abstract

A Brownian particle hopping in a symmetric double-well potential can be statistically confined into either well by the action of two periodic input signals that rock the potential simultaneously. The underlying harmonic mixing mechanism exhibits resonant behavior responsible for asymmetry inversion. Asymmetric confinement through harmonic mixing can be conveniently controlled by tuning the input signal parameters (frequencies, relative phase, and amplitudes).

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To analyze the role of correlations in networks, in particular, assortativity and dissortativity, we introduce two algorithms which respectively produce assortative and dissortative mixing to a desired degree. In both procedures this degree is governed by a single parameter \(p\). Varying this parameter, one can change correlations in networks without modifying their degree distribution to produce new versions ranging from fully random (\(p = 0\)) to totally assortative or dissortative (\(p = 1\)), depending on the algorithm used. We discuss the properties of networks emerging when applying our algorithms to a Barabási–Albert scale-free construction. In spite of having exactly the same degree distribution, different correlated networks exhibit different geometrical and transport properties. Thus, the average path length and clustering coefficient, as well as the shell structure and percolation properties change significantly when modifying correlations.

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Symbolic dynamics, in which the system trajectory is represented as a string of symbols, appears as a convenient method for the analysis of properties of chaotic attractors. In this paper, we show that, using a non-canonical coding scheme based on a moving partition point, we are able to access such properties of the phase space of a dynamical system as the localisation of unstable periodic orbits and of their stable invariant manifolds. Applying different coding schemes enables us to extract different information about the phase space structure from the chaotic trajectory. A judicial choice of the method of symbolic coding allows to obtain information which may be missing in the symbolic dynamics from the generating partition. We present results for the 1-D case taking the logistic map as a numerical example. The extension to higher dimension is also discussed. The theoretical background of the methods used is also given.

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In this paper we introduce a generalization of the De Vylder approximation of the ruin probability. Here the risk process is described in the language of a continuous time random walk. Our idea of approximation is to replace the risk process with the one with gamma claims, matching first four moments. We compare the two approximations studying mixture of exponentials and lognormal claims. We show that the proposed 4-moment gamma approximation works better than the original one.

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The chemical equilibrium state is treated as a fundamental “reference frame” in description of chemical reaction. In a definition of reactive absolute activities for components in chemical reaction the difference of chemical potential and its value in the equilibrium is used. The chemical reaction rate is shown to be proportional to the force \(X_{\rm new}\) defined as the difference of reactive absolute activities of reactants and products. The force \(X_{\rm new}\) is shown to be equivalent to the force following from chemical kinetics equations and compared with the reduced affinity \(X\) as well as with the force of Ross and Mazur \(X_{\rm RM} = 1 - {\rm exp}(-X)\). The force \(X_{\rm new}\) coincides with \(X\) and \(X_{\rm RM}\) near to the chemical equilibrium state. A range of the molar fraction of product, in which a difference between the forces \(X_{\rm new}\) and \(X\) is relatively small, is larger than it would be for the forces \(X_{\rm RM}\) and \(X\). It means that for some chemical reactions the formalism of linear nonequilibrium thermodynamics can be used in wider ranges than usually expected. Particular analysis is presented for simple reactions.

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We consider models for control of epidemics on local, global, small-world and scale-free networks, with only partial information accessible about the status of individuals and their connections. The effectiveness of local (e.g. ring vaccination or culling) vs global (e.g. random vaccination) control measures is evaluated, with the aim of minimising the total cost of an epidemic. The costs include direct costs of treating infected individuals as well as costs of treatment. We first consider a random (global) vaccination strategy designed to stop any potential outbreak. We show that if the costs of the preventive vaccination are ignored, the optimal strategy is to vaccinate the whole population, although most of the resources are wasted on preventing a small number of cases. If the vaccination costs are included, or if a local strategy (within a certain neighbourhood of a symptomatic individual) is chosen, there is an optimum number of treated individuals. Inclusion of non-local contacts (‘small-worlds’ or scale-free networks) increases the levels of preventive (random) vaccination and radius of local treatment necessary for stopping the outbreak at a minimal cost. The number of initial foci also influences our choice of optimal strategy. The size of epidemics and the number of treated individuals increase for outbreaks that are initiated from a larger number of initial foci, but the optimal radius of local control actually decreases. The results are important for designing control strategies based on cost effectiveness.

