### Regular Series

#### XXIV Marian Smoluchowski Symposium on Statistical Physics Insights into Stochastic Nonequilibrium

Szilard-machine-like Features in a Processive Motor Protein

Acta Phys. Pol. B 43, 889 (2012)

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abstract

The motor protein kinesin literally walks on two legs along the biopolymer microtubule as it hydrolyzes ATP for its fuel supply. The fraction of accidental backsteps that kinesin takes appears to be about seven orders of magnitude larger than what one would expect given the amount of free energy that ATP hydrolysis makes available. This is puzzling, as more than a billion years of natural selection should have optimized the motor protein for its speed and efficiency. With an imagined device, Szilard has shown that the dissipation of information can drive motion. A higher backstepping probability creates more randomness in the walk and, consequently, leads to production of more entropy. If the product state of a transition has a higher entropy, then the free energy of that product state is lower. With the free energy that is made available by the production of “backstepping entropy”, the catalytic cycle of the kinesin can be speeded up. We show quantitatively how the actually measured backstepping rate represents an optimum at which maximal net forward speed is achieved. We, furthermore, show how this thermodynamic mechanism can realistically operate on a biomolecular level. The results suggest that kinesin uses backstepping as a source of energy and that natural selection has manipulated the backstepping rate to optimize kinesin’s speed.

Pattern–Non-pattern Transition for a Nonlocal Population Dynamics

Acta Phys. Pol. B 43, 909 (2012)

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abstract

In this article, we study pattern formation for one-species population in nonlocal domains. The nonlocal growth and competition terms are defined from the parameters $\alpha$ and $\beta$ ranging in length $L$. In this space $(\alpha ,\beta )$ we have analyzed a coexistence curve $\alpha ^{*}=\alpha ^{*}(\beta )$ which delimits domains for the existence (or not) of pattern formation in population dynamics systems. Pattern–no-pattern transition emerges from this model when nonlocal interaction among the individuals are present.

Geometric Phase of Open Two-level Systems

Acta Phys. Pol. B 43, 921 (2012)

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abstract

Geometric phase of open quantum systems is reviewed. An emphasis is given on specific features of the geometric phase which can serve as an indicator of type and strength of interaction between two-level system (qubit) and its bosonic environment. We study three examples: (i) a single qubit dephasingly coupled to the environment, (ii) a qubit being a part of quantum register, and (iii) a neutrino interacting with matter and environment.

Transient Dynamics of Verhulst Model with Fluctuating Saturation Parameter

Acta Phys. Pol. B 43, 935 (2012)

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abstract

The stochastic Verhulst equation for the population density with fluctuating volume of resources is considered. Using the exact solution of this equation, the conditional probability density function is calculated for the excitation in the form of Lévy white noise with one-sided stable distribution. The phenomenon of transient bimodality and non-monotonic relaxation of mean population density for the white noise with Lévy–Smirnov stable distribution are found. An exact expression for the transitional time from bimodality to unimodality is obtained. It is interesting that for such a case the correlation function of population density in a steady state has a simple exponential form, and the correlation time does not depend on noise parameters.

Fuzzy Analysis of the Cancer Risk Factor

Acta Phys. Pol. B 43, 947 (2012)

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abstract

A system which allows to predict type of cancer on the basis of the largest risk factors for particular patient, was created using fuzzy set theory. Lung, colon, breast, colorectal, stomach, cervical and prostate cancer were considered. The Mamdani model, implemented in the Fuzzy Logic Toolbox in Matlab, was used for data analysis. As inputs to the system genetic, biological (race, age, sex) and behavioral (overweight, alcohol consumption, tobacco smoke) risk factors were taken. The output was “the kind of cancer”. Obtained results show that fuzzy logic can be an effective tool in dealing with this kind of medical problem.

