Regular Series


Vol. 44 (2013), No. 5, pp. 795 – 1233

XXV Marian Smoluchowski Symposium on Statistical Physics Fluctuation Relations in Nonequilibrium Regime

Kraków, Poland; September 10–13, 2012

Optimizing Non-ergodic Feedback Engines

abstract

Maxwell’s demon is a special case of a feedback controlled system, where the information gathered by measurement is utilized by driving a system along a thermodynamic process that depends on the measurement outcome. The demon illustrates that with feedback one can design an engine that performs work by extracting energy from a single thermal bath. Besides the fundamental questions posed by the demon — the probabilistic nature of the Second Law, the relationship between entropy and information, etc. — there are other practical problems related to feedback engines. One of those is the design of optimal engines, protocols that extract the maximum amount of energy given some amount of information. A refinement of the second law to feedback systems establishes a bound to the extracted energy, a bound that is met by optimal feedback engines. It is also known that optimal engines are characterized by time reversibility. As a consequence, the optimal protocol given a measurement is the one that, run in reverse, prepares the system in the post-measurement state (preparation prescription). In this paper, we review these results and analyze some specific features of the preparation prescription when applied to non-ergodic systems.


Time-reversal Symmetry Relations for Fluctuating Currents in Nonequilibrium Systems

abstract

Fluctuation relations for currents are established in several classes of systems. For the effusion of noninteracting particles through a small hole in a thin wall, a fluctuation relation for the particle current is directly proved from the Hamiltonian microdynamics by constructing the exact invariant probability measure, which is shown to break time-reversal symmetry under nonequilibrium conditions. Current fluctuation relations are also obtained for the stochastic processes ruled by the Smoluchowski and Fokker–Planck master equations by modifying the master operator with the current counting parameters. Finally, the same method is applied to the coarse-grained master equation of the fluctuating Boltzmann equation to establish the fluctuation relation for the currents in dilute or rarefied gases.


From Adiabatic Piston to Non-equilibrium Hydrodynamics

abstract

Based on the new concept of the momentum transfer deficit due to dissipation (MDD), the physical basis of the mechanism of “adiabatic piston” is explained. The implication of MDD in terms of hydrodynamics under non-equilibrium steady state is also discussed.


Onsagers Fluctuation Theory and New Developments Including Non-equilibrium Lévy Fluctuations

abstract

The first part of the paper briefly reviews and explains basic ideas of the theory of Gaussian fluctuations and their relaxation developed in 1931 by Lars Onsager in the context of a general theory of irreversible processes. Motivated by Onsager’s approach, we extend the theory to fluctuations including Lévy processes. We assume that deviations from Gaussian distributions, which are often observed in non-equilibrium systems, may be described by convoluted Gauss–Lévy distributions and their relation to stationary states by generalized Smoluchowski equations. The central part of the distributions we study here is determined by the Gaussian core with the wings decaying according to a power law characteristic for a Lévy-type contribution to statistics. Furthermore, we develop a generalization of Onsager’s theory of linear relaxation processes to those which include statistically independent Gaussian fluctuations and (non-equilibrium) Lévy noises. We apply the generalized version of the fluctuation-dissipation theorem (FDT) to analyze regime of the linear response of the non-equilibrium system driven by Lévy (Cauchy) white noise and subject to thermal (Gaussian) fluctuations. In the last part, applications to non-Maxwellian velocity fluctuations and their relaxations are investigated.


A Novel Approach to the KPZ Dynamics

abstract

We discuss a tentative path-integral approach to numerically follow the scaling properties of the mean rugosity (and other typical averages) of an interface whose growth is described by the Kardar–Parisi–Zhang equation. It resorts to functional minimization and a cellular automata-like algorithm, and can be regarded as a kind of importance-sampling approach. This method is intended to predict the crossover time as a function of the coefficient of the nonlinear term, through the comparison of the weight of the different terms in the “stochastic action”.


Einstein Relation in Systems with Anomalous Diffusion

abstract

We discuss the role of non-equilibrium conditions in the context of anomalous dynamics. We study in detail the response properties in different models, featuring subdiffusion and superdiffusion: in such models, the presence of currents induces a violation of the Einstein relation. We show how in some of them it is possible to find the correlation function proportional to the linear response, in other words, we have a generalized fluctuation-response relation.