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Trajectories of on–off events are the output of many single molecule experiments. Usually, one assumes that the underlying mechanism that generates the trajectory can be described by a kinetic scheme, and by analyzing the trajectory aims at deducing this scheme. In a previous work (O. Flomenbom, J. Klafter and A. Szabo, Biophys. J., in press) we showed that when successive events along a trajectory are uncorrelated, all the information in the trajectory is contained in two basic functions, which are the waiting time probability density functions (PDFs) of the on state and off state. The kinetic schemes that lead to such uncorrelated trajectories were termed reducible. The topologies of reducible kinetic schemes were then given. Here, we provide the mathematical steps that were used to find theses topologies.

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In this study, a comprehensive view of a model crystal formation in a complex fluctuating medium is presented. The model incorporates Gaussian curvature effects at the crystal boundary as well as the possibility for super-diffusive motion near the crystal surface. A special emphasis is put on the finite-size effect of the building blocks (macroions, or the aggregates of macroions) constituting the crystal. From it an integrated static–dynamic picture of the crystal formation in terms of mesoscopic nonequilibrium thermodynamics (MNET), and with inclusion of the physically sound effects mentioned, emerges. Its quantitative measure appears to be the overall diffusion function of the formation which contains both finite-size curvature-inducing effects as well as a time-dependent super-diffusive part. A quite qualitative agreement with experiments, mostly those concerning investigations of dynamic growth layer of (poly)crystalline aggregation, exemplified by non-Kossel crystals and biomolecular spherulites, has been achieved.

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Shannon information entropy is a natural measure of probability (de)localization and thus (un)predictability in various procedures of data analysis for model systems. We pay particular attention to links between the Shannon entropy and the related Fisher information notion, which jointly account for the shape and extension of continuous probability distributions. Classical, dynamical and random systems in general give rise to time-dependent probability densities and associated information measures. The induced dynamics of Shannon and Fisher functionals reveals an interplay among various characteristics of the considered diffusion-type systems: information, uncertainty and localization while put against mean energy and its balance.

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The phenomenon of epidemic spreading in a population with a hierarchical structure of interpersonal interactions is described and investigated numerically. The SIS model with temporal immunity to a disease and a time of incubation is used. In our model spatial localization of individuals belonging to different social groups, effectiveness of different interpersonal interactions and the mobility of a contemporary community are taken into account. The structure of interpersonal connections is based on a scale-free network. The influence of the structure of the social network on typical relations characterizing the spreading process, like a range of epidemic and epidemic curves, is discussed. The probability that endemic state occurs is also calculated. Surprisingly it occurs, that less contagious diseases has greater chance to survive. The influence of preventive vaccinations on the spreading process is investigated and critical range of vaccinations that is sufficient for the suppression of an epidemic is calculated. Our results of numerical calculations are compared with the solutions of the master equation for the spreading process, and good agreement is found.

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1D diffusion in glassy polymers with no stresses considered explicitly is studied. A class of different uncorrelated and correlated random walks (RW), described by suitable master equations, is presented. The limiting processes which lead to the set of partial differential equations (PDEs) of parabolic and hyperbolic types generating the “travelling waves” solutions are discussed. Finally, some numerical solutions addressed to diffusion in glassy polymers are discussed.

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Transport of over-damped Brownian particles in periodic double-barrier potentials is studied in the absence and under the influence of a constant tilting force. Depending on the value of the tilt the transport of particles in potentials with two barriers per period has in general character similar to that in simple potentials, exhibiting at the same time in certain parameter regions qualitatively different features. As the most unexpected result it is found that diffusion coefficient can have two maxima. It is also shown that in the wide range of tilting force the transport can be realized through two different Poissonian processes, having at a certain tilt a resonant-like enhancement of the coherence.

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We have studied variability and predictability of population behaviour in a simple model of exponential growth. Population variability is related to uncertainty of prediction for the dynamics conditioned upon the initial state only. We contrasted it with replicate variability, defined in terms of short-term predictability along a single realisation of a stochastic process. We show that for exponential growth, the population variance increases proportionally to the square of the current population size, whereas the replicate variance is a linear function of the population size. Thus, for large population sizes, the relative predictability for a single population is much better than for an ensemble of realisations. This stands in contrast with the behaviour of a simple stochastic process (Ornstein–Uhlenbeck process), where the population and the replicate variances have similar behaviour. The results have profound consequences for parameter estimation and prediction for many stochastic population models based on the exponential formula.

