We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional
O(3) and $O(\ensuremath{\infty})$ \ensuremath{\sigma} models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.
We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional O(3) and O(\\infty) \\sigma-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.
We show that the conventional power law of the inverse of the particle number (
N−1) for extrapolation of total energies into the thermodynamic limit does not reproduce the Gell-Mann and Brueckner results for the three-dimensional high density electron gas, even when system sizes of
N∼2000 electrons are used. Instead, a power law of
N−1/3 fits the data beyond 1000 electrons, which has also been found by other authors. The
N−1/3 power law also appears to fit data from other densities better than
N−1. To explain this observation, we develop a novel analysis of the low-
G components of the wavefunction, and show they are not well-sampled enough, even at high electron numbers, to yield a straight-forward
N−1 convergence to the thermodynamic limit (TDL). We show that, for most practical calculations, the TDL correlation energy lies between the extrapolation from
N−1 and the extrapolation from
N−1/3. Furthermore, the energy extrapolation from an
N−1/3 power law tends to either increase or decrease with increasing system size, but when energies increase is an indication that
N−1 can effectively be used for extrapolation.
A new field opened when modern computers offered the possibility of performing extensive simulations of large systems. This allows known behaviours to be checked and provides an exploratory guide in circumstances where theoretical tools are absent. Measurements of observables can be compared both to existing theoretical expectations – providing a crucial test for their validity – and to experiments – checking the modelling of a physical system –. This chapter presents the background material needed to design simulations on a (relatively) large scale. Some numerical examples have already been presented in previous chapters, and we only give a few further illustrations, pertaining mainly to lattice gauge theory, the usefulness of which relies extensively on this method. We also describe a practical implementation of real space renormalization, known as the Monte Carlo Renormalization Group method (Ma, Swendsen, Wilson). Finally, we discuss specific issues relevant to the extension of the simulations to fermionic systems. Algorithms Generalities Systems with up to 10 6 to 10 7 variables can be handled by computers, and these numbers may soon be significantly increased. The measurements can be sufficiently numerous to allow statistical accuracy. Although simulated systems still have a modest size as compared to macroscopic systems, collective effects already clearly appear and accurate results about critical phenomena emerge from the numerical simulations. It turns out, when investigating more closely the available numerical methods, that one gets a better insight into the foundations of equilibrium statistical physics, the ergodicity problems, the meaning of probabilities and, last but not least, ways to approach equilibrium.
The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.