Coarse-graining and Thermodynamics in Far-from-equilibrium Systems

abstract

Lying at the core of statistical physics is the need to reduce the number of degrees of freedom in a system. Coarse-graining is a frequently-used procedure to bridge molecular modeling with experiments. In equilibrium systems, this task can be readily performed; however in systems outside equilibrium, a possible lack of equilibration of the eliminated degrees of freedom may lead to incomplete or even misleading descriptions. Here, we present some examples showing how an improper coarse-graining procedure may result in linear approaches to nonlinear processes, miscalculations of activation rates and violations of the fluctuation-dissipation theorem.


Defining Chaos in the Logistic Map by Sharkovskii’s Theorem

abstract

The fixed points of 3-cycle in the logistic map are obtained by solving a sextic polynomial analytically. Therewith, the domain of chaos is established by Sharkovskii’s theorem. A fix-point spectrum is then constructed in the chaotic domain. By Sharkovskii’s theorem, a chaotic trajectory is shown to be a superposition of all finite cycles, termed an aleph cycle. An aleph cycle means chaos and it defines chaos in the logistic map in an absolute sense. In particular, a trajectory which is ergodic is aleph-cyclic, hence it is also chaotic.


Domain Structure Created by Irreversible Adsorption of Dimers

abstract

Structure of monolayers built during adsorption process is strongly related to the properties of adsorbed particles. The most important factor is their shape. For example, adsorption of elongated molecules on patterned surfaces may produce certain orientational order inside a covering layer. This study, however, focuses on random adsorption of dimers on flat, homogeneous surfaces. It has been observed that despite the lack of global orientational ordering, adsorbed dimers may form local, orientationally ordered structures. Our investigations focus on the dependence between domain size distribution and environmental parameters such as ionic strength, which affects the range of electrostatic interaction between molecules.


Competitive Adsorption of Bimodal Latex Suspension

abstract

Competitive deposition of binary mixture of small and large latex particles is studied using Random Sequential Adsorption (RSA) algorithm. We designated the saturated random coverage ratio dependence on small-to-large particle concentration ratio and on particles size ratio. Moreover, the deposition kinetics of the process was calculated numerically. To check validity of the numerical model, the saturated random coverage ratio for 1:1 binary latex particle mixture was measured experimentally using Scanning Electron Microscopy (SEM) and Atomic Force Microscopy (AFM).


all authors

A. Strzelewicz, M. Krasowska, G. Dudek, A. Rybak, R. Turczyn, M. Cieśla

Anomalous Diffusion on Fractal Structure of Magnetic Membranes

abstract

The concept of diffusion on fractal structure of polymeric membrane with magnetic powder is presented. The fractal characteristics, i.e. static fractal dimension \(d_{\rm f}\) and fractal dimension of the trajectory of the random walk \(d_{\rm w}\), were evaluated for qualitative and quantitative description of membrane structures. The way of introducing the fractal dimension and anomalous-diffusion exponent into the generalized diffusion equation on fractal geometries obtained by Metzler et al. has been shown. The results showed that the random walk within the membranes of the smallest granulation of magnetic powder was of the most subdiffusive character.


The Solution to Subdiffusion-reaction Equation for the System with One Mobile and One Static Reactant

abstract

We theoretically study the subdiffusion of \(B\) particles which can chemically react with \(A\) particles according to the formula \(A+B\rightarrow \emptyset {\rm (inert)}\). The \(A\) particles are static and located at the wall which bounds the system. To describe the process, we use a fractional subdiffusion-reaction equation in which the character of the transport process is included in the reaction term. We find the exact solution to the equation for arbitrarily chosen initial conditions in terms of the Laplace transform. The inverse Laplace transform of this solution is calculated over a long time limit. We also briefly discuss the possibility of experimental verification of the model.


all authors

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

Stochastic Dynamics of Two Picophytoplankton Populations in a Real Marine Ecosystem

abstract

A stochastic reaction-diffusion-taxis model is analyzed to get the stationary distribution along water column of two species of picophytoplankton, that is picoeukaryotes and Prochlorococcus. The model is valid for weakly mixed waters, typical of the Mediterranean Sea. External random fluctuations are considered by adding a multiplicative Gaussian noise to the dynamical equation of the nutrient concentration. The statistical tests show that shape and magnitude of the theoretical concentration profile exhibit a good agreement with the experimental findings. Finally, we study the effects of seasonal variations on picophytoplankton groups, including an oscillating term in the auxiliary equation for the light intensity.


