Statistical-Temperature Monte Carlo and Molecular Dynamics Algorithms

A simulation method is presented that achieves a flat energy distribution by updating the statistical temperature instead of the density of states in Wang-Landau sampling. A novel molecular dynamics algorithm (STMD) applicable to complex systems and a Monte Carlo algorithm are developed from this point of view. Accelerated convergence for large energy bins, essential for large systems, is demonstrated in tests on the Ising model, the Lennard-Jones fluid, and bead models of proteins. STMD shows a superior ability to find local minima in proteins and new global minima are found for the 55 bead $AB$ model in two and three dimensions. Calculations of the occupation probabilities of individual protein inherent structures provide new insights into folding and misfolding.

Figure 1

The $32\times 32$ Ising model: (a) convergent $\tilde{T}(E)$ and histograms (inset); (b) entropy estimates $\tilde{S}(E)$ and absolute errors $\delta S(E)$ (inset) for energy bins $\Delta =8$, 32, and 64. The notation for curves is $(f_{d,0}\Delta )$.

Figure 2

The $N=110$ LJ fluid: (a) convergent $\tilde{T}(E)$ and histograms (inset); (b) statistical errors of STMC and WL in specific heats as a function of MC steps with various values of $(f_{d,0},\Delta )$. Reweighted results indicated by * have been obtained from the initial simulation data of $1.5\times {10}^6$ MC steps with $\Delta =1$ and ${10}^6$ MC steps with $\Delta =16$.

Figure 3

(a) Specific heat $C_V$ for BLN 46-mer and 69-mer, with peaks corresponding to the collapse transition. The mean inherent structure occupation probability $p_i(T)$ for the $i$th inherent structure for (b) 46-mer and (c) 69-mer ($p_0$ corresponds to the global minimum).

Figure 4

Structures of new global minima of $S_{55}$ in $d=2$ and 3 dimensions. Gray and black balls correspond to hydrophilic and hydrophobic monomers, respectively.