Regular Series

Vol. 40 (2009), No. 5, pp. 1257 – 1547

XXI Marian Smoluchowski Symposium on Statistical Physics

Zakopane, Poland; September 13–18, 2008

Stochastic Diffusion: From Markov to Non-Markov Modeling


We briefly discuss omnipresence of stochastic modeling in physical science by recalling definitions of Markovian diffusion and generally, non-Markovian continuous time random walks (CTRW). If the motion of an idealized system can be described by a sum of independent displacements whose statistic over short time intervals has a well defined variance, the resulting random walk converges to a normal diffusion process. In turn, if formulation of such motion assumes the idea of distribution of waiting times between subsequent steps, the CTRW scenario emerges, which typically violates the Markovian property.

Itô Formula for Subordinated Langevin Equation. A Case of Time Dependent Force


A century after Paul Langevin’s landmark paper (1908) we derive here an analog of the Itô formula for subordinated Langevin equation. We show that for any subdiffusion process \(Y_t\) with time-dependent force its image \(f(t,Y_t)\) by any function \(f\in C^{1,2}(\mathbb {R}_+\times \mathbb {R})\) is given again by a stochastic differential equation of Langevin type.

Polylogarithms and Logarithmic Diversion in Statistical Mechanics


Sommerfeld’s work on the low temperature theory of an ideal Fermi gas cannot be easily extended nor generalized. By recognizing an underlying logarithmic structure in ideal quantum gases, a new approach has been developed and this work is reviewed with some new insights. A unified formulation of the grand partition function provides several special solutions such as thermodynamic equivalence in two dimensions and \(\mu \) singularity in null dimension.

Statistical Analysis of the Maximum Energy in Solar X-Ray Flare Activity


We analyze the time series of soft X-ray emission registered in the years 1974–2007 by the GOES spacecrafts. We show that in the periods of high solar activity, namely 1977–1981, 1988–1992, 1999–2003 the maximum energy of soft X-ray solar flares exhibits both heavy tails and long-range dependence. We investigate the presence of long-range dependence by means of different self-similarity estimators. This analysis gives a promising start to model the appearance of such solar events during solar maxima.

all authors

R. Metzler, V. Tejedor, J.-H. Jeon, Y. He, W.H. Deng, S. Burov, E. Barkai

Analysis of Single Particle Trajectories: From Normal to Anomalous Diffusion


With modern experimental tools it is possible to track the motion of single nanoparticles in real time, even in complex environments such as biological cells. The quest is then to reliably evaluate the time series of individual trajectories. While this is straightforward for particles performing normal Brownian motion, interesting subtleties occur in the case of anomalously diffusing particles: it is no longer granted that the long time average equals the ensemble average. We here discuss for two different models of anomalous diffusion the detailed behaviour of time averaged mean squared displacement and related quantities, and present possible criteria to analyse single particle trajectories. An important finding is that although the time average may suggest normal diffusion the actual process may in fact be subdiffusive.

Random Walks with Bivariate Lévy-Stable Jumps in Comparison with Lévy Flights


In this paper we compare the Lévy flight model on a plane with the random walk resulting from bivariate Lévy-stable random jumps with the uniform spectral measure. We show that, in general, both processes exhibit similar properties, i.e. they are characterized by the presence of the jumps with extremely large lenghts and uniformly distributed directions (reflecting the same heavy-tail behavior and the spherical symmetry of the jump distributions), connecting characteristic clusters of short steps. The bivariate Lévy-stable random walks, belonging to the class of the well investigated stable processes, can enlarge the class of random-walk models for transport phenomena if other than uniform spectral measures are considered.

Subdynamics of Financial Data From Fractional Fokker–Planck Equation


In exhibition of many real market data we observe characteristic traps. This behavior is especially noticeable for processes corresponding to stock prices. Till now, such economic systems were analyzed in the following manner: before the further investigation trap-data were removed or omitted and then the conventional methods used. Unfortunately, for many observations this approach seems not to be reasonable therefore, we propose an alternative attitude based on the subdiffusion models that demonstrate such characteristic behavior and their corresponding probability distribution function (pdf) is described by the fractional Fokker–Planck equation. In this paper we model market data using subdiffusion with a constant force. We demonstrate properties of the considered systems and propose estimation methods.