Synchronization of Kuramoto Oscillators with Distance-dependent Delay

abstract

We investigate the synchronization process in a Kuramoto model of phase-coupled oscillators with distance-dependent delay. The oscillators occupy the nodes of a two-dimensional square lattice subjected to periodic boundary conditions. The mean-field interactions with velocity-dependent delays propagate along the lattice sites. This gives rise to a non-uniform distribution of delays and lattice dimensionality dependence, which is not present in mean-field models without delays. We find that the ‘coupling strength-delay’ phase diagram does not show up reentrant behavior present in models with uniform delay. A number of dynamic patterns, reported earlier for a generalized Kuramoto model with non-mean-field distance-dependent interactions, is also found.


The Role of Non-Gaussian Sources in the Transient Dynamics of Long Josephson Junctions

abstract

We analyze the effects of different non-Gaussian noise sources on the transient dynamics of an overdamped long Josephson junction. We find nonmonotonic behavior of the mean escape time as a function of the noise intensity and frequency of the external driving signal for all the noise sources investigated.

Version corrected according to Erratum Acta Phys. Pol. B 47, 1177 (2016)


Competing of Sznajd and Voter Dynamics in the Watts–Strogatz Network

abstract

We investigate the Watts–Strogatz network with the clustering coefficient \(C\) dependent on the rewiring probability. The network is an area of two opposite contact processes, where nodes can be in two states, S or D. One of the processes is governed by the Sznajd dynamics: if there are two connected nodes in D-state all their neighbors become D with probability \(p\). For the opposite process, it is sufficient to have only one neighbor in state S; this transition occurs with probability 1. The concentration of S-nodes changes abruptly at given value of the probability \(p\). The result is that for small \(p\), in clusterized networks the activation of S nodes prevails. This result is explained by a comparison of two limit cases: the Watts–Strogatz network without rewiring, where \(C=0.5\), and the Bethe lattice, where \(C=0\).


Regime Variance Testing — a Quantile Approach

abstract

In this paper, we examine time series that exhibit behavior related to two or more regimes with different statistical properties. The motivation of our study are two real data sets from plasma physics with an observable two-regimes structure. In this paper, we develop a procedure to estimate the critical point of the division in a structural change in a time series. Moreover, we propose three tests to recognize such specific behavior. The presented methodology is based on the empirical second moment and its main advantage is the assumption of a lack of distribution. Moreover, the examined statistical properties are expressed in the language of empirical quantiles of the squared data, therefore, the methodology is an extension of the approach known from the literature. Theoretical results are confirmed by simulations and analysis of real data of turbulent laboratory plasma.


Overdamped Dynamics in Septate Channels

abstract

I summarize recent results on Brownian transport in hard-geometry compartmentalized channel, investigating mobility and diffusion as functions of the driving force and the geometry of the cells. Numerical calculations are performed using parallel algorithms on graphics cards. The large drive limit can be fairly well understood in terms of transverse diffusion, while the low drive one can be explained using renewal theory or the random walker jump approximation.


A Method of Mechanical Control of Structure-property Relationship in Grains-containing Material Systems

abstract

Material systems can grow indefinitely large unless there appear some physical restrictions superimposed on them. At first approximation, they can be restricted in their evolution by means of an external mechanical factor such as pressure. If the material system’s evolution can be described by a Smoluchowski type equation in the phase space of grain sizes, the average grain size magnitude can be suitably controlled by such mechanical factor. Both magnitude and direction of the factor can render the grain sizes either bigger or smaller than the one expected without any influence of the factor. We propose to embody the mechanical factor in the drift term of the corresponding Smoluchowski equation, in general, derivable from the entropy production principle. Such an embodiment allows one to modify controllably (over certain distinguished time scales of the process) the grain sizes, thus, the mechanically affected material properties such as superplasticity or superconductivity. A simple econophysical example, addressing investment strategies upon random-market tensions, is added to support the overall rationale of the method thus developed.


Random Walk, Diffusion and Wave Equation

abstract

One-dimensional random walk is analyzed. First, it is shown that the classical interpretation of random walk reaching Lord Rayleigh’s analysis should be completed. Further, an attention is called to the fact that the parabolic diffusion is not an unique interpretation, but also the wave (or hyperbolic) equation can be deduced. It depends on the accepted scale of the length of step \(h\) and duration of the step \(\tau \) in the walk, whether Fick–Smoluchowski’s diffusion or a wave process is obtained. Only additional arguments, such as positivity of distribution function or positivity of the entropy growth, can help to choose the proper physical model. Also, the infinite diffusion velocity paradox in connection with Einstein’s formula is explained.


