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Ensemble-based control of evolutionary optimization algorithms.

This paper presents a study on how the intrinsic search parameters of an evolutionary optimization algorithm can be automatically controlled. It will be shown that only a small search parameter window ensures good optimization results. This evolutionary window, enclosing effective values for the mutation rate and temperature, can be adapted to by carefully steering the ensemble's fitness dispersion. A control sensor based on an entropy measure is introduced to achieve this goal. The efficiency of the proposed control method will be tested by optimizing artificial sequences such as the well-known low autocorrelated binary strings and natural sequences including RNA.

Algorithms↗

Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a microwave billiard with threefold symmetry.

Recently it has been shown that time-reversal invariant systems with discrete symmetries may display, in certain irreducible subspaces, spectral statistics corresponding to the Gaussian-unitary ensemble (GUE) rather than to the expected orthogonal one (GOE). A Kramers-type degeneracy is predicted in such situations. We present results for a microwave billiard with a threefold rotational symmetry and with the option to display or break a reflection symmetry. This allows us to observe the change from GOE to GUE statistics for one subset of levels. Since it was not possible to separate the three subspectra reliably, the number variances for the superimposed spectra were studied. The experimental results are compared with a theoretical and numerical study considering the effects of level splitting and level loss.

Journal Article↗

Stable heteroclinic cycles for ensembles of chaotic oscillators.

We study the formation of synchronous clusters in ensembles of globally coupled chaotic oscillators. We reveal that at least three clusters of identical synchronization are formed in such a system for large enough values of coupling strength. Our main result is an unexpected intermittent process of clusterization. This process gives strong indication to the existence of a stable heteroclinic cycle.

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Dynamical origin of uniform sampling in multicanonical ensemble.

The stochastic model describing the sampling process in multicanonical ensemble has been derived by considering the sampling process as an overdamped Brownian motion on the free energy surface. The essential dynamics of the multicanonical sampling has been characterized by a Langevin equation in a piecewise multivalleyed free energy landscape, modulated by a temperature-dependent curvature. Based on the stochastic model we showed that the multicanonical weight can be determined by interpolating maximum probability energy points of the canonical samplings at different temperatures.

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Stiff polymer in monomer ensemble.

We employ an ordered monomer ensemble formalism in order to develop techniques to investigate a stiff polymer chain which is confined to a certain region. In particular, we calculate the segment density for a given location and segment orientation distribution within the confining geometry. With this method the role of the stiffness can be examined by means of differential equations, integral equations, or recursive relations for both continuum and lattice models. A suitable choice of lattice model permits an exact analytical solution for the segment location and orientation density for a chain between two parallel plates. For the stiff polymer in a spherical cavity we develop an integral equation formalism which is treated numerically, and in the same spherical geometry, a different model of the polymer displays a solution of a differential equation.

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Equivalence of ensembles in creation-annihilation nonequilibrium models.

We consider a class of stochastic models on a lattice in which particles are created and annihilated with given rates. The class include very distinct models such as the contact process and the stochastic Ising model. We show how to set up stochastic models with the dynamics that conserve the number of particles that are equivalent, in the thermodynamic limit, to the creation-annihilation models. We also obtain a formula that allows the calculation of the rates with respect to the ensemble of constant particle number.

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Radiative scattering properties of an ensemble of variously shaped small particles.

This paper presents a rigorous solution to the scattering of a monochromatic plane wave by an arbitrary configuration of wavelength-sized small particles that can be of different shape, structure, size, and composition. A T-matrix formulation is developed for the calculation of optical cross sections and the asymmetry parameter of such an ensemble of scatterers in both fixed and random orientations. The solution is based on the T matrix T(jl), that is, the inverse of the coefficient matrix of boundary condition equations. A linear system containing T(jl) is derived to efficiently solve the T matrix, which is required in the practical implementation of the solution.

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Detection of mutual phase synchronization in multivariate signals and application to phase ensembles and chaotic data.

This work presents a method for the detection of mutual phase synchronization in nonstationary time series. We show how the application of a cluster algorithm that considers spatiotemporal structures of data follows from the general condition of phase-synchronized data. In view of the topology of phasic data, we reformulate the K-means cluster algorithm on a flat torus and apply a segmentation index derived in an earlier work [A. Hutt and H. Riedel, Physica D 177, 203 (2003)]. This index is extended by means of averaging in order to reflect phase synchronization in ensembles of multivariate time series. The method is illustrated using simulated multivariate phase dynamics and arrays of chaotic systems, in which temporal segments of phase-synchronized states are registered. A comparison with results from an existing bivariate synchronization index reveals major advantages of our method.

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Reaction ensemble Monte Carlo technique and hypernetted chain approximation study of dense hydrogen.

In spite of the simple structure of hydrogen, up to now there is no unified theoretical and experimental description of hydrogen at high pressures. Recent results of Z-pinch experiments show a large deviation from those obtained by laser driven ones. Theoretical investigations including ab initio computer simulations show considerable differences at such extreme conditions from each other and from experimental values. We apply the reaction ensemble Monte Carlo technique on one hand and a combination of the hypernetted chain approximation with the mass action law on the other to study the behavior of dense hydrogen at such conditions. The agreement between both methods for the equation of state and for the Hugoniot curve is excellent. Comparison to other methods and experimental results is also performed.

