Equation of State Calculations by Fast Computing Machines

Inside the paper

The problem

Calculating the properties of interacting particles requires averaging over a vast space of configurations. Direct numerical integration is difficult because many possible configurations contribute little to the equilibrium average.

The idea

Use a random walk through particle configurations that spends more time in configurations with higher statistical weight. Estimate physical quantities from the configurations visited by the walk.

How it works

• Place a finite collection of particles in a square and use periodic copies of the square to reduce boundary effects.
• Propose a small random displacement for a particle and calculate the resulting change in interaction energy.
• Accept an energy-lowering move; accept an energy-raising move with a probability that decreases with its energy cost and depends on temperature.
• Keep the old configuration when a move is rejected and accumulate averages over the resulting sequence of configurations.

What it showed

The authors use their method to calculate the equation of state of a two-dimensional rigid-sphere system. They compare the computed pressures with approximations and report agreement between runs with different numbers of particles within statistical error.

Read it for

Read Section II for the sampling rule and Section III for its rigid-sphere specialization. Section IV and Figure 4 show the numerical comparison; the detailed virial-coefficient calculation in Section V can be skimmed initially.

Written in our own words from the paper itself. A draft the editors have not yet reviewed.