Lévy Flights, Dynamical Duality and Fractional Quantum Mechanics


We discuss dual time evolution scenarios which, albeit running according to the same real time clock, in each considered case may be mapped among each other by means of a suitable analytic continuation in time procedure. This dynamical duality is a generic feature of diffusion-type processes. Technically that involves a familiar transformation from a non-Hermitian Fokker–Planck operator to the Hermitian operator (e.g. Schrödinger Hamiltonian), whose negative is known to generate a dynamical semigroup. Under suitable restrictions upon the generator, the semigroup admits an analytic continuation in time and ultimately yields dual motions. We analyze an extension of the duality concept to Lévy flights, free and with an external forcing, while presuming that the corresponding evolution rule (fractional dynamical semigroup) is a dual counterpart of the quantum motion (fractional unitary dynamics).

Dynamics of a Three-Dimensional Inextensible Chain


In the first part of this work the classical and statistical aspects of the dynamics of an inextensible chain in three dimensions are investigated. In the second part the special case of a chain admitting only fixed angles with respect to the \(z\)-axis is studied using a path integral approach. It is shown that it is possible to reduce this problem to a two-dimensional case, in a way which is similar to the reduction of the statistical mechanics of a directed polymer to the random walk of a two-dimensional particle.

Diffusion of Brownian Particles and Liouville Field Theory


In this work we review a recently proposed transformation which is useful in order to simplify non-polynomial potential given in the form of an exponential. As an application, it is shown that the Liouville field theory may be mapped into a field theory with a polynomial interaction between two scalar fields and a massive vector field. The used methodology is illustrated with the help of the simple case of a particle performing a random walk in a delta function potentials.

Series Decomposition of Fractional Brownian Motion and Its Lamperti Transform


The Lamperti transformation of a self-similar process is a stationary process. In particular, the fractional Brownian motion transforms to the second order stationary Gaussian process. This process is represented as a series of independent processes. The terms of this series are Ornstein–Uhlenbeck processes if \(H\lt 1\)/2, and linear combinations of two dependent Ornstein–Uhlenbeck processes whose two dimensional structure is Markovian if \(H\gt 1\)/2. From the representation effective approximations of the process are derived. The corresponding results for the fractional Brownian motion are obtained by applying the inverse Lamperti transformation. Implications for simulating the fractional Brownian motion are discussed.

First Passage Time in a System with Subdiffusive Membrane


We study the first passage time (FPT) of a particle passing through a subdiffusive membrane; the membrane separates the media where normal diffusion occurs. The transport inside the membrane is described by the subdiffusion equation with fractional time derivative. Outside the membrane the normal diffusion equation is used. Starting with the solutions of the equations, we find the probability density of FPT and discuss its properties.

“Smoluchowski type” Equations for Modelling of Air Separation by Membranes with Various Structure


The problem of a membrane air separation in the presence of a magnetic field, is considered. Paramagnetism of oxygen and diamagnetic behaviour of nitrogen form the basis for air separation. A new concept of polymer membranes filled with neodymium powder and magnetized (“magnetic membranes”), was applied. The Smoluchowski equation for oxygen, and simple diffusion equation for nitrogen behaviour in the air have been used. Multifractal analysis of structure and morphology of membranes were applied to optical microscopy images.

Magnetism of Frustrated Regular Networks


We consider a regular random network where each node has exactly three neighbours. Ising spins at the network nodes interact antiferromagnetically along the links. The clustering coefficient \(C\) is tuned from zero to 1/3 by adding new links. At the same time, the density of geometrically frustrated links increases. We calculate the magnetic specific heat, the spin susceptibility and the Edwards–Anderson order parameter \(q\) by means of the heat-bath Monte Carlo simulations. The aim is the transition temperature \(T_x\) dependence on the clustering coefficient \(C\). The results are compared with the predictions of the Bethe approximation. At \(C=0\), the network is bipartite and the low temperature phase is antiferromagnetic. When \(C\) increases, the critical temperature falls down towards the values which are close to the theoretical predictions for the spin-glass phase.

