Ryszard Kutner

Chemical diffusion in the lattice gas of non-interacting particles

Abstract The occupancy correlation function of non-interacting particles hopping on a lattice, with exclusion of double occupancy, is rederived in a concise way. The resulting chemical diffusion coefficient is discussed.

Thermal neutron scattering from a hydrogen—Metal system in terms of a general multi-sublattice jump diffusion model. Part I. Theory

A Multi-Sublattice Jump Diffusion Model (MSJD) for hydrogen diffusion through interstitial-site lattices is presented. The MSJD approach may, in principle, be considered as an extension of the Rowe et al.[1] model. Jump diffusion to any neighbours with different jump times which may be asymmetric in space is discussed. On the basis of the model a new method of calculating the diffusion tensor is advanced. The quasielastic, double differential cross section for thermal neutron scattering is obtained in terms of the MSJD model. The model can be used for systems in which interstitial jump diffusion of impurity particles occurs. In Part II the theoretical results are compared with those for quasielastic neutron scattering from the αNbHx system.

Structural and topological phase transitions on the German Stock Exchange

We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two transitions of the topology of a complex network representing the FSE were found. The first transition is from a hierarchical scale-free MST representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the market’s state for the period containing the crash. Subsequently, a transition is observed from this transient, (meta)stable state of the crash to a hierarchical scale-free MST decorated by several star-like trees after the worldwide financial crash. The phase transitions observed are analogous to the ones we obtained earlier for the Warsaw Stock Exchange and more pronounced than those found by Onnela–Chakraborti–Kaski–Kertesz for the S&P 500 index in the vicinity of Black Monday (October 19, 1987) and also in the vicinity of January 1, 1998. Our results provide an empirical foundation for the future theory of dynamical, structural and topological phase transitions on financial markets.

Modelling of income distribution in the European Union with the Fokker–Planck equation

Herein, we applied statistical physics to study incomes of three (low-, medium- and high-income) society classes instead of the two (low- and medium-income) classes studied so far. In the frame of the threshold nonlinear Langevin dynamics and its threshold Fokker–Planck counterpart, we derived a unified formula for description of income of all society classes, by way of example, of those of the European Union in years 2006 and 2008. Hence, the formula is more general than the well known formula of Yakovenko et al.. That is, our formula well describes not only two regions but simultaneously the third region in the plot of the complementary cumulative distribution function vs. an annual household income. Furthermore, the known stylised facts concerning this income are well described by our formula. Namely, the formula provides the Boltzmann–Gibbs income distribution function for the low-income society class and the weak Pareto law for the medium-income society class, as expected. Importantly, it predicts (to satisfactory approximation) the Zipf law for the high-income society class. Moreover, the region of medium-income society class is now distinctly reduced because the bottom of high-income society class is distinctly lowered. This reduction made, in fact, the medium-income society class an intermediate-income society class.

Hierarchical spatio-temporal coupling in fractional wanderings. (I) Continuous-time Weierstrass flights

The one-dimensional continuous-time Weierstrass flights (CTWF) model is considered in the framework of the nonseparable continuous-time random walks formalism (CTRW). A novel spatio-temporal coupling is introduced by assuming that in each scale the probability density for the flight and for waiting are joined. Hence, we treat the spatio-temporal relations in terms of the self-similar structure of the Weierstrass process.This (stochastic) structure is characterized by the spatial fractional dimension 1/β representing the flights and the temporal one 1/α representing the waiting. Time was assumed here as the only independent truncation range. In this work we study the asymptotic properties of the CTWF model. For example, by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form, P(X,t)∼t−η(α,ptβ)/2F(ξ), where the scalingfunction F(ξ)=ξν(α,ptβ)exp(−const.(α,β)ξν(α,ptβ)), while the scaling variable ξ=|X|/tη(α,ptβ)/2 is large. The principal result of our analysis is that the exponents ν and ν depend on more fundamental ones, α and β, what leads to a novel scaling. As a result of competition between exponents α and βan enhanced, dispersive or normal diffusion was recognized in distinction from the prediction of the separable CTRW model where the enhanced diffusion is lost and the dispersive one is strongly limited. It should be noted that we compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes. Having the propogator we calculated, for example the particle mean-square displacement and found its novel asymptotic scaling with time for enhanced diffusion, given by ∼t1+α(2/β−1), in distinction from its diverging for β<2 within the separable CTRW model. Our version of the nonseparable CTRW approach, i.e. the CTWF model or renewed continuous-time Levy flights (CTLF), offers a possibility to properly model the time-dependence for any fractional (critical) wandering of jump type. A similar, though mathematically more complicated analysis is also applied in Part II to the Weierstrass walks (WW) which leads to a novel version of the Levy walks (LW) process.

The positive muon as a tracer for the study of dynamic correlation effects in metal hydrogen systems

In order to investigate whether the positive muon (μ+) can be used as a tracer in hydrogen diffusion studies, we have made what we believe to be the first systematic study of muon–hydrogen correlations in a metal hydride (NbHx) over a large range of hydrogen concentