Monte Carlo Methods: Basics

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Monte Carlo Methods: Basics. Charusita Chakravarty Department 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:

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## Monte Carlo Methods: Basics

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### Monte Carlo Methods:Basics

Charusita Chakravarty

Department 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 Carlo

Matrix Inversion techniques

Other uses of stochastic simulation techniques

See J. S. Liu, Monte Carlo Strategies in Scientific Computing

Quantum Many-Body Systems

• electrons in atoms, molecules and solids
• bosonic superfluid: liquid helium
• semiclassical systems: liquid H2,water
• Quark-Gluon plasma

References:

1. J. W. Negele and H. Orland, Quantum Many-Particle Systems

2. 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 wavefunction
• Diffusion and Variational Monte Carlo
• Nuclear/Atomic Motion
• Energy 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
Grid-Based

Define a set of grid points and associated weights

Error

Computational efficiency will grow exponentially with dimensionality

Stochastic

Sample points randomly and uniformly in the interval [a:b]

Error: Central Limit Theorem

Computational efficiency

Integration:
Importance Sampling:Reducing the Variance

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

Uniform Distributions

Linear

Congruential

Generator

Non-uniform Distributions:von Neumann Rejection Method

Convenient

Sampling

Distribution,f(x)

• Sample xifrom the distribution f(xi) and compute p(xi).
• Sample a random number xuniformly distributed between 0 and 1.
• 3. If p(xi)/f(xi) > x, accept xi; otherwise reject xi.

Desired

distribution, p(x)

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 positionto some neighbouring position.The transition matrix must satisfy the requirements:
Classical Monte Carlo

Generating a set of configurations, {xi, i=1,N} distributed according to their Boltzmann weights, P(xi)=(exp(-bV(xi))

P(x)=(exp(-bV(x))

xold

x’new

xnew

Generating the Boltzmann Distribution

Current Configuration

Reject new configuration

Trial Configuration

No

Compute ratio of Boltzmann weights

Yes

Is x < w?

Accept new configuration

Yes

Is w> 1?

Generate uniform random no. x between 0 and 1

No

Random Walks and Differential Equations

One-dimensional Diffusion Equation

Random Walks