Modeling of Short Term Interest Rate Based on Tempered Fractional Langevin Equation

Acta Phys. Pol. B 43, 961 (2012)

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abstract

We develop a new class of continuous-time models based on the solutions of tempered fractional Langevin equations for Ornstein–Uhlenbeck process driven by Lévy noise. We present methods of simulation of sample paths of such processes. We show how to use such models in modeling short term interest rate. We develop tempered Vasiček interest rate model by finding explicit solutions of tempered fractional Langevin equations.

Dynamics of Confined Lévy Flights in Terms of (Lévy) Semigroups

Acta Phys. Pol. B 43, 977 (2012)

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abstract

The master equation for a probability density function (PDF) driven by Lévy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic Lévy-stable semigroups, for all values of the stability index $\mu \in (0,2)$. That is known to resolve the problem about an invariant PDF for confined Lévy flights. Jeopardies of the procedure are discussed, with a focus on: (i) when an invariant PDF actually is an asymptotic one, (ii) subtleties of the PDF $\mu$-dependence in the vicinity and sharply at the boundaries $0$ and $2$ of the stability interval, where jump-type scenarios cease to be valid.

Anomalous Diffusion Models: Different Types of Subordinator Distribution

Acta Phys. Pol. B 43, 1001 (2012)

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abstract

Subordinated processes play an important role in modeling anomalous diffusion-type behavior. In such models the observed constant time periods are described by the subordinator distribution. Therefore, on the basis of the observed time series, it is possible to conclude on the main properties of the subordinator. In this paper, we analyze the anomalous diffusion models with three types of subordinator distribution: $\alpha$-stable, tempered stable and gamma. We present similarities and differences between the analyzed processes and point at their main properties (like the behavior of moments or the mean square displacement).

Poling in Two Dimensions by a Purely Octupolar Electrostatic Potential: Homogeneity of the Ground State

Acta Phys. Pol. B 43, 1017 (2012)

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abstract

We investigate an arbitrary set of point charges in two dimensions, which models a rigid molecule subject to a purely octupolar electric field poling potential imposed by a system of symmetry-adapted cylindrical electrodes. We formulate the conditions which guarantee that the minimum of electrostatic potential energy of the molecule is homogeneous, that is independent of its localization inside the poling cell. These conditions state that the molecule must be charge neutral, apolar and with an isotropic charge quadrupole moment. This result constitutes a first step towards a comprehensive classification scheme for the ground states ($T=0$) of a set of multipolar molecules poled by electric potentials of different multipolar symmetries.

Random Local Field Method for the Description of Physical Properties of Disordered Solids

Acta Phys. Pol. B 43, 1027 (2012)

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abstract

We suggest an approach to describe the physical properties of disordered dielectric and/or magnetic systems. These systems are characterized by randomly positioned and oriented spins (dipoles) in a host crystal lattice. The ensemble of these spins or dipoles create the random magnetic or electric fields in a host lattice. Their distribution function, defined as an average (over spatial and orientational fluctuations) of Dirac delta contributions of each spin (dipole), enables us to obtain the self-consistent equations for order parameters like average magnetization (polarization) $\langle S\rangle$, and/or general quantities like $\langle S^n\rangle$. We calculate explicitly the above distribution functions for different types of interactions and show that, in general, they are not Gaussian. Our theory delivers pretty good description of experiments in disordered ferroelectrics, multiferroics, magnets and diluted magnetic semiconductors.

The Nonextensive Entropy Approach Versus the Stochastic in Describing Subdiffusion

Acta Phys. Pol. B 43, 1043 (2012)

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abstract

We have proposed a new stochastic interpretation of the sudiffusion described by the Sharma–Mittal entropy formalism which generates a non-linear subdiffusion equation with natural order derivatives. We have shown that the solution to the diffusion equation generated by Gauss entropy (which is the particular case of Sharma–Mittal entropy) is the same as the solution of the Fokker–Planck (FP) equation generated by the Langevin generalised equation, where the ‘long memory effect’ is taken into account. The external noise which pertubates the subdiffusion coefficient (occurring in the solution of FP equation) according to the formula $D_\alpha \rightarrow D_\alpha /u$, where $u$ is a random variable described by the Gamma distribution, provides us with solutions of equations obtained from Sharma–Mittal entropy. We have also shown that the parameters $q$ and $r$ occurring in Sharma–Mittal entropy are controlled by the parameters $\alpha$ and $\langle u\rangle$, respectively.

