Biomedical subjects
Publications and source records attributed to James L Monroe.
We investigate the Wu conjecture for the critical temperature of the Potts model on the kagomé lattice. Previous testing of this conjecture has been for small values of q. Our emphasis is on large values of q where we are able to obtain very accurate estimates of the critical temperature which show the conjecture itself to be very accurate for these q values but not exact.
We present estimates of the critical temperature of both higher-spin Ising and Potts models on the square and simple cubic lattices based on a Husimi tree, dynamical systems approach. For the square lattice, exact results are available for the Potts model but we can refine and improve some previous estimates based on series expansions for the higher-spin Ising model. For the simple cubic lattice, based on the systematics of this approach, we are able to present accurate estimates for the critical temperature of spin systems with spin values, where estimates by other methods are unavailable or much less accurate. The same is true for the Potts model. All our estimates can be obtained using a personal computer.
The Bulirsch-Stoer extrapolation algorithm was used in a statistical mechanics setting in 1984 by Henkel and Patkos. Since then it has been used numerous times in a large variety of settings to extrapolate from finite size systems to the infinite system in a large variety of situations in statistical mechanics. We investigate some of its characteristics by using it in situations where the behavior of the infinite system is known. One characteristic is the error involved with the algorithm. More importantly we investigate the dependence of the effectiveness of the algorithm on the size and number of systems used as input and find that a larger number of smaller systems results in better results than a few much larger systems.
We generalize a systematic approximation method presented by the present author earlier [Monroe, Phys. Rev. E 64, 016126 (2001)], and which was applied to Ising models with spin one-half. The generalization allows one to consider higher spin systems. In particular we consider the spin-one, Blume-Capel model on a square lattice. We obtain an approximation to the phase diagram of the system that we show is as or more accurate than any presently available. This we are able to do with a rather modest effort thereby illustrating the fact that the method gives one rather accurate results without requiring too extensive computer calculations.