Phase changes in 38 atom Lennard-Jones clusters. II: A parallel tempering study of equilibrium and dynamic properties in the molecular dynamics and microcanonical

We study the 38-atom Lennard-Jones cluster with parallel tempering Monte Carlo methods in the microcanonical and molecular dynamics ensembles. A new Monte Carlo algorithm is presented that samples rigorously the molecular dynamics ensemble for a system at constant total energy, linear and angular momenta. By combining the parallel tempering technique with molecular dynamics methods, we develop a hybrid method to overcome quasi-ergodicity and to extract both equilibrium and dynamical properties from Monte Carlo and molecular dynamics simulations. Several thermodynamic, structural and dynamical properties are investigated for LJ$_{38}$, including the caloric curve, the diffusion constant and the largest Lyapunov exponent. The importance of insuring ergodicity in molecular dynamics simulations is illustrated by comparing the results of ergodic simulations with earlier molecular dynamics simulations.Comment: Journal of Chemical Physics, accepte

Parallel strategy for optimal learning in perceptrons

We developed a parallel strategy for learning optimally specific realizable rules by perceptrons, in an online learning scenario. Our result is a generalization of the Caticha–Kinouchi (CK) algorithm developed for learning a perceptron with a synaptic vector drawn from a uniform distribution over the N-dimensional sphere, so called the typical case. Our method outperforms the CK algorithm in almost all possible situations, failing only in a denumerable set of cases. The algorithm is optimal in the sense that it saturates Bayesian bounds when it succeeds

Phase Changes in 38-Atom Lennard-Jones Clusters. I. A Parallel Tempering Study in the Canonical Ensemble

The heat capacity and isomer distributions of the 38-atom Lennard-Jones cluster have been calculated in the canonical ensemble using parallel tempering Monte Carlo methods. A distinct region of temperature is identified that corresponds to equilibrium between the global minimum structure and the icosahedral basin of structures. This region of temperatures occurs below the melting peak of the heat capacity and is accompanied by a peak in the derivative of the heat capacity with temperature. Parallel tempering is shown to introduce correlations between results at different temperatures. A discussion is given that compares parallel tempering with other related approaches that ensure ergodic simulations

The Approach to Ergodicity in Monte Carlo Simulations

The approach to the ergodic limit in Monte Carlo simulations is studied using both analytic and numerical methods. With the help of a stochastic model, a metric is defined that enables the examination of a simulation in both the ergodic and non-ergodic regimes. In the non-ergodic regime, the model implies how the simulation is expected to approach ergodic behavior analytically, and the analytically inferred decay law of the metric allows the monitoring of the onset of ergodic behavior. The metric is related to previously defined measures developed for molecular dynamics simulations, and the metric enables the comparison of the relative efficiencies of different Monte Carlo schemes. Applications to Lennard-Jones 13-particle clusters are shown to match the model for Metropolis, J-walking and parallel tempering based approaches. The relative efficiencies of these three Monte Carlo approaches are compared, and the decay law is shown to be useful in determining needed high temperature parameters in parallel tempering and J-walking studies of atomic clusters.Comment: 17 Pages, 7 Figure

Phase changes in 38 atom Lennard-Jones clusters; 1, A parallel tempering study in the canonical ensemble

The heat capacity and isomer distributions of the 38 atom Lennard-Jones cluster have been calculated in the canonical ensemble using parallel tempering Monte Carlo methods. A distinct region of temperature is identified that corresponds to equilibrium between the global minimum structure and the icosahedral basin of structures. This region of temperatures occurs below the melting peak of the heat capacity and is accompanied by a peak in the derivative of the heat capacity with temperature. Parallel tempering is shown to introduce correlations between results at different temperatures. A discussion is given that compares parallel tempering with other related approaches that ensure ergodic simulations

Dynamical replica theoretic analysis of CDMA detection dynamics

We investigate the detection dynamics of the Gibbs sampler for code-division multiple access (CDMA) multiuser detection. Our approach is based upon dynamical replica theory which allows an analytic approximation to the dynamics. We use this tool to investigate the basins of attraction when phase coexistence occurs and examine its efficacy via comparison with Monte Carlo simulations.Comment: 18 pages, 2 figure

Inference by replication in densely connected systems

An efficient Bayesian inference method for problems that can be mapped onto dense graphs is presented. The approach is based on message passing where messages are averaged over a large number of replicated variable systems exposed to the same evidential nodes. An assumption about the symmetry of the solutions is required for carrying out the averages; here we extend the previous derivation based on a replica symmetric (RS) like structure to include a more complex one-step replica symmetry breaking (1RSB)-like ansatz. To demonstrate the potential of the approach it is employed for studying critical properties of the Ising linear perceptron and for multiuser detection in Code Division Multiple Access (CDMA) under different noise models. Results obtained under the RS assumption in the non-critical regime give rise to a highly efficient signal detection algorithm in the context of CDMA; while in the critical regime one observes a first order transition line that ends in a continuous phase transition point. Finite size effects are also observed. While the 1RSB ansatz is not required for the original problems, it was applied to the CDMA signal detection problem with a more complex noise model that exhibits RSB behaviour, resulting in an improvement in performance.Comment: 47 pages, 7 figure

Computational capabilities of multilayer committee machines

We obtained an analytical expression for the computational complexity of many layered committee machines with a finite number of hidden layers (L < 8) using the generalization complexity measure introduced by Franco et al (2006) IEEE Trans. Neural Netw. 17 578. Although our result is valid in the large-size limit and for an overlap synaptic matrix that is ultrametric, it provides a useful tool for inferring the appropriate architecture a network must have to reproduce an arbitrary realizable Boolean function

Opinion Dynamics of Learning Agents: Does Seeking Consensus Lead to Disagreement?

We study opinion dynamics in a population of interacting adaptive agents voting on a set of complex multidimensional issues. We consider agents which can classify issues into for or against. The agents arrive at the opinions about each issue in question using an adaptive algorithm. Adaptation comes from learning and the information for the learning process comes from interacting with other neighboring agents and trying to change the internal state in order to concur with their opinions. The change in the internal state is driven by the information contained in the issue and in the opinion of the other agent. We present results in a simple yet rich context where each agent uses a Boolean Perceptron to state its opinion. If there is no internal clock, so the update occurs with asynchronously exchanged information among pairs of agents, then the typical case, if the number of issues is kept small, is the evolution into a society thorn by the emergence of factions with extreme opposite beliefs. This occurs even when seeking consensus with agents with opposite opinions. The curious result is that it is learning from those that hold the same opinions that drives the emergence of factions. This results follows from the fact that factions are prevented by not learning at all from those agents that hold the same opinion. If the number of issues is large, the dynamics becomes trapped and the society does not evolve into factions and a distribution of moderate opinions is observed. We also study the less realistic, but technically simpler synchronous case showing that global consensus is a fixed point. However, the approach to this consensus is glassy in the limit of large societies if agents adapt even in the case of agreement.Comment: 16 pages, 10 figures, revised versio

Improved message passing for inference in densely connected systems

An improved inference method for densely connected systems is presented. The approach is based on passing condensed messages between variables, representing macroscopic averages of microscopic messages. We extend previous work that showed promising results in cases where the solution space is contiguous to cases where fragmentation occurs. We apply the method to the signal detection problem of Code Division Multiple Access (CDMA) for demonstrating its potential. A highly efficient practical algorithm is also derived on the basis of insight gained from the analysis