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Simulations of a small protein in a specifically designed generalized ensemble.

We describe a variant of the generalized-ensemble approach that allows faster simulations for special classes of proteins. We test this technique for an all-atom model of the 36-residue protein HP-36. The dependence of various thermodynamic quantities on small modifications of the solvent representation is explored. Configurations with a root-mean square deviation of less than 4 A to the experimentally determined structure are observed.

Algorithms↗

Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems.

In this paper, we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems (flows or maps) with diagonal diffusive coupling. The condition of chaotic synchronization is determined only by two "parameters of order," i.e., the largest Lyapunov exponent and the coupling coefficient. Our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular, arbitrarily assumed structure of coupling. The main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems. Numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision.

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Optimizing the ensemble for equilibration in broad-histogram Monte Carlo simulations.

We present an adaptive algorithm which optimizes the statistical-mechanical ensemble in a generalized broad-histogram Monte Carlo simulation to maximize the system's rate of round trips in total energy. The scaling of the mean round-trip time from the ground state to the maximum entropy state for this local-update method is found to be O ( [N ln N](2) ) for both the ferromagnetic and the fully frustrated two-dimensional Ising model with N spins. Our algorithm thereby substantially outperforms flat-histogram methods such as the Wang-Landau algorithm.

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Spatially adaptive grand canonical ensemble Monte Carlo simulations.

A spatially adaptive Monte Carlo method is introduced directly from the underlying microscopic mechanisms, which satisfies detailed balance, gives the correct noise, and describes accurately dynamic and equilibrium states for adsorption-desorption (grand canonical ensemble) processes. It enables simulations of large scales while capturing sharp gradients with molecular resolution at significantly reduced computational cost. A posteriori estimates, in the sense used in finite-elements methods, are developed for assessing errors (information loss) in coarse-graining and guiding mesh generation.

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Synchronization and coordination of sequences in two neural ensembles.

There are many types of neural networks involved in the sequential motor behavior of animals. For high species, the control and coordination of the network dynamics is a function of the higher levels of the central nervous system, in particular the cerebellum. However, in many cases, especially for invertebrates, such coordination is the result of direct synaptic connections between small circuits. We show here that even the chaotic sequential activity of small model networks can be coordinated by electrotonic synapses connecting one or several pairs of neurons that belong to two different networks. As an example, we analyzed the coordination and synchronization of the sequential activity of two statocyst model networks of the marine mollusk Clione. The statocysts are gravity sensory organs that play a key role in postural control of the animal and the generation of a complex hunting motor program. Each statocyst network was modeled by a small ensemble of neurons with Lotka-Volterra type dynamics and nonsymmetric inhibitory interactions. We studied how two such networks were synchronized by electrical coupling in the presence of an external signal which lead to winnerless competition among the neurons. We found that as a function of the number and the strength of connections between the two networks, it is possible to coordinate and synchronize the sequences that each network generates with its own chaotic dynamics. In spite of the chaoticity, the coordination of the signals is established through an activation sequence lock for those neurons that are active at a particular instant of time.

Action Potentials↗

Overcoming the slowing down of flat-histogram Monte Carlo simulations: cluster updates and optimized broad-histogram ensembles.

We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform suboptimally in comparison to an unbiased Markovian random walk in energy space. For the d = 1, 2, 3 Ising model, the mean first-passage time tau scales with the number of spins N = L(d) as tau proportional N2L(z). The exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z > 0 for finite d can be overcome by two complementary approaches--cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that tau proportional N2 up to logarithmic corrections for the d = 1, 2 Ising model.

Journal Article↗

Refined evaluation of the level-spacing distribution of symplectic ensembles: moments and implications.

To obtain a more precise value for the variance sigma2 of the joint probability distribution of a symplectic ensemble, we extend previous numerical evaluations of a power series. Our result sigma2 approximately 0.1041 shows that the excellent approximation using the analytically simple Wigner surmise fractionally overestimates this value. This behavior is important in establishing the trend of a generalization of the surmise to describe the terrace-width distribution on vicinal surfaces. We also obtain precise estimates of the skewness and the kurtosis of the exact distribution, as well as the related moments.

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Nonlinear dynamics and self-organization of rotary molecular motor ensembles.

The coupled nonlinear dynamics of rotary motor ensembles and an embedding fluid medium are simulated on the basis of Navier-Stokes equations. The model description considers space-dependent rotation of motors and a spatially inhomogeneous motor distribution as well as spatial fluctuations in molecular properties. The mutual influence of motor rotations and fluid dynamics as well as spatial inhomogeneities of their molecular properties are self-consistently included. Space-time simulations visualize the complex interplay between the fluid dynamics and motor rotation and show that the dynamic coupling to the environment via the velocity field as well as the spatial inhomogeneity and distribution of rotors determines the overall behavior of the motor complex. Depending on the initial molecular distribution, rotation frequency, and fluid dynamics a spatial self-organization or a chaotic behavior may arise.

Computer Simulation↗