General Scaling Relations in Anomalous Diffusion

abstract

Diffusion regimes most frequently found in nature are described in terms of asymptotic behaviors. In this work, we use a generalization of the final-value theorem for Laplace transform in order to investigate the anomalous diffusion phenomenon for asymptotic times. We generalize the concept of the diffusion exponent, including a wide variety of asymptotic behaviors than the power law. A method is proposed to obtain the diffusion coefficient analytically through the introduction of a time scaling factor, \(\lambda \). We obtain as well an exact expression for \(\lambda \), which makes possible to describe all diffusive regimes featuring a universal parameter determined by the diffusion exponent. We show the existence of two kinds of ballistic diffusion, ergodic and non-ergodic. The method is general and may be applied to many types of stochastic problem.


Reinterpreting Polymer Unfolding Effect Induced by a Spatially Correlated Noise

abstract

This paper provides additional insight into the effect of spontaneous unfolding of the model polymeric chain driven by spatially correlated noise, described in M. Majka, P.F. Góra, Phys. Rev. E86, 051122 (2012). We examine the statistical data on the linearized chain substructures to find that the global unfolding effect arises mainly from the cumulation of short, 2-segment-long fragments, scattered along the chain. This supports an alternative view of spatially correlated noise as both the source of disturbance and the conformation preserving factor.


Trajectory Statistics of Confined Lévy Flights and Boltzmann-type Equilibria

abstract

We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise, where the Langevin representation is absent. In view of the Lévy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) \(\rho _*(x) \sim \exp [-{\mit \Phi } (x)]\). Here, we infer pdf \(\rho (x,t)\) based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for \(\rho (x,t)\) and its target pdf \(\rho _*(x)\). To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems, where it can be useful.


Ornstein–Uhlenbeck Process with Non-Gaussian Structure

abstract

In this paper, we examine the Ornstein–Uhlenbeck process, i.e. one of the most famous example of continuous time models. Because many studies indicate that the classic version of the Ornstein–Uhlenbeck process is insufficient to description of examined phenomenon, there is a need to consider various modifications of the conventional process. We introduce generalized versions of the classic process in which the standard Brownian motion (Wiener process) is replaced by \(\alpha \)-stable and variance gamma processes. We analyze similarities and differences between Gaussian and considered non-Gaussian versions of the Ornstein–Uhlenbeck process. We point at testing and estimation procedures which we illustrate by simulated data. In order to illustrate theoretical results, we examine a real financial data set in the context of presented methodology.


Anomalous Weakly Nonergodic Brownian Motions in Nonuniform Temperatures

abstract

It is shown that ergodicity breaking and anomalous Brownian motion are not limited to composite and/or disordered systems, but can be generated also in simple fluids or solids with time-dependent and/or spatially nonuniform temperature \(T(t,{\bf r})\). A few examples of simple arrangements easy for experimental realization are discussed in detail. Proposed measurements will enable also the observation of transitions from ergodic to weakly nonergodic and from normal to anomalous diffusion. Similar behaviour can be expected in inflationary systems with time-dependent metric, and in expanding gases.


Fractional Fokker–Planck Equation with Space Dependent Drift and Diffusion: the Case of Tempered \(\alpha \)-stable Waiting-times

abstract

In this paper, we propose a stochastic process with space dependent force and diffusion coefficients. We show that PDF of this process satisfies a generalized tempered fractional Fokker–Planck equation. Thus we obtain a complete description of subdiffusion with tempered \(\alpha \)-stable waiting-times and with space dependent force and diffusion. Based on derived stochastic representation, we simulate paths of the underlying process. Moreover, we approximate the solution of the proposed system via Monte Carlo methods.


