Multicanonical Method Research Articles

The Multicanonical Weighted Histogram Analysis Method for the Free Energy Landscape along Structural Transition Paths

Abstract A new algorithm for revealing the free-energy barrier among stable states was developed by combining the multicanonical ensemble with the weighted histogram analysis method (WHAM). The method, called multicanonical WHAM, was applied to the cis / trans -imide isomerization of a blocked peptide dimer, –Ala–Pro–. Many conformations around the transition states with high potential energies were sampled. From the reweighted canonical ensemble at 300 K, the landscape of the potential of mean force covering the transition states and the two stable cis - and trans -isomer states was observed. The structural transition paths between the cis - and trans -isomer states are discussed.

Long-range interactions in a lattice gas automaton model for immiscible fluids

Generalized-ensemble Monte Carlo simulations such as the multicanonical method and similar techniques are among the most efficient approaches for simulations of systems undergoing discontinuous phase transitions or with rugged free-energy landscapes. As Markov chain methods, they are inherently serial computationally. It was demonstrated recently, however, that a combination of independent simulations that communicate weight updates at variable intervals allows for the efficient utilization of parallel computational resources for multicanonical simulations. Implementing this approach for the many-thread architecture provided by current generations of graphics processing units (GPUs), we show how it can be efficiently employed with of the order of 104 parallel walkers and beyond, thus constituting a versatile tool for Monte Carlo simulations in the era of massively parallel computing. We provide the fully documented source code for the approach applied to the paradigmatic example of the two-dimensional Ising model as starting point and reference for practitioners in the field.Program Title: cudamucaProgram Files doi: http://dx.doi.org/10.17632/tzhfpdymv9.1 Licensing provisions: Creative Commons Attribution license (CC BY 4.0)Programming language: C++, CUDAExternal routines/libraries: NVIDIA CUDA Toolkit 6.5 or newerNature of problem: The program determines weights for a multicanonical simulation of the 2D Ising model to result in a flat energy histogram. A final production run with these weights provides an estimate of the density of states of the model.Solution method: The code uses a parallel variant of the multicanonical method employing many parallel walkers that accumulate a common histogram. The resulting histogram is used to determine the weight function for the next iteration. Once the iteration has converged, simulations visit all possible energies with the same probability.Additional comments including restrictions and unusual features: The system size and size of the population of replicas are limited depending on the memory of the GPU device used. Code repository at https://github.com/CQT-Leipzig/cudamuca.

Multicanonical jump walking: A method for efficiently sampling rough energy landscapes

The multicanonical sampling and jump walking methods are combined to provide a new, effective means of overcoming quasiergodicity in Monte Carlo simulations. In this new method, configurations generated during a long multicanonical sampling are stored infrequently and a modified jump walking procedure is implemented using this set of configurations to sample phase space at low temperature. Multicanonical jump walking, as this new method is called, is compared with regular jump walking and with straight multicanonical ensemble sampling on two systems: a one-dimensional random potential and an Ar13 cluster. It is shown that for the same number of MC steps, the multicanonical jump walking method more efficiently samples the phase space than either the regular jump walking or the pure multicanonical ensemble sampling method.

Formation of Helical States in Wormlike Polymer Chains

We propose a potential for wormlike polymer chains which can be used to model the low-temperature conformational structures. We successfully reproduced helix ground states up to 6.5 helical loops, using the multicanonical Monte Carlo simulation method. We demonstrate that the coil-helix transition involves four distinct phases: coil(gaslike), collapsed globular(liquidlike), and two helical phases I and II (both solidlike). The helix I phase is characterized by a helical structure with dangling loose ends, and the helix II phase corresponds to a near perfect helix ordering in the entire crystallized chain.

A selectively enhanced multicanonical molecular dynamics method for conformational sampling of peptides in realistic water molecules

We propose an efficient conformational sampling method for biological macromolecules in explicit water molecules, called the selectively enhanced multicanonical method. This method selectively enhances the sampling efficiency in the energy space for the solute–solute and solute–water interactions by using the multicanonical algorithm, while it restrains the sampling of the water–water energy space. We applied the current method and the conventional multicanonical method to a model peptide in water and showed that much less computation time was required for the current method than for the conventional one and the distributions at 300 K obtained by these two methods, are in good agreement.