Cyclic Solutions in Chaos and the Sharkowskii Theorem

Acta Phys. Pol. B 43, 1053 (2012)

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abstract

The fixed points of the logistic map at full chaos are the roots of a special class of polynomials. These polynomials are solvable by a method of multiple angles. The solutions are expressible in cyclic form. By using the theorem due to Sharkovskii we show that the fixed point spectrum has a finite measure. We argue that chaos in 1d, defined through a finite spectral measure, is superior to any phenomenological definitions of chaos such as the Lyapunov exponent.

The Perturbation Method to Solve Subdiffusion-reaction Equations

Acta Phys. Pol. B 43, 1065 (2012)

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abstract

We use the perturbation method to approximately solve subdiffusion-reaction equations. Within this method we obtain the solutions of the zeroth and the first order. The comparison our analytical solutions with the numerical results shown that the perturbation method can be useful to find approximate solutions of nonlinear subdiffusion-reaction equations.

Understanding Operating Principles and Processivity of Molecular Motors

Acta Phys. Pol. B 43, 1073 (2012)

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abstract

Motor proteins, sometimes referred to as mechanoenzymes, are a group of proteins that maintain a large part of intracellular motion. Being enzymes, they undergo chemical reactions leading to energy conversion and changes of their conformation. Being mechanodevices, they use the chemical energy to perform mechanical work, leading to the phenomena of motion. Over the past 20 years a series of novel experiments (e.g. single molecule observations) has been performed to gain the deeper knowledge about chemical states of molecular motors as well as their dynamics in the presence or absence of an external force. At the same time, many theoretical models have been proposed, offering various insights into the nano-world dynamics. They can be divided into three main categories: mechanochemical models, ratchet models and molecular dynamics simulations. We demonstrate that by combining those complementary approaches a deeper understanding of the dynamics and chemistry of the motor proteins can be achieved. As a working example, we choose kinesin — a motor protein responsible for directed transport of organelles and vesicles along microtubule tracts.

Anomalous Dynamics of Black–Scholes Model Time-changed by Inverse Subordinators

Acta Phys. Pol. B 43, 1093 (2012)

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abstract

In this paper we consider a generalization of one of the earliest models of an asset price, namely the Black–Scholes model, which captures the subdiffusive nature of an asset price dynamics. We introduce the geometric Brownian motion time-changed by infinitely divisible inverse subordinators, to reflect underlying anomalous diffusion mechanism. In the proposed model the waiting times (periods when the asset price stays motionless) are modeled by general class of infinitely divisible distributions. We find the corresponding Fractional Fokker–Planck equation governing the probability density function of the introduced process. We prove that considered model is arbitrage-free, construct corresponding martingale measure and show that the model is incomplete. We also find formulas for values of European call and put option prices in subdiffusive Black–Scholes model and show how one can approximate them based on Monte Carlo methods. We present some Monte Carlo simulations for the particular case of tempered $\alpha$-stable distribution of waiting times. We compare obtained results with the classical and subdiffusive $\alpha$-stable Black–Scholes prices.

Scaling Limits of Overshooting Lévy Walks

Acta Phys. Pol. B 43, 1111 (2012)

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abstract

In this paper, we obtain the scaling limits of one-dimensional overshooting Lévy walks. We also find the limiting processes for extensions of Lévy walks, in which the waiting times and jumps are related by power-law, exponential and logarithmic dependence. We find that limiting processes of overshooting Lévy walk are characterized by infinite mean-square-displacement. It also occurs that introducing different dependence between waiting times and jumps of Lévy walks results in subdiffusive properties.

