Complex temporal patterns in molecular dynamics:a direct measure of the phase-space exploration by the trajectory at macroscopic time scales

Computer simulated trajectories of bulk water molecules form complex spatiotemporal structures at the picosecond time scale. This intrinsic complexity, which underlies the formation of molecular structures at longer time scales, has been quantified using a measure of statistical complexity. The method estimates the information contained in the molecular trajectory by detecting and quantifying temporal patterns present in the simulated data (velocity time series). Two types of temporal patterns are found. The first, defined by the short-time correlations corresponding to the velocity autocorrelation decay times (â‰0.1â€ps), remains asymptotically stable for time intervals longer than several tens of nanoseconds. The second is caused by previously unknown longer-time correlations (found at longer than the nanoseconds time scales) leading to a value of statistical complexity that slowly increases with time. A direct measure based on the notion of statistical complexity that describes how the trajectory explores the phase space and independent from the particular molecular signal used as the observed time series is introduced. © 2008 The American Physical Society.

Molecular phase space transport in water:non-stationary random walk model

Molecular transport in phase space is crucial for chemical reactions because it defines how pre-reactive molecular configurations are found during the time evolution of the system. Using Molecular Dynamics (MD) simulated atomistic trajectories we test the assumption of the normal diffusion in the phase space for bulk water at ambient conditions by checking the equivalence of the transport to the random walk model. Contrary to common expectations we have found that some statistical features of the transport in the phase space differ from those of the normal diffusion models. This implies a non-random character of the path search process by the reacting complexes in water solutions. Our further numerical experiments show that a significant long period of non-stationarity in the transition probabilities of the segments of molecular trajectories can account for the observed non-uniform filling of the phase space. Surprisingly, the characteristic periods in the model non-stationarity constitute hundreds of nanoseconds, that is much longer time scales compared to typical lifetime of known liquid water molecular structures (several picoseconds).

Computational mechanics of molecular systems:quantifying high-dimensional dynamics by distribution of Poincaré recurrence times

A framework that connects computational mechanics and molecular dynamics has been developed and described. As the key parts of the framework, the problem of symbolising molecular trajectory and the associated interrelation between microscopic phase space variables and macroscopic observables of the molecular system are considered. Following Shalizi and Moore, it is shown that causal states, the constituent parts of the main construct of computational mechanics, the e-machine, define areas of the phase space that are optimal in the sense of transferring information from the micro-variables to the macro-observables. We have demonstrated that, based on the decay of their Poincare´ return times, these areas can be divided into two classes that characterise the separation of the phase space into resonant and chaotic areas. The first class is characterised by predominantly short time returns, typical to quasi-periodic or periodic trajectories. This class includes a countable number of areas corresponding to resonances. The second class includes trajectories with chaotic behaviour characterised by the exponential decay of return times in accordance with the Poincare´ theorem.

Transition from time-continuous systems to discrete mappings

Detailed transport studies in plasmas require the solution of the time evolution of many different initial positions of test particles in the phase space of the systems to be investigated. To reduce this amount of numerical work, one would like to replace the integration of the time-continues system with a mapping.

Quantifying the complexity of chaos in multibasin multidimensional dynamics of molecular systems

The simulated classical dynamics of a small molecule exhibiting self-organizing behavior via a fast transition between two states is analyzed by calculation of the statistical complexity of the system. It is shown that the complexity of molecular descriptors such as atom coordinates and dihedral angles have different values before and after the transition. This provides a new tool to identify metastable states during molecular self-organization. The highly concerted collective motion of the molecule is revealed. Low-dimensional subspaces dynamics is found sensitive to the processes in the whole, high-dimensional phase space of the system. © 2004 Wiley Periodicals, Inc.