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We study a transport in composite system where the subdiffusive solvent (as for example gel) is separated by a thin membrane from the region where normal diffusion occurs. The solutions of the diffusion equation with fractional derivative are found in the system of interest. We also discuss the dependence of mean square displacement \(\sigma ^{2}\) on time in long-time approximation.

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Experimental evidence shows significant effect of ionic concentration of aqueous buffers on the membrane stability, for example during electroporation experiments. Also, the lateral diffusion coefficient is sensitive to ionic strength. We study the effect of ionic strength on lipid membrane by modified Pink model, applying Monte Carlo simulations for a lattice of 50 lipid molecules. The study is provided for the model dipalmitoyl-phosphatidylcholine (DPPC) membrane in the gel phase at the fixed value of dielectric constant. The study in the range of 10–3000 mM, shows a decrease of repulsive interactions between polar heads, accompanied with an increase of attracting van der Waals interactions between acyl chains. Additionally, the chains assume more stable conformation with lower conformational energy. Simulations show rising number of standing polar heads, which may indicate better accessibility of ions from the solution to the polar part of the molecules and their consequent binding.

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New experimental data are presented on the gel electrophoresis of DNA. Experiment was made for molecules of length 173 kbp, in 1 percent agarose gel, in TAE \(1 \times \) buffer and the field intensity between 5 and 9 V/cm. The results are compared with our computer simulations, performed within the repton model of Duke and Rubinstein. The ranges of field and molecule length are determined, where the geometration effect appears. We investigate also the field dependence of the velocity and the diffusion coefficient at the border of the geometration regime.

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Each biological molecular machine can be effectively considered a common chemochemical enzymatic machine occurring, however, in multitude rather than a few conformational substates distinguished by the conventional kinetics. The cases of the proton pump cytochrome bc\(_1\) and the actomyosin motor are considered in some detail. In the steady state, a slow character of conformational transition dynamics causes the necessity of replacing conventional reaction rate constants by more sophisticated quantities, the mean first-passage times between some distinguished conformational substates of the machine. The most important results obtained for the flux-force dependence of the actomyosin motor are noted.

abstract

The Smoluchowski approach to the kinetics of fluorescence quenching reaction in liquids is tested by comparing the results of molecular dynamics simulations for spherical molecules with the Smoluchowski–Collins–Kimball model and the Step Function Nonradiative Lifetime model. The reaction cross-sections used in the simulations are exactly the same as assumed in the models and the quencher concentration is very low. The discrepancies between the simulations and the models give a general indication on the scale of errors of the Smoluchowski approach. A large number of particles used (typically \(N = 681472\)) allow us to obtain quantitative results. The simulations show a decisive influence of the distribution function of the reagents, \(g_{\rm AB}(r)\), on the accuracy of results. If the liquid structure is ignored \((g_{\rm AB}(r)\equiv 1)\) the discrepancies between the model and the simulations are large especially for a very short times for which the models fail to match simulations even qualitatively. An inclusion of the distribution function significantly improves the description of the quenching process. For short time stages of the quenching the model excellently agrees with the simulations, if the characteristic reaction time is long. If it is very short (the SCK model), significant discrepancies appear due to ballistic motion of the reactants but the quantitative agreement is still good. For a long time the model that takes into account the liquid structure is typically burdened with a few times smaller errors than the model that assumes \(g_{\rm AB}(r)\equiv 1\).

abstract

Our objective here is the study of the noise-assisted generation of magnetic flux in a collection of identical mesoscopic cylinders which are coupled via mutual inductances. With thermal (Johnson–Nyquist)-fluctuations acting at finite temperature, the system can be modeled in terms of a set of Langevin equations with a corresponding Fokker–Planck equation. In the limit of infinitely many constituents, the steady-state of the system is determined by a mean-field-like, nonlinear Fokker–Planck equation. The rich complexity of the generated average flux through each cylinder and its characteristic fluctuations are investigated as a function of various parameters such as the temperature, the coupling strength and an externally applied, uniform magnetic field.

abstract

Effects of network topology are studied in a system of cellular automata driven by a totalistic rule. In particular, propagation of a signal is considered in the directed network obtained from a flat (square) lattice by adding directed connections. The model is motivated by features found in human neural system. Cooperation between local dynamics and network organization results in fast stabilization of the system. Simple model of neural pyramidal cell is proposed to stabilize the automata in the oscillating firing patterns form.

