Monte Carlo Methods:Basics

Charusita ChakravartyDepartment of Chemistry

Indian Institute of Technology Delhi

Flavours of Monte Carlo• Metropolis Monte Carlo Methods:

Generation of multidimensional probability distributions

Multidimensional integration• Projector Monte Carlo Methods:

Solution of partial differential equations in many-dimensions

• Green’s function Monte CarloMatrix Inversion techniques

Other uses of stochastic simulation techniquesSee J. S. Liu, Monte Carlo Strategies in Scientific Computing

• electrons in atoms, molecules and solids• bosonic superfluid: liquid helium

• semiclassical systems: liquid H2,water

• Quark-Gluon plasma

Quantum Many-Body Systems

References:1. J. W. Negele and H. Orland, Quantum Many-Particle Systems2. W. M. C. Foulkes, L. Mitas, R. J. Needs and G. Rajagopal, Rev. Mod. Phys.,73, 33 (2001)3. D. M. Ceperley and B. J. Alder, Ground State of the Electron Gas by a Stochastic Method, Phys. Rev. Lett., 45, 567 (1980)4. D. M. Ceperley, Rev. Mod. Phys.,67, 279 (1995)5. C. Chakravarty, Int. Rev. Phys. Chem., 16, 421 (1997)6. B. L. Hammond, W. A. Lester and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry

Atomic and Molecular Systems

•Born-Oppenheimer approximation: Adiabatic separation of electronic and nuclear motion

• Electronic motion:Energy scales e.g chemical binding energies (1-5 eV)Excited state occupancy at temperatures of interest (< 1000K) is negligible.Aim is to obtain many-body ground state wavefunctionDiffusion and Variational Monte Carlo

• Nuclear/Atomic MotionEnergy scales: rotations and vibrations (0.1eV)Finite-temperature methods: Path-integral Monte Carlo

Lattice/Continuum Hamiltonians

Organization

• Basics of Monte Carlo Methods• Continuum systems: interacting electrons and/or

atoms

– Diffusion Monte Carlo– Path integral Monte Carlo

• Lattice Hamiltonians

)(2

ˆ 2

1

2

RVm

H i

N

i i

Integration:

Grid-Based• Define a set of grid points

and associated weights

• Error

• Computational efficiency will grow exponentially with dimensionality

StochasticSample points randomly and

uniformly in the interval [a:b]

Error: Central Limit Theorem

Computational efficiency

b

adxxfI )(

i

ii xfwI )(

kch

kdc cT /)/(

N

iixfN

abI

1

)(

NffN

/22

2

220

tTc

Importance Sampling:Reducing the Variance

b

adxxfI )(

b

a

dxxpxp

xfI )(

)(

)(

2

2

22

p

f

p

f

Sampling distribution p(x) must be finite and non-negative

Uniform Distributions

mIx

mqpII

mp,q

jj

jj

/

mod)(

and :parametersinteger Three

11

1

Linear CongruentialGenerator

Non-uniform Distributions:von Neumann Rejection Method

Desired distribution, p(x)

ConvenientSampling Distribution,f(x)

)()( xpxf 1. Sample xi from the

distribution f(xi) and compute p(xi).

2. Sample a random number uniformly distributed between 0 and 1.

3. If p(xi)/f(xi) > , accept xi; otherwise reject xi.

xi

What is a Random Walk?

A random walk is a set of probabilistic rules which allow for the motion of the state point of the system through some conguration space; the moving state point is referred to as a walker. All random walks must have the following features:– The available configuration space and attributes of walker must

be defined. An ensemble of walkers is the probability distribution of the system at some point in time.

– Source distribution at time t=0 must be defined.– The system's kinetics will be determined by the

transition matrix, T, which is a set of probabilistic rules which govern a single move of the walker from its current position to some neighbouring position.The transition matrix must satisfy the requirements:

1 and 0 '' dxTT xxxx

Markov Chains

Metropolis Algorithm

Classical Monte CarloGenerating a set of configurations, {xi, i=1,N} distributed

according to their Boltzmann weights, P(xi)=(exp(-V(xi))

P(x

)=(e

xp(-V

(x))

xold x’newxnew

Generating the Boltzmann Distribution

)( and nold

nold rVVrr

)( and newnewn

new rVVrr

)exp(/)exp( oldnew VVw

Current Configuration

Trial Configuration

Compute ratio of Boltzmann weights

Is w> 1?

Accept new configuration

Generate uniform random no. between 0 and 1

Is < w?

Reject new configuration

Yes

No

Yes

No

Random Walks and Differential Equations

One-dimensional Diffusion Equation

Random Walks