Aplicação do algoritmo de metropolis a discos rígidos

Abstract

This work applies the Metropolis–Hastings algorithm to the simulation of a two-dimensional hard-disk system, assessing the method’s accuracy in estimating thermodynamic quantities. The model consists of 224 identical rigid disks of radius σ confined in a unit-square box with periodic boundary conditions. The radial distribution function g(r) is estimated from samples generated by a Markov chain in which successive random displacements are applied individually to each particle. By extrapolating g(r) to the contact distance d0, one obtains g(d0), which is required to compute the equation of state. The resulting values agree with free-volume theory at high densities, whereas the error increases at low densities. The behavior of g(r) is further examined for r ≫σ: it exhibits oscillations about unity that decay with distance from the reference particle. Low-density systems display stronger damping, while high densities retain oscillations of larger amplitude, indicating greater short-range order. The near-ideal-gas behavior at low densities explains the reduced accuracy, owing to the lower collision frequency between particles.