Polymer Shape Dynamics Induced by Spatially Correlated Noise

Acta Phys. Pol. B 43, 1133 (2012)

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abstract

In this paper, we examine the influence of a spatially correlated noise on a 2D polymer-like particle. The molecule is modeled with harmonic potential for bonds and angular interactions and a global Lennard–Jones potential. We present a method for generating a spatially correlated noise if the time and spatial terms in the correlation function factorize. The dynamics of polymer’s shape transformation process is investigated by means of Fourier analysis. An increase in correlation length results in the environmentally induced stiffening of the chain.

Maximal Entropy Random Walk: Solvable Cases of Dynamics

Acta Phys. Pol. B 43, 1143 (2012)

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abstract

We focus on the study of dynamics of two kinds of random walk: generic random walk (GRW) and maximal entropy random walk (MERW) on two model networks: Cayley trees and ladder graphs. The stationary probability distribution for MERW is given by the squared components of the eigenvector associated with the largest eigenvalue $\lambda _0$ of the adjacency matrix of a graph, while the dynamics of the probability distribution approaching to the stationary state depends on the second largest eigenvalue $\lambda _1$. Firstly, we give analytic solutions for Cayley trees with arbitrary branching number, root degree, and number of generations. We determine three regimes of a tree structure corresponding to strongly, critically, and weakly branched roots. Each of them results in different statics and dynamics of MERW. We show how the relaxation times, generically shorter for MERW than for GRW, scale with the graph size. Secondly, we give numerical results for ladder graphs with symmetric defects. MERW shows a clear exponential growth of the relaxation time with the size of defective regions, which indicates trapping of a particle within highly entropic intact region and its escaping that resembles quantum tunneling through a potential barrier. GRW shows standard diffusive dependence irrespective of the defects.

Anomalous Diffusion — The Thinning Property of Fractional Brownian Motion

Acta Phys. Pol. B 43, 1157 (2012)

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abstract

We show that thinning of increments of the fractional Brownian motion with Hurst exponent $H\neq 1/2$ breaks its $H$-self-similarity property. As a result, we obtain a new Gaussian process with stationary increments which is not the fractional Brownian motion for any $H.$ Moreover, in the subdiffusion case ($H\lt 1/2$), the new process statistically resembles the classical Brownian motion ($H=1/2$). To this end, we study analytically the second moment of such processes. Finally, Monte Carlo simulations show that the $H$ estimator obtained by mean square displacement is close to the Brownian motion case with $H=1/2.$ These results show that stationary data describing anomalous diffusion phenomenon can lead to different statistical conclusions for different resolution of measurement. Therefore, one should be very careful in statistical inference, especially in strong subdiffusion regimes ($H\approx 0$).

full authors' list

B. Spagnolo, P. Caldara, A. La Cognata, G. Augello, D. Valenti, A. Fiasconaro, A.A. Dubkov, G. Falci

Relaxation Phenomena in Classical and Quantum Systems

Acta Phys. Pol. B 43, 1169 (2012)

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abstract

Relaxation phenomena in three different classical and quantum systems are investigated. First, the role of multiplicative and additive noise in a classical metastable system is analyzed. The mean lifetime of the metastable state shows a nonmonotonic behavior with a maximum as a function of both the additive and multiplicative noise intensities. In the second system, the simultaneous action of thermal and non-Gaussian noise on the dynamics of an overdamped point Josephson junction is studied. The effect of a Lévy noise generated by a Cauchy–Lorentz distribution on the mean lifetime of the superconductive metastable state, in the presence of a periodic driving, is investigated. We find resonant activation and noise enhanced stability in the presence of Lévy noise. Finally, the time evolution of a quantum particle moving in a metastable potential and interacting with a thermal reservoir is analyzed. Within the Caldeira–Legget model and the Feynman–Vernon functional approach, we obtain the time evolution of the population distributions in the position eigenstates of the particle, for different values of the thermal bath coupling strength.