Dynamic complexity of chaotic transitions in high-dimensional classical dynamics:Leu-Enkephalin folding

Leu-Enkephalin in explicit water is simulated using classical molecular dynamics. A ß-turn transition is investigated by calculating the topological complexity (in the "computational mechanics" framework [J. P. Crutchfield and K. Young, Phys. Rev. Lett., 63, 105 (1989)]) of the dynamics of both the peptide and the neighbouring water molecules. The complexity of the atomic trajectories of the (relatively short) simulations used in this study reflect the degree of phase space mixing in the system. It is demonstrated that the dynamic complexity of the hydrogen atoms of the peptide and almost all of the hydrogens of the neighbouring waters exhibit a minimum precisely at the moment of the ß-turn transition. This indicates the appearance of simplified periodic patterns in the atomic motion, which could correspond to high-dimensional tori in the phase space. It is hypothesized that this behaviour is the manifestation of the effect described in the approach to molecular transitions by Komatsuzaki and Berry [T. Komatsuzaki and R.S. Berry, Adv. Chem. Phys., 123, 79 (2002)], where a "quasi-regular" dynamics at the transition is suggested. Therefore, for the first time, the less chaotic character of the folding transition in a realistic molecular system is demonstrated. © Springer-Verlag Berlin Heidelberg 2006.

Threshold-induced phase transitions in perceptrons

Error rates of a Boolean perceptron with threshold and either spherical or Ising constraint on the weight vector are calculated for storing patterns from biased input and output distributions derived within a one-step replica symmetry breaking (RSB) treatment. For unbiased output distribution and non-zero stability of the patterns, we find a critical load, α p, above which two solutions to the saddlepoint equations appear; one with higher free energy and zero threshold and a dominant solution with non-zero threshold. We examine this second-order phase transition and the dependence of α p on the required pattern stability, κ, for both one-step RSB and replica symmetry (RS) in the spherical case and for one-step RSB in the Ising case.

Robust control of nonlinear stochastic systems by modelling conditional distributions of control signals

We introduce a novel inversion-based neuro-controller for solving control problems involving uncertain nonlinear systems that could also compensate for multi-valued systems. The approach uses recent developments in neural networks, especially in the context of modelling statistical distributions, which are applied to forward and inverse plant models. Provided that certain conditions are met, an estimate of the intrinsic uncertainty for the outputs of neural networks can be obtained using the statistical properties of networks. More generally, multicomponent distributions can be modelled by the mixture density network. In this work a novel robust inverse control approach is obtained based on importance sampling from these distributions. This importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The performance of the new algorithm is illustrated through simulations with example systems.

Phase transitions and memory effects in the dynamics of Boolean networks

The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between Boolean networks and general Boolean formulas are identified and results common to both system types are highlighted. © 2012 Copyright Taylor and Francis Group, LLC.

The effect of interocular phase difference on perceived contrast

Binocular vision is traditionally treated as two processes: the fusion of similar images, and the interocular suppression of dissimilar images (e.g. binocular rivalry). Recent work has demonstrated that interocular suppression is phase-insensitive, whereas binocular summation occurs only when stimuli are in phase. But how do these processes affect our perception of binocular contrast? We measured perceived contrast using a matching paradigm for a wide range of interocular phase offsets (0–180°) and matching contrasts (2–32%). Our results revealed a complex interaction between contrast and interocular phase. At low contrasts, perceived contrast reduced monotonically with increasing phase offset, by up to a factor of 1.6. At higher contrasts the pattern was non-monotonic: perceived contrast was veridical for in-phase and antiphase conditions, and monocular presentation, but increased a little at intermediate phase angles. These findings challenge a recent model in which contrast perception is phase-invariant. The results were predicted by a binocular contrast gain control model. The model involves monocular gain controls with interocular suppression from positive and negative phase channels, followed by summation across eyes and then across space. Importantly, this model—applied to conditions with vertical disparity—has only a single (zero) disparity channel and embodies both fusion and suppression processes within a single framework.

Phase transitions in wave turbulence

We consider turbulence within the Gross-Pitaevsky model and look into the creation of a coherent condensate via an inverse cascade originating at small scales. The growth of the condensate leads to a spontaneous breakdown of statistical symmetries of overcondensate fluctuations: First, isotropy is broken, then a series of phase transitions marks the changing symmetry from twofold to threefold to fourfold. We describe respective anisotropic flux flows in the k space. At the highest level reached, we observe a short-range positional and long-range orientational order (as in a hexatic phase). In other words, the more one pumps the system, the more ordered the system becomes. The phase transitions happen when the system is pumped by an instability term and does not occur when pumped by a random force. We thus demonstrate nonuniversality of an inverse-cascade turbulence with respect to the nature of small-scale forcing.

