Fast Adaptive Flat-histogram Ensemble for Calculating Density of States and Enhanced Sampling in Large Systems

We presented an efficient algorithm, fast adaptive flat-histogram ensemble (FAFE), to estimate the density of states (DOS) and to enhance sampling in large systems. FAFE calculates the means of an arbitrary extensive variable $U$ in generalized ensembles to form points on the curve $\beta_{s}(U) \equiv \frac{\partial S(U)}{\partial U}$, the derivative of the logarithmic DOS. Unlike the popular Wang-Landau-like (WLL) methods, FAFE satisfies the detailed-balance condition through out the simulation and automatically generates non-uniform $(\beta_{i}, U_{i})$ data points to follow the real change rate of $\beta_{s}(U)$ in different $U$ regions and in different systems. Combined with a $U-$compression transformation, FAFE reduces the required simulation steps from $O(N^{3/2})$ in WLL to $O(N^{1/2})$, where $N$ is the system size. We demonstrate the efficiency of FAFE in Lennard-Jones liquids with several $N$ values. More importantly, we show its abilities in finding and identifying different macroscopic states including meta-stable states in phase co-existing regions.