptance probability for the new postion:

Measurements:Energy, specific heat, spatial correlation functions, structure function

Equlibration! Note: Gives the same results as molecular dynamics

Repeat many times

Discrete systems: Ising spins

System of N interacting Ising spins Si {+1,-1}, placed on the nodes of a d-dimensional latticea = (S1,S2,…,SN): spin configurations

Energy:

Jij = coupling strengths, e.g. Jij = J > 0 for all (i,j) ferromagneth = external field strength

For instance 1d: with periodic poundary conditions

(i,j)

Quantities of interest / Measurements

Magnetization

Susceptibility

Average energy

Specific heat

How to compute:

where at are the configurations generated by theMarkov chain (the simulation) at time step t.

Ising spins: Metropolis update

for

for

Procedure Ising Metropolis:Initialize S = (S1,…,SN)

label Generate new configuration S‘Calculate H= H(S,S‘)if H 0 accept S‘ (i.e. S‘S)else generate random numer x[0,1] if x<exp(-H) accept S‘ (i.e. S‘S)compute O(S)goto label

H(S,S‘) = H(S‘)-H(S)

Single spin flip Metropolis for 2d IsingProcedure single spin flip Input L, T, N=L*L Define arrays: S[i], i=1,…,N, h[i], i=1,…,N, etc. Initialize S[i], nxm[i], nxp[i],…., h[i] step = 0 while (step<max_step) choose random site i calculate dE = 2*h[i]*S[i] if ( dE <= 0 )

S[i]=-S[i]; update h[nxm[i]], h[nxp[i]],… else p = exp(-dE/T) x = rand() if ( x<p) S[i]=-S[i]; update h[nxm[i]], h[nxp[i]],… compute M(S), E(S), … accumulate M, E, … step++ Output m, e, …

Implementation issues

Periodic boundary conditions Neighbor tables

if if

e.g.:

Implementation issues (2)

With single spin flip E(S) and E(S‘) differ only by 4 terms in 2d (6 terms in 3d):

Flip spin i means Si‘ = -Si, all other spins fixed, i.e. Sj‘=Sj for all ji

Tabulate exponentials exp(-4), exp(-8) to avoid transcendental functions in the innermost loop

Use array h[i] for local fields, if move (flip is rejected nothing to be done, if flip accepted update Si and hnxm[i], hnxp[i], etc.

Study of phase transitions with MC

Ising model in d>1 has a 2nd order phase transition at h=0, T=Tc

Magnetization (order parameter):

Phase diagram

T<Tc

m

h

h0

h=0

1st order phase transitionas a functionof h!

2nd orderphase transition as a function of Tat h=0!

Critical behavior

Magnetization:

Susceptibility:

Specific heat:

Correlation function:

Correlation length:

Scaling relations:

Singularities at Tc in the thermodynamic limit (N):

Finite size behavior

w. Periodic b.c.

Finite Size Scaling

FSS forms:

4th order cumulant:

Dimensionless (no L-dependent prefactor)- Good for the localization of the critical point

Critical exponents of the d-dim. Ising model

Slowing down at the critical point

Quality of the MC estimats of therodynamic expectation values dependson the number of uncorrelated configurations –

Need an estimate for the correlation time of the Markov process!

Autocorrelation-function

Schematically:

for TTc Configurations should decorrelatefaster than with single spin-flip!

Solution: Cluster Moves

Cluster Algorithms

Construction process of the clusters in the Wolff algorith:Start from an initial + site, include other + sites with prbability p (left).The whole cluster (gray) is then flipped

p(b)

Here c1=10, c2=14

Detailled balance condition: p(a) A(ab) w(ab) = p(b) A(ba) w(ba)

p(a)

„A priori“ or construction probability:

Wolff algorithm (cont.)

Once the cluster is constructed with given p, one gets c1 and c2 ,with which one can compute the acceptance probability w(ab)

But with p = 1-e-2 the acceptance probability w(ab) becomes 1!

Thus with p=1-e-2 the constructed cluster is always flipped!

Remarkable speed up, no critical slowing down at the critical point!

Wolff cluster flipping for Ising

(1) Randomly choose a site i

(2) Draw bonds to all nearest neighbors j

with probability

(3) If bonds have been drawn to any site j draw bonds to all

nearest neighbors k of j with probability

(4) Repeat step (3) until no more bonds are created

(5) Flip all spins in the cluster

(6) Got to (1)

(Note for S=S‘, and = 0 for SS‘, such that p=0 for SjSk)

Swendsen-Wang algorithm

Similar to Wolff, but

(1) Draw bonds between ALL nearest neighbors

with probability

(2) Identify connected clusters

(3) Flip each individual cluster with probability 1/2