Narrow Results

Monte Carlo simulation techniques, like simulated annealing and parallel tempering, are often used to evaluate low-temperature properties and find ground states of disordered systems. Here we compare these methods using direct calculations of ground states for three-dimensional Ising diluted antiferromagnets in a field (DAFF) and three-dimensional Ising spin glasses (ISG). For the DAFF, we find that, with respect to obtaining ground states, parallel tempering is superior to simple Monte Carlo and to simulated annealing. However, equilibration becomes more difficult with increasing magnitude of the externally applied field. For the ISG with bimodal couplings, which exhibits a high degeneracy, we conclude that finding true ground states is easy for small systems, as is already known. But finding each of the degenerate ground states with the same probability (or frequency), as required by Boltzmann statistics, is considerably harder and becomes almost impossible for larger systems.

We compare the efficiency of two prominent techniques for simulation of complex systems: parallel tempering and Wang–Landau sampling. We show that both methods are of comparable efficiency but are optimized for different platforms. Parallel tempering should be chosen on multi-processor system while Wang–Landau sampling is easier to implement on a single-processor computer.

We have performed parallel tempering simulations of a 13-residue peptide fragment of ribonuclease-A, c-peptide, in implicit solvent with constant dielectric permittivity. This peptide has a strong tendency to form α-helical conformations in solvent as suggested by circular dichroism (CD) and nuclear magnetic resonance (NMR) experiments. Our results demonstrate that 5th and 8–12 residues are in the α-helical region of the Ramachandran map for global minimum energy state in solvent environment. Effects of salt bridge formation on stability of α-helix structure are discussed.

We explore the potential of parallel tempering as a combinatorial optimization method, applying it to the traveling salesman problem. We compare simulation results of parallel tempering with a benchmark implementation of simulated annealing, and study how different choices of parameters affect the relative performance of the two methods. We find that a straightforward implementation of parallel tempering can outperform simulated annealing in several crucial respects. When parameters are chosen appropriately, both methods yield close approximation to the actual minimum distance for an instance with 200 nodes. However, parallel tempering yields more consistently accurate results when a series of independent simulations are performed. Our results suggest that parallel tempering might offer a simple but powerful alternative to simulated annealing for combinatorial optimization problems.

Some recent progress in Monte Carlo simulations of spin glasses will be presented. The problem of slow dynamics at low temperatures is partially alleviated by use of the parallel tempering (replica exchange) method. A useful technique to check for equilibration (applicable only for a Gaussian distribution) will be discussed. It will be argued that a finite size scaling analysis of the scaled correlation length of the system is a good approach with which to investigate phase transitions in spin glasses. This method will be used to study two questions:

(i) whether there is a phase transition in zero field in the Heisenberg spin glass in three dimensions, and

(ii) whether there is phase transition in a magnetic field in an Ising spin glass, also in three dimensions.

This paper describes an algorithm for selecting parameter values (e.g. temperature values) at which to measure equilibrium properties with Parallel Tempering Monte Carlo simulation. Simple approaches to choosing parameter values can lead to poor equilibration of the simulation, especially for Ising spin systems that undergo first-order phase transitions. However, starting from an initial set of parameter values, the careful, iterative respacing of these values based on results with the previous set of values greatly improves equilibration. Example spin systems presented here appear in the context of Quantum Monte Carlo.

The effective temperature configuration of parallel tempering (PT) in finite-time optimization is studied for the solution of the traveling salesman problem. An experimental analysis is conducted to decide the relative importance of the two characteristic temperatures, the specific-heat-peak temperature referred to in the general guidelines and the effective intermediate temperature identified in the recent study on simulated annealing (SA). The results show that the operation near the former has no notable significance contrary to the conventional belief but that the operation near the latter plays a crucial role in fulfilling the optimization function of PT. The method shares the same origin of effectiveness with the SA and SA-related algorithms.

Monte Carlo simulations based on simulated annealing and multicanonical algorithm have been performed to predict the secondary and tertiary structures of oligopeptide systems. Two oligopeptides, C-peptide of ribonuclease A and the fragment BPTI(16-36) of bovine pancreatic trypsin inhibitor, were studied. Only the amino-acid sequence information was used as input and initial conformations were randomly generated. The lowest-energy conformations obtained have α-helix structure and β-sheet structure for C-peptide and BPTI(16-36), respectively, in remarkable agreement with experimental results.