External Noise Effects in Silicon MOS Inversion Layer

abstract

The effect of the addition of an external source of correlated noise on the electron transport in silicon MOS inversion layer, driven by a static electric field, has been investigated. The electron dynamics is simulated by a Monte Carlo procedure which takes into account non-polar optical and acoustic phonons. In our modelling of the quasi-two-dimensional electron gas, the potential profile, perpendicular to the MOS structure, is assumed to follow the triangular potential approximation. We calculate the changes in both the autocorrelation function and the spectral density of the velocity fluctuations, at different values of noise amplitude and correlation time. The findings indicate that, the presence of a fluctuating component added to the static electric field can affect the total noise power, i.e. the variance of the electron velocity fluctuations. Moreover, this effect critically depends on both the amplitude of the driving electric field and the noise parameters.


Diffusion with Breaks

abstract

In this paper, we present a stochastic model which is a normal diffusion interrupted by events lasting some period of time during which particle does not move. We assume, that waiting time is described by a one-sided Lèvy \(\alpha \)-stable distribution. For large times, we derive fractional differential equation (FDE) describing evolution of probability density. This asymptotic form is determined by parameters describing underlying stochastic motion. We also show density evolution according to fractional differential equation for asymptotic model and obtain a solution for various model parameters.


all authors

L. Magazzù, D. Valenti, P. Caldara, A. La Cognata, B. Spagnolo, G. Falci

Transient Dynamics and Asymptotic Populations in a Driven Metastable Quantum System

abstract

The transient dynamics of a periodically driven metastable quantum system, interacting with a heat bath, is investigated. The time evolution of the populations, within the framework of the Feynman–Vernon influence functional and in the discrete variable representation, is analyzed by varying the parameters of the external driving. The results display strong non-monotonic behaviour of the populations with respect to the driving frequency.


A Numerical Technique for Preserving the Topology of Polymer Knots: The Case of Short-range Attractive Interactions

abstract

The statistical mechanics of single polymer knots is studied using Monte Carlo simulations. The polymers are considered on a cubic lattice and their conformations are randomly changed with the help of pivot transformations. After each transformation, it is checked if the topology of the knot is preserved by means of a method called Pivot Algorithm and Excluded Area (in short PAEA) and described in a previous publication of the authors. As an application of this method the specific energy, the radius of gyration and heat capacity of a few types of knots are computed. The case of attractive short-range forces is investigated. The sampling of the energy states is performed by means of the Wang–Landau algorithm. The obtained results show that the specific energy and heat capacity increase with increasing knot complexity.


Ambidextrous Chiral Domains in Nonchiral Liquid-crystalline Materials

abstract

Recently, we studied equilibrium properties of the Lebwohl–Lasher model with quadrupolar and octupolar interactions in the large twist limit. A complete mean field analysis of the model and Monte Carlo simulations were presented to show a global stabilization of new structures like tetrahedratic, nematic tetrahedratic, and chiral nematic tetrahedratic phases of \(T_{d}\), \(D_{2d}\), and \(D_2\) symmetry, respectively. Here, by means of Monte Carlo simulation on two-dimensional system, we show that the model can also give a molecular interpretation of macroscopic regions with opposite optical activity (ambidextrous chirality), observed in achiral bent-core systems, and recently in ferrocenomesogens and flexible liquid crystal dimers. The resulting superstructures include short- and long-range twist deformations.


Effects of Confinement on a Two-dimensional System of the Lennard-Jones Particles in Spherical Geometry

abstract

Influence of confinement on a two-dimensional Lennard-Jones system of spherical particles has been studied by means of Molecular Dynamics simulations. High Resolution Density Map (HRDM) method has been applied to study of inhomogeneous configurations in a circular geometry. Solidification has been shown to depend strongly as well on the structure as on the type of constituting particles of the surrounding wall. Within the liquid state, for certain parameters of density and temperature, configurations occur that remind of the structure of node lines characteristic for the Bessel equation, which are argued to play the role of the seeds for solidification.


all authors

D. Makowiec, Z. Struzik, B. Graff, J. Wdowczyk-Szulc, M. Żarczyńska-Buchowiecka, S. Kryszewski

Community Structure in Network Representation of Increments in Beat-to-beat Time Intervals of the Heart in Patients After Heart Transplantation

abstract

The network representations in the characterization of time series complexity is a relatively new but quickly developing branch of time series analysis. The changes in beat-to-beat time intervals, called \(RR\)-increments, can be mapped into the directed and weighted network. The vertices in this network represent \(RR\)-increments and edges correspond to pairs of subsequent increments. We show that community structure analysis, called \(p\)-core analysis, is an effective measure which allows the evaluation of the information on dynamical processes represented by networks constructed from \(RR\)-increments.


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