Multicanonical Monte Carlo simulations

Canonical Monte Carlo simulations of disordered systems like spin glasses and systems undergoing first-order phase transitions are severely hampered by rare event states which lead to exponentially diverging autocorrelation times with increasing system size and hence to exponentially large statistical errors. One possibility to overcome this problem is the multicanonical reweighting method. Using standard local update algorithms it could be demonstrated that the dependence of autocorrelation times on the system size V is well described by a less divergent power law, τ ∝ V α , with 1< α<3, depending on the system. After a brief review of the basic ideas, combinations of multicanonical reweighting with non-local update algorithms will be discussed. With the multibondic algorithm, which combines multicanonical reweighting with cluster updates, the dynamical exponent α can be reduced to unity, the optimal value one would expect from a random walk argument. Asymptotically for large system sizes the multibondic algorithm therefore always performs better than the standard multicanonical method. Finally it is shown that a combination with multigrid update techniques improves the performance of multicanonical simulations by roughly one order of magnitude, uniformly for all system sizes.

Algorithmic aspects of multicanonical simulations

Monte Carlo (MC) simulations of many systems, can be considerably speeded up by using multicanonical or related methods. I shall focus on two aspects: (i) Opinions about the optimal choice of weights. (ii) Recursive weight factor estimates.

Multicanonical methods, molecular dynamics, and Monte Carlo methods: Comparison for Lennard-Jones glasses

We applied a multicanonical algorithm (entropic sampling) to a two-dimensional and a three-dimensional Lennard-Jones system with quasicrystalline and glassy ground states. Focusing on the ability of the algorithm to locate low lying energy states, we compared the results of the multicanonical simulations with standard Monte Carlo simulated annealing and molecular dynamics methods. We find slight benefits to using entropic sampling in small systems (less than 80 particles), which disappear with larger systems. This is disappointing as the multicanonical methods are designed to surmount energy barriers to relaxation. We analyze this failure theoretically, and show (1) the multicanonical method is reduced in the thermodynamic limit (large systems) to an effective Monte Carlo simulated annealing with a random temperature vs. time, and (2) the multicanonical method gets trapped by unphysical entropy barriers in the same metastable states whose energy barriers trap the traditional quenches. The performance of Monte Carlo and molecular dynamics quenches were remarkably similar.

First-order phase transition in a stiff polymer chain

We studied the transition between the elongated coil and collapsed globule states in a single polymer chain, avoiding kinetic effects such as supercooling, or kinetic trapping into local minima, using a multicanonical method. It has become clear that single stiff chains exhibits a first-order phase transition. Reflecting the finiteness of the number of elements (or the number of segments) in a chain, the two states, coil and globule, coexist within a finite width of temperature for the ensemble of monodisperse chains, even under true thermodynamic equilibrium.

Exchange Monte Carlo Method and Application to Spin Glass Simulations

We propose an efficient Monte Carlo algorithm for simulating a ``hardly-relaxing" system, in which many replicas with different temperatures are simultaneously simulated and a virtual process exchanging configurations of these replica is introduced. This exchange process is expected to let the system at low temperatures escape from a local minimum. By using this algorithm the three-dimensional $\pm J$ Ising spin glass model is studied. The ergodicity time in this method is found much smaller than that of the multi-canonical method. In particular the time correlation function almost follows an exponential decay whose relaxation time is comparable to the ergodicity time at low temperatures. It suggests that the system relaxes very rapidly through the exchange process even in the low temperature phase.

Comparative study of multicanonical and simulated annealing algorithms in the protein folding problem

We compare a few variants of the recently proposed multicanonical method with the well known simulated annealing for the effectiveness in search of the energy global minimum of a biomolecular system. For this we study in detail Met-enkephalin, one of the simplest peptides. We show that the new method not only outperforms simulated annealing in the search of the energy groundstate but also provides more statistical-mechanical information about the system.

Sampling Ground-State Configurations of a Peptide by Multicanonical Annealing

We test a variant of the multicanonical method, multicanonical annealing , to determine the ground-state configuration of biological molecules. As a simple example, we study Met-enkephalin, one of the simplest peptides. Our results are compared with those of simulated annealing .

Optimization by multicanonical annealing and the traveling salesman problem.

We propose a powerful and general simulated annealing method to study combinatorial optimization problems. It combines the multicanonical method, which samples directly the microcanonical entropy of the system, with an elaborate but straightforward annealing scheme. The information about the local entropy obtained during short Monte Carlo simulations is fully utilized for optimization in an iterative fashion. We present results of an extensive investigation of the traveling salesman problem in a unit square. We estimate the optimal length in the limit of a large number of cities.