The influence of the chemical nature of dispersed phase on stability in oil-in-water emulsions

Some of the problems arising from the inherent instability of emulsions are discussed. Aspects of emulsion stability are described and particular attention is given to the influence of the chemical nature of the dispersed phase on adsorbed film structure and stability, Emulsion stability has been measured by a photomicrographic technique. Electrophoresis, interfacial tension and droplet rest-time data were also obtained. Emulsions were prepared using a range of oils, including aliphatic and aromatic hydrocarbons, dispersed In a solution of sodium dodecyl sulphate. In some cases a small amount of alkane or alkanol was incorporated into the oil phase. In general the findings agree with the classical view that the stability of oil-in-water emulsions is favoured by a closely packed interfacial film and appreciable electric charge on the droplets. The inclusion of non-ionic alcohol leads to enhanced stability, presumably owing to the formation of a "mixed" interfacial film which is more closely packed and probably more coherent than that of the anionic surfactant alone. In some instances differences in stability cannot he accounted for simply by differences in interfacial adsorption or droplet charge. Alternative explanations are discussed and it is postulated that the coarsening of emulsions may occur not only hy coalescence but also through the migration of oil from small droplets to larger ones by molecular diffusion. The viability of using the coalescence rates of droplets at a plane interface as a guide to emulsion stability has been researched. The construction of a suitable apparatus and the development of a standard testing procedure are described. Coalescence-time distributions may be correlated by equations similar to those presented by other workers, or by an analysis based upon the log-normal function. Stability parameters for a range of oils are discussed in terms of differences in film drainage and the natl1re of the interfacial film. Despite some broad correlations there is generally poor agreement between droplet and emulsion stabilities. It is concluded that hydrodynamic factors largely determine droplet stability in the systems studied. Consequently droplet rest-time measurements do not provide a sensible indication of emulsion stability,

The Riesz transform and simultaneous representations of phase, energy and orientation in spatial vision

To represent the local orientation and energy of a 1-D image signal, many models of early visual processing employ bandpass quadrature filters, formed by combining the original signal with its Hilbert transform. However, representations capable of estimating an image signal's 2-D phase have been largely ignored. Here, we consider 2-D phase representations using a method based upon the Riesz transform. For spatial images there exist two Riesz transformed signals and one original signal from which orientation, phase and energy may be represented as a vector in 3-D signal space. We show that these image properties may be represented by a Singular Value Decomposition (SVD) of the higher-order derivatives of the original and the Riesz transformed signals. We further show that the expected responses of even and odd symmetric filters from the Riesz transform may be represented by a single signal autocorrelation function, which is beneficial in simplifying Bayesian computations for spatial orientation. Importantly, the Riesz transform allows one to weight linearly across orientation using both symmetric and asymmetric filters to account for some perceptual phase distortions observed in image signals - notably one's perception of edge structure within plaid patterns whose component gratings are either equal or unequal in contrast. Finally, exploiting the benefits that arise from the Riesz definition of local energy as a scalar quantity, we demonstrate the utility of Riesz signal representations in estimating the spatial orientation of second-order image signals. We conclude that the Riesz transform may be employed as a general tool for 2-D visual pattern recognition by its virtue of representing phase, orientation and energy as orthogonal signal quantities.

In situ studies of phase separation and crystallization directed by Marangoni instabilities during spin-coating

Results of a pioneering study are presented in which for the first time, crystallization, phase separation and Marangoni instabilities occurring during the spin-coating of polymer blends are directly visualized, in real-space and real-time. The results provide exciting new insights into the process of self-assembly, taking place during spin-coating, paving the way for the rational design of processing conditions, to allow desired morphologies to be obtained. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.