Performances of Wang-Landau algorithms for continuous systems

The relative performances of different implementations of the Wang-Landau method are assessed on two classes of systems with continuous degrees of freedom, namely, two polypeptides and two atomic Lennard-Jones clusters. Parallel tempering Monte Carlo simulations serve as a reference, and we pay particular attention to the variations of the multiplicative factor $f$ during the course of the simulation. For the systems studied, the Wang-Landau method is found to be of comparable accuracy as parallel tempering, but has significant difficulties in reproducing low-temperature transitions exhibited by the Lennard-Jones clusters at low temperature. Using a complementary order parameter and calculating a two-dimensional joint density of states significantly improves the situation, especially for the notoriously difficult ${\mathrm{LJ}}_{38}$ system. However, while parallel tempering easily converges for ${\mathrm{LJ}}_{31}$, we have not been able to get data of comparable accuracy with Wang-Landau multicanonical sampling.

Figure 1

(Color online) Schematic schedule of the variations of the modification factor $f$ during the course of different Wang-Landau simulations. (a) Standard WL; (b) WL with Zhou-Bhatt condition (WL-ZB); (c) WL with Jayasri-Sastry-Murthy (WL-JSM) schedule; (d) WL-annealing schedule. See text for details about the ZB condition and JSM method.

Figure 2

(Color online) Heat capacity curves of gas-phase polypeptides, as a function of temperature. The results of various implementations of the Wang-Landau method are compared to the data obtained with parallel tempering Monte Carlo. (a) ${\mathrm{Ala}}_8$, (b) ${\mathrm{AGWLK}}^{+}$.

Figure 3

(Color online) Thermal average $⟨d⟩$ of the distance between the tryptophan indolic nitrogen and the protonated nitrogen of the lysine amino acid of the ${\mathrm{AGWLK}}^{+}$, obtained from parallel tempering and multicanonical simulations. The density of states of the multicanonical simulation is provided by the WL-annealing method. For the parallel tempering data, both the direct results at the replica temperatures (symbols) and the continuous variations after histogram analysis are shown.

Figure 4

(Color online) Thermodynamical and structural data for the ${\mathrm{LJ}}_{38}$ cluster obtained from parallel tempering, from multicanonical simulations using the density of states from parallel tempering or from WL-annealing simulations, and from two-dimensional WL-annealing simulations. (a) Heat capacity, (b) $⟨Q_4⟩$ order parameter. The temperature is given in reduced LJ units of $\epsilon ∕k_{\mathrm{B}}$.

Figure 5

(Color online) Thermodynamical and structural data for the ${\mathrm{LJ}}_{31}$ cluster obtained from parallel tempering, from multicanonical simulations using the density of states from WL-annealing, and from two-dimensional WL-annealing simulations. (a) Heat capacity, (b) $⟨Q_6⟩$ order parameter. The temperature is given in reduced LJ units of $\epsilon ∕k_{\mathrm{B}}$.

Figure 6

(Color online) Accumulated number of $\mathrm{L}\to \mathrm{H}$ tunneling events for different implementations of the Wang-Landau method, during the course of the simulations. (a) ${\mathrm{AGWLK}}^{+}$, (b) ${\mathrm{LJ}}_{13}$.