Improved Wang-Landau algorithm for the joint density of states of continuous models

The joint density of states of a statistical physical system is the key to calculating thermodynamic observables at all temperatures and external fields. For example, \textit{$\rho $}($M$, $E)$ of a Heisenberg ferromagnet is a generalization of \textit{$\rho $}($E)$, from which magnetization and susceptibility at all temperatures can be obtained. A well-known method to calculate \textit{$\rho $}($E)$ is the Wang-Landau algorithm [1], which can be in principle extended to the joint density of states. Unfortunately, a straightforward application of the Wang-Landau algorithm to this type of problem turns out to be inefficient. We thus adopt a number of strategies to accelerate the simulation and to increase the performance of the algorithm in low-density regimes. In particular, we replace the conventional binning scheme with kernel density estimation, so that the algorithm is intrinsically suitable for continuous systems. This version of the Wang-Landau algorithm is also generally applicable to classical statistical physics models, including discrete models with large size. We also discuss other promising applications to magnetic nano-particles and in biophysics. [1] F. Wang and D. P. Landau, Phys. Rev. Lett. \textbf{86}, 2050 (2001). *This research is supported by the Department of Energy through the Laboratory Technology Research Program of OASCR and the Computational Materials Science Network of BES under Contract No. DE-AC05-00OR22725 with UT-Battelle LLC, and by NSF DMR-0341874.