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In the classical theory of diffusion limited growth, it is assumed that the concentration field of solution is described by the standard diffusion equation. It means that particles of the solution undergo a random walk described by the Wiener process. In turn, it means that the velocity of particles is a stochastic process being Gaussian white noise. In consequence, the velocity–velocity correlation function is the Dirac \(\delta \)-function and velocity correlation time is zero. In many cases such modeling is insufficient and one should consider models in which velocity is correlated in space and/or time. The question is whether correlations of velocity can change the kinetics of growth, in particular, whether the long-time asymptotics of the growth kinetics displays the power-law time dependence with the classical exponent \(1/2\). How to model such processes is a subject of this paper.

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In a two-dimensional (2D) equilibrium Lennard–Jones (LJ) liquid, coarse-grained time-averaged spatial distribution of local solid-like structures is studied in order to estimate the hypothetical local-structure-based long time scale. Standard NVE molecular dynamics simulation method is used. Time-averaged distributions indicate a structural slow mode with a characteristic time-scale at least two orders of magnitude larger than the local oscillation period.

abstract

We use the cellular-automaton Duke–Rubinstein model to simulate gel electrophoresis of DNA in periodically changing electric field. The field is dichotomic and its time average is zero. We observe non-vanishing current of molecules, what is known as the ratchet effect. We calculate the drift velocity and the diffusion coefficient for large field amplitude, where nonlinear effects can be observed. The results indicate that tuning the amplitude and frequency of the applied field for a given range of the molecule length can improve the resolving power of the separation of DNA.

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Starting from the Volterra model of dissociated dislocations, a dislocation dissociated into two Shockley partials under the action of the periodic Peierls potential and a general external stress is modeled as a pair of coupled Brownian particles in a washboard potential. It is found that mobility shows a sensitive dependence on the parameters of the interaction between partials. In particular, a resonant-like behavior of the average velocity, for equilibrium separation distances close to half-integer multiples of the period of the Peierls potential, is observed.

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We study the motion of active Brownian particles in 2d-external potentials. We give the stationary probability distribution in the four-dimensional phase space in several representations and show that it is maximized above the deterministic integrals of motion.

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We have examined a topology of 21 public transport networks in Poland. Our data exhibit several universal features in considered systems when they are analyzed from the point of view of evolving networks. Depending on the assumed definition of a network topology the degree distribution can follow a power law \(p(k)\!\sim \!k^{-\gamma }\) or can be described by an exponential function \(p(k)\!\sim \!\exp (-\alpha k)\). In the first case one observes that mean distances between two nodes are a linear function of logarithms of their degrees product.

abstract

The simple cubic lattice model of polymer chains was used to study the properties of adsorbed macromolecules with different internal architectures: linear chains and star-branched chains with 3 arms of equal lengths. The polymer chains were modeled with the excluded volume interactions only, i.e. in good solvent conditions. The chains were placed on an impenetrable surface and a contact potential between polymer segments and this surface was assumed. The strength of this potential was chosen to emulate the conditions of a weak adsorption regime. The Metropolis-like sampling Monte Carlo algorithm was used to determine the properties of the adsorbed polymer film. The size and the internal structure of adsorbed chains were described. The size, distribution and lifetimes of structural elements such as tails, loops and trains were also determined. The differences between the structure of films consisting of star-branched and linear chains were described and discussed.

abstract

In this work we studied a simple model of a copolymer (polypeptide) chain in a confined space. The model chain was restricted to a flexible [310] lattice. It was represented as a sequence of united atoms located at the positions of alpha carbons. The force field introduced into the model consisted of the long-range contact potential between amino acid residues and a local helical potential. The chain was built of hydrophilic and hydrophobic segments. The properties of such chains were determined by means of the Monte Carlo simulations using a Metropolis-like algorithm. During the simulations we observed and tracked the translocation of the chain during its passage through a hole in an impenetrable wall. The influence of the length of the chain and the structure of the polymer film on the translocation process were investigated. The dynamic properties of the system such as the translocation time were also studied and discussed.

