On the Bethe ansatz for random directed polymers

We show that the inclusion of the (gapless) center-of-mass motion together with a functional integral representation of the Bethe wave function allows one to predict exactly the critical exponents for random directed polymers in (1+1) dimensions. The corresponding amplitudes are computed; they compare satisfactorily with existing numerical data. Within a replica-symmetric theory, we find that the Green function of the polymer has the form recently proposed by Parisi.     

Nonuniversality and analytical continuation in moments of directed polymers on hierarchical lattices

We prove the moments of the directed polymer partition function GZ, using an exact position space renormalization group scheme on a hierarchical lattice. After sufficient iteration the characteristic functionf(n)=lnGZⁿ of the probability (Z) converges to a stable limitf ^*(n). For smalln the limiting behavior is independent of the initial distribution, while for largen,f ^*(n) is completely determined by it and is thus nonuniversal. There is a smooth crossover between the two regimes for small effective dimensions, and the nonlinear behavior of the small moments can be used to extract information on the universal scaling properties of the distribution. For large effective dimensions there is a sharp transition between the two regimes, and analytical continuation from integer moments ton0 is not possible. Replica arguments can account for most features of the observed results.     

Kinetic theory of long time tails in velocity correlation functions in a moderately dense electron gas

The long time tails of the correlation functions that determine the self-diffusion coefficient and the kinetic parts of the shear viscosity and heat conductivity in a one-component plasma are calculated using a systematic kinetic theory. The results are in agreement with those obtained from the phenomenological mode coupling theory. The formal kinetic theory calculations of previous workers, who obtained incomplete long time tail results, are also discussed.     

Directed polymers in a random environment: Some results on fluctuations

We consider a polymer model on ℤ ₊ ^d where to each edgee is associated a random variable v(e). A polymer configuration is represented by a directed pathr and has a weight exp[-β ∑ _e ∈r ν(e)], withβ=1/T the inverse temperature. We extend some rigorous results that have been obtained for the ground state of this model to finite temperatures. In particular we obtain some upper and lower bounds on sample-to-sample free energy fluctuations, and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations     

Diffusion of directed polymers in a random environment

We consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time. In spatial dimensionsd>2, we establish that the behavior is diffusive with probability one. The diffusion constant is not renormalized by the interaction.     

Effect of polymerization and depolymerization reactions on molecular size distribution in living polymers

In this paper we have derived an exact expression for the molecular size distribution in living polymers for batch polymerization when depropagation is simultaneously operative. This expression is valid at a point of time when inhomogeneity is slightly greater than half that of the final equilibrium distribution. The authors felicitate Prof. D S Kothari on his eightieth birthday and dedicate this paper to him on this occasion.     

Diffusion of directed polymers in a strong random environment

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. It was shown by Imbrie and Spencer that in spatial dimensions three or above the behavior is diffusive if the directed polymer interacts weakly with the environment and if the random environment follows the Bernoulli distribution. Under the same assumption on the random environment as that of Imbrie and Spencer, we establish that in spatial dimensions four or above the behavior is still diffusive even when the directed polymer interacts strongly with the environment. More generally, we can prove that, if the random environment is bounded and if the supremum of the support of the distribution has a positive mass, then there is an integerd ₀ such that in dimensions higher thand ₀ the behavior of the random polymer is always diffusive.     

A remark on diffusion of directed polymers in random environments

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one ifd>2 and <₀, where ₀ is defined in terms of the probability that the symmetric nearest neighbor random walk on thed-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system.     

A statistical physics view of financial fluctuations: Evidence for scaling and universality

The unique scaling behavior of financial time series have attracted the research interest of physicists. Variables such as stock returns, share volume, and number of trades have been found to display distributions that are consistent with a power-law tail. We present an overview of recent research joining practitioners of economic theory and statistical physics to try to understand better some puzzles regarding economic fluctuations. One of these puzzles is how to describe outliers, i.e. phenomena that lie outside of patterns of statistical regularity. We review recent research, which suggests that such outliers may not in fact exist and that the same laws seem to govern outliers as well as day-to-day fluctuations.     

Monomolecular film and absorption behavior of polymers having carboxyl groups

π-A isotherms of monolayers of various copolymers containing carboxyl groups derived from acrylic acid (AA) were measured. The conformation of polymer adsorbed on the water surface and interaction forces between polymer and water surface (considered to be a model for the surface of inorganic powders) owing to AA groups is discussed based on the π-A isotherms.

Polyacrylic esters having a small amount of AA groups formed the expanded type of monolayer. AA groups cause an increase in the interaction force between the polymer and water surface. Polymethyl methacrylate having a small amount of AA groups formed condensed-type monolayers, while with a large amount of AA groups it showed two types of monolayer collapse. Though polystyrene could not spread as a mono-layer, poly (styrene-acrylic acid) [p(St-AA)] copolymer formed a monolayer when the AA content was over 4 mol %. The adsorbance of P(St-AA) on water, calculated from π-A isotherm, is the same as that obtained on α-Fe₂O₃ independently. This means the conformation of the polymer on the solid surface and the interaction force can be discussed by using the π-A isotherm.     

Novel scaling behavior of the Ising model on curved surfaces

We demonstrate the nontrivial scaling behavior of Ising models defined on (i) a donut-shaped surface and (ii) a curved surface with a constant negative curvature. By performing Monte Carlo simulations, we find that the former model has two distinct critical temperatures at which both the specific heat C(T) and magnetic susceptibility χ(T) show sharp peaks. The critical exponents associated with the two critical temperatures are evaluated by the finite-size scaling analysis; the result reveals that the values of these exponents vary depending on the temperature range under consideration. In the case of the latter model, it is found that static and dynamic critical exponents deviate from those of the Ising model on a flat plane; this is a direct consequence of the constant negative curvature of the underlying surface.     

Road traffic: A case study of flow and path-dependency in weighted directed networks

How much can we tell about flows through networks just from their topological properties? Whereas flow distributions of river basins, trees or cardiovascular systems come naturally to mind, more complex topologies are not so immediate, especially if the network is large and heterogeneously directed. Our study is motivated by the question of how the distribution of path-dependent trails in directed networks is correlated to the distribution of network flows. As an example we have studied the path-dependencies in closed trails in four metropolitan areas in England and the USA and computed their global and spatial correlations with measured traffic flows. We have found that the heterogeneous distribution of traffic intensity is mirrored by the distribution of agglomerate path-dependency and that high traffic roads are packed along corridors at short-to-medium trail lengths from the ensemble of nodes.     

Critical behavior of sandpile models with sticky grains

We revisit the question whether the critical behavior of sandpile models with sticky grains is in the directed percolation universality class. Our earlier theoretical arguments in favor, supported by evidence from numerical simulations [P.K. Mohanty, D. Dhar, Phys. Rev. Lett. 89 (2002) 104303], have been disputed by Bonachela et al. [Phys. Rev. E 74 (2006) 050102] for sandpiles with no preferred direction. We discuss possible reasons for the discrepancy. Our new results of longer simulations of the one-dimensional undirected model fully support our earlier conclusions.     

Adaptive approach to global synchronization of directed networks with fast switching topologies

Global synchronization of directed networks with switching topologies is investigated. It is found that if there exists at least one directed spanning tree in the network with the fixed time-average topology and the time-average topology is achieved sufficiently fast, the network will reach global synchronization for appreciate coupling strength. Furthermore, this appreciate coupling strength may be obtained by local adaptive approach. A sufficient condition about the global synchronization is given. Numerical simulations verify the effectiveness of the adaptive strategy.     

Consumer expenditure distribution in India, 1983-2007: Evidence of a long Pareto tail

This work presents a comprehensive study of the evolution of the expenditure distribution in India. The consumption process is theoretically modeled based on certain physical assumptions. The proposed statistical model for the expenditure distribution may follow either a double Pareto distribution or a mixture of log-normal and Pareto distribution. The goodness-of-fit tests with the Indian data, collected from the National Sample Survey Organisation Reports for the years of 1983-2007, validate the proposal of a mixture of log-normal and Pareto distribution. The relative weight of the Pareto tail has a remarkable magnitude of approximately 10%-20% of the population. Moreover, though the Pareto tail is widening over time for the rural sector only, there is no significant change in the overall inequality measurement across the entire period of study.     

The scaling behavior of the molecular parameters of the hyperbranched polymers made by self-condensing vinyl polymerization

The hyperbranched polymers can be made by self-condensing vinyl polymerization without gelation transition. The average molecular weights, as well as the average sizes, can reach infinite values as the reaction is quantitatively completed, and the scaling forms of the molecular parameters should exist. In the paper, based on a recursion formula, the scaling form of the number fraction distribution and the number of the n-mers are given analytically as the conversion of double bonds is near 1. The mean square radius of gyration for very large hyperbranched polymers is calculated explicitly to give a scaling exponent. Finally, a scaling relation associated with the fractal dimension and the polydispersity exponent is given clearly.     

A modified fractional model to describe the viscoelastic behavior of solid amorphous polymers: The effect of physical aging

Amorphous polymers are viscoelastic materials. When they are subjected to dynamical loads, their behavior can be modeled by transform functions of stress and strain in the complex plane. In our work, a model based on the fractional calculus concept is proposed in order to predict the viscoelastic behavior of polymethylmethylacrylate (PMMA) over a wide temperature range between (T_g -190°C) and (T_g +15°C). The extended fractional solid model is shown to be capable of describing experimentally observed dynamic viscoelastic behavior over a wide temperature range, including multiple relaxations, using a limited number of free parameters. Structural recovery of PMMA was studied by dynamic mechanical spectrometry, and its effect on the different parameters is also discussed. Furthermore, from the fractional differential and fractional integral formulations. most of the relevant viscoelastic functions that quantify the degree of molecular mobility of amorphous polymers, like E(t), E?(τ). and H(t), can be derived analytically.     

Estimation of the effective and functional human cortical connectivity with structural equation modeling and directed transfer function applied to high-resolution EEG

Different brain imaging devices are presently available to provide images of the human functional cortical activity, based on hemodynamic, metabolic or electromagnetic measurements. However, static images of brain regions activated during particular tasks do not convey the information of how these regions are interconnected. The concept of brain connectivity plays a central role in the neuroscience, and different definitions of connectivity, functional and effective, have been adopted in literature. While the functional connectivity is defined as the temporal coherence among the activities of different brain areas, the effective connectivity is defined as the simplest brain circuit that would produce the same temporal relationship as observed experimentally among cortical sites. The structural equation modeling (SEM) is the most used method to estimate effective connectivity in neuroscience, and its typical application is on data related to brain hemodynamic behavior tested by functional magnetic resonance imaging (fMRI), whereas the directed transfer function (DTF) method is a frequency-domain approach based on both a multivariate autoregressive (MVAR) modeling of time series and on the concept of Granger causality.

This study presents advanced methods for the estimation of cortical connectivity by applying SEM and DTF on the cortical signals estimated from high-resolution electroencephalography (EEG) recordings, since these signals exhibit a higher spatial resolution than conventional cerebral electromagnetic measures. To estimate correctly the cortical signals, we used a subject's multicompartment head model (scalp, skull, dura mater, cortex) constructed from individual MRI, a distributed source model and a regularized linear inverse source estimates of cortical current density. Before the application of SEM and DTF methodology to the cortical waveforms estimated from high-resolution EEG data, we performed a simulation study, in which different main factors (signal-to-noise ratio, SNR, and simulated cortical activity duration, LENGTH) were systematically manipulated in the generation of test signals, and the errors in the estimated connectivity were evaluated by the analysis of variance (ANOVA). The statistical analysis returned that during simulations, both SEM and DTF estimators were able to correctly estimate the imposed connectivity patterns under reasonable operative conditions, that is, when data exhibit an SNR of at least 3 and a LENGTH of at least 75 s of nonconsecutive EEG recordings at 64 Hz of sampling rate.

Hence, effective and functional connectivity patterns of cortical activity can be effectively estimated under general conditions met in any practical EEG recordings, by combining high-resolution EEG techniques and linear inverse estimation with SEM or DTF methods. We conclude that the estimation of cortical connectivity can be performed not only with hemodynamic measurements, but also with EEG signals treated with advanced computational techniques.