Metropolis sampling

Molecular Thermodynamics Lesson 1

Stochastic integration and importance sampling in high-dimensional configuration space.

The Sampling Challenge

The partition function \( Z \) requires integrating over all possible configurations:

\[ Z = \int e^{-\beta U(\mathbf{r})} d\mathbf{r}^N \]

For \( N=100 \) particles, this is a 300-dimensional integral.
Uniform grid sampling fails: \( 10^{300} \) points needed if we take just 10 points per dimension.

The Problem

Most of configuration space has very high energy \( U(\mathbf{r}) \), meaning the Boltzmann factor \( e^{-\beta U} \) is essentially zero.

We need to sample where the action is!

Importance Sampling

Instead of sampling uniformly, pick points according to a probability distribution \( P(\mathbf{r}) \).

"Spend time in regions that contribute most to the average."

Monte Carlo Integration

If we sample from \( P(\mathbf{r}) \), the average of an observable \( A \) is:

\[ \langle A \rangle \approx \frac{1}{M} \sum_{i=1}^M A(\mathbf{r}_i) \]

No need to divide by the integral of \( P \)!

The Metropolis Insight

We can sample from a distribution \( P \) even if we only know its ratios, not its normalization constant.

The Metropolis Loop

Algorithm 1.1

1

Generate a trial move

Displace a particle: \( \mathbf{r}' = \mathbf{r} + \Delta \)
2

Calculate energy change

\( \Delta U = U(\mathbf{r}') - U(\mathbf{r}) \)
3

Acceptance Test

Is \( \Delta U < 0 \)? Accept immediately.

If not, accept with probability \( P = e^{-\beta \Delta U} \).
4

Update State

If rejected, the system stays in the current configuration for the next step.

Markov Chain Property

The future depends only on the present configuration, not the history.

Detailed Balance

To ensure the simulation reaches the correct equilibrium distribution \( \mathcal{N} \), we must satisfy:

\( \mathcal{N}(o) \cdot \alpha(o \to n) \cdot P_{acc}(o \to n) = \mathcal{N}(n) \cdot \alpha(n \to o) \cdot P_{acc}(n \to o) \)

"The flux of probability from state \( o \) to \( n \) must equal the flux from \( n \) back to \( o \)."

Thermalization

System must 'forget' the initial lattice before sampling.

Correlation

Successive steps are correlated. Need long runs for error estimation.

Acceptance Rate

Target ~30-50% for optimal phase space exploration.

Computational Molecular Thermodynamics

Metropolis Monte Carlo Integration

Molecular Dynamics and Ensembles

Free Energy and Widom Insertion

Phase Equilibria Gibbs Ensemble

Lennard-Jones Fluid Project

Metropolis Slides Presentation

Metropolis Blueprint Worksheet

Metropolis Implementation Guide Teacher Resource

MD Mechanics Slides Presentation

MD Time Step Challenge Problem Set

MD Simulation Lab Student Guide

Free Energy Slides Presentation

Widom Math Derivation Worksheet

Gibbs Ensemble Slides Presentation

Coexistence Plotter Activity Sheet

Lennard-Jones Project Slides Presentation

Lennard-Jones Project Report Template

Lennard-Jones Project Rubric Teacher Resource

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