Version corrected according to Erratum Acta Phys. Pol. B 47, 1179 (2016)

Effect of a Fluctuating Electric Field on Electron Spin Dephasing Time in III–V Semiconductors

Acta Phys. Pol. B 43, 1191 (2012)

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abstract

We investigate the electron spin dephasing in low $n$-doped GaAs semiconductor bulks driven by a correlated fluctuating electric field. The electron dynamics is simulated by a Monte Carlo procedure which keeps into account all the possible scattering phenomena of the hot electrons in the medium and includes the evolution of spin polarization. Spin relaxation times are computed through the D’yakonov–Perel process, which is the only relevant relaxation mechanism in zinc-blende semiconductors. The decay of initial spin polarization of conduction electrons is calculated for different values of field strength, noise intensity and noise correlation time. For values of noise correlation time comparable to the spin lifetime of the system, we find that spin relaxation times are significantly affected by the external noise. The effect increases with the noise amplitude. Moreover, for each value of the noise amplitude, a nonmonotonic behaviour of spin relaxation time as a function of the noise correlation time is found.

Control of Transport Characteristics in Two Coupled Josephson Junctions

Acta Phys. Pol. B 43, 1203 (2012)

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abstract

We report on a theoretical study of transport properties of two coupled Josephson junctions and compare two scenarios for controlling the current-voltage characteristics when the system is driven by an external biased DC current and unbiased AC current consisting of one harmonic. In the first scenario, only one junction is subjected to both DC and AC currents. In the second scenario the signal is split — one junction is subjected to the DC current while the other is subjected to the AC current. We study DC voltages across both junctions and find diversity of anomalous transport regimes for the first and second driving scenarios.

Revised Approach to Statistical Analysis of Ionic Current Fluctuations

Acta Phys. Pol. B 43, 1215 (2012)

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abstract

We study properties of ion current through a high conductance locust potassium channel. Applying the $p$-variation test to the current signal we exclude the continuous time random walk (CTRW) as an underlying mechanism of the current fluctuations. Instead, we show that a fractional Brownian motion (FBM) should be considered as the most suitable stochastic model of the ionic channel action. Using the sample mean square displacement (sample MSD) test we bring to light the superdiffusive, persistent properties of the studied current. To illustrate the obtained results, we propose a simple hydrodynamic approximation of the effective charge transport.

full authors' list

D. Valenti, G. Denaro, A. La Cognata, B. Spagnolo, A. Bonanno, G. Basilone, S. Mazzola, S. Zgozi, S. Aronica

Picophytoplankton Dynamics in Noisy Marine Environment

Acta Phys. Pol. B 43, 1227 (2012)

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abstract

We present a stochastic reaction-diffusion-taxis model to describe the picophytoplankton dynamics along a water column. The model, which is valid for poorly mixed waters, typical of the Mediterranean Sea, considers intraspecific competition of picophytoplankton for light and nutrients. Random fluctuations of environmental variables are taken into account by adding a source of multiplicative noise to the diffusion equation for the picophytoplankton biomass concentration, whose distribution along the water column shows a maximum at a certain depth. After converting our results into chlorophyll a concentrations, we compare theoretical distributions, obtained for different noise intensities, with the experimental chlorophyll a distribution sampled in a site of the Strait of Sicily. Specifically, we find that position and height of the chlorophyll a peak concentration obtained from the model are in a very good agreement with field observations. Finally, we consider the effects of seasonal variations on phytoplankton dynamics by adding an oscillating term in the equation for the light intensity.

How to Identify the Proper Model?

Acta Phys. Pol. B 43, 1241 (2012)

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abstract

One of the important steps towards constructing an appropriate mathematical model for the real-life data is to determine the structure of dependence. A conventional way of gaining information concerning the dependence structure (in the second-order case) of a given set of observations is estimating the autocovariance or the autocorrelation function (ACF) that can provide useful guidance in the choice of satisfactory model or family of models. As in some cases, calculations of ACF for the real-life data may turn out to be insufficient to solve the model selection problem, we propose to consider the autocorrelation function of the squared series as well. Using this approach, in this paper we investigate the dependence structure for several cases of time series models. In order to illustrate theoretical results, we calibrate one of the examined process to real data set that presents CO$_2$ concentration in the indoor air.

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