Kramers–Moyal Expansion of Heart Rate Variability


The first six Kramers–Moyal coefficients were extracted from human heart rate variability recordings. The method requires the determination of the Markov time and of the proper conditional probability densities. We analyzed heart rate data recorded in a group of ten young, healthy subjects. We obtained non-negligible higher order Kramers–Moyal (K–M) terms in 6 h nighttime parts of the 24 h recordings. This indicates that the data is a non-Gaussian process and probably a correlated signal. The analysis yielded important new insights into the character and distribution of the stochastic processes measured in healthy group. In the night hours, the dominant oscillation in the heart rate is the so called respiratory sinus arrhythmia (RSA) — a physiological phenomenon in which respiration acts as a drive for the heart rate. Certain kinds of pathology may disrupt RSA. We compared nighttime recordings of the healthy group with those recorded in six patients with hypertrophic cardiomyopathy (HCM). HCM is generally a pathology of heart cells but abnormalities in autonomic regulation are also observed. Using the higher order Kramers–Moyal coefficients, we analyzed the skewness and kurtosis in the nighttime recordings for the normal subjects.

Pattern Formation in a Nonlocal Convective Fisher Equation


We investigate the nonlocal convective Fisher equation and the conditions for pattern formation. We observe that the width of the influence function completely determines whether the pattern is present.

On the Relation Between Lacunarity and Fractal Dimension


I discuss the relation between fractal dimension and lacunarity. Commenting the known results, I propose a method for estimation of the scaling constant in the power law dependency. Additionally, I provide a simple new derivation of a known experimental relation for lacunarity and fractal dimension.

Version corrected according to Erratum Acta Phys. Pol. B 42, 2045 (2011)

Hyperbolic Subdiffusion in a Membrane System


We discuss the physical meaning of three different version of hyperbolic anomalous diffusion equations with fractional time derivatives, which were derived in the paper A. Compte, R. Metzler, J. Phys. A 30, 7277 (1997). We find that only one of them has clear physical interpretation and can be used to describe subdiffusion. We obtain the solutions of this equation and of the parabolic subdiffusion equation for a one-dimensional system with a thin membrane, where the flux flowing through the membrane is proportional to the concentration difference between membrane surfaces. We compare the solutions of hyperbolic and parabolic equations and briefly discuss their properties.

Bose–Einstein Condensation of Noninteracting Free or Trapped Particles: A Common Treatment of the Both Theories


The Bose–Einstein condensation of noninteracting identical particles is investigated, both in the case of free gas in a box, and the trapped particles confined in a harmonic potential. The importance of the low temperature behavior of the exact single-particle canonical partition function is pointed out. The elementary proof of the phase-transition in the thermodynamical limit is presented.

A Multiplicative Law of Network Traffic and Its Consequences


A basic law of Network traffic is derived. The appearance of long tails and self-similar processes is clarified. The theory is confirmed by empirical measurements.

Monofractality in RR Heart Rate by Multifractal Tools


Multifractal formalism is tested if it can work as a robust estimator of monofractals when scaling intervals are fixed. Intervals for scaling are selected to be consistent with known frequency bands of power spectral analysis used in estimates of heart rate variability: low frequency (LF), very low frequency (VLF), and ultra low frequency (ULF). Tests on fractional Brownian motions and a binomial cascade are performed to validate popular multifractal methods: Wavelet Transform Modulus Maxima and Multifractal Detrended Fluctuation Analysis. Then the methods are applied to identify monofractal elements of control processes driving the heart rate. A transition is found in the dynamic organization of autonomic nervous system control of the heart rate related to the change in scaling intervals. The control over the diurnal heart rate is of a multifractal type when considered in LF and of a monofractal type when observed in ULF. Additionally, this transition affects on a switch in a relation between widths of diurnal and nocturnal multifractal spectra.


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