abstract

Chaotic time series obtained from simple dynamical systems (the tent map and the logistic map) are analyzed by means of Detrended Fluctuation Analysis (DFA) — a widely used method for quantifying long-range correlations in time series obtained from complex systems. The first conclusion is that time series obtained from stochastic (noise-driven) and deterministic systems may be indistinguishable using the DFA method. We introduce the adaptive DFA exponent and find that it is related to the structure of the periodic orbit. We show that persistence detected in deterministic series by DFA has a different interpretation than that used in the context of stochastic series analysis. For chaotic time series, we find that only a large level of dynamic additive noise can alter the short-range DFA exponent. Finally, a relation between the DFA exponents and the control parameter of the map is studied. The short-range DFA exponent is sensitive to different kinds of nonlinear transitions — we show that the exponent decreases with the merging of chaotic bands and increases as the natural measure becomes more symmetric. If periodic windows occur in the bifurcation diagram, they can be also detected by DFA as an abrupt decrease of the short-range exponent to a value close to 0. An interior crisis occurs at the end of each periodic window — as a result, the DFA exponent increases as a function of the control parameter until the next band-merging point. As the periodic windows are dense in the bifurcation diagram, the relation of the DFA exponent on the control parameter is more complex for this case.

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We show how the Cole–Davidson relaxation response, characteristic of alcoholic systems, can be derived within the framework of the continuous-time random walk (CTRW). Using the random-variable formalism, we indicate that the high-frequency power law of dielectric spectra is determined by the heavy-tailed distribution of quantities that provide the spatio-temporal coupling in the random-walk process. As an illustration, we present the dielectric permittivity spectra of several butanediol isomers.

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A projection of a Markov process onto the dynamics of its metastable states is performed by means of conveniently defined site localizing functions. The method is illustrated by a simple model with time dependent transition rates. In this particular case an alternative method is available. The results of both methods are compared and found to agree with each other.

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The rigidity spectra for two consecutive Forbush effects (October– November 2003) of the galactic cosmic ray intensity estimated by use of the experimental data from neutron monitors are significantly different. The rigidity spectrum gradually hardens in the course of the first Forbush effect (October 22–28). On the contrary, it is very hard at the beginning of the second Forbush effect (October 29 to November 10). The rigidity spectrum progressively softens during the recovery phase of the galactic cosmic ray intensity. It is assumed that the peculiarities of the rigidity spectrum of the Forbush effects are related to the enhancement of the power within the energy range of the interplanetary magnetic field turbulence during the major phase of the Forbush effects. The additional large scale irregularities of the interplanetary magnetic field should be created due to the interaction of the extending high speed disturbances with the background solar wind. This assumption is confirmed by the analysis of the interplanetary magnetic field data. During the major phase of the Forbush effects the power spectral densities of the components of the interplanetary magnetic field are significantly larger than before and after the Forbush effects.

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We show how to modify the random-walk scenario underlying the classical, exponential relaxation response in order to derive the empirical Havriliak–Negami function, commonly used to fit the dielectric permittivity of complex-material data. The turnover from the exponential Debye to the power-law Havriliak–Negami relaxation response is associated with a new type of a coupled memory continuous-time random walk (CTRW) driving a fractional dynamics.

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Effects of quantum statistics are clearly seen in the final states of high-energy multiparticle production processes. These effects are being widely used to obtain information about the regions where the final state hadrons are produced. Here we briefly present and discuss the assumptions underlying most of these analyses.

abstract

We investigate nonlinear instabilities in human heart rate variability. We focus on phenomena with characteristic, easily recognizable features which are well known in physics. In the past we were able to show two groups of evidence. The first was an ever expanding roster of such cases of heart rate variability pathology which exhibit type I intermittency. This phenomenon occurs in those dynamical systems which have come close to a saddle-node bifurcation. The second were observations of homoclinic orbits and the gluing bifurcation in measured heart rate variability. We present here two cases of 24-hour recordings of human heart rate which exhibit special, regular patterns. We show that period-1, period-2 orbits and homoclinic orbits may be found in return maps formed using this data. Using a pair of coupled modified van der Pol–Duffing oscillators, we are able to model the behavior of the sino-atrial node and of the atrio-ventricular node (elements of the conduction system of the heart) in such a way as to obtain orbits similar to those measured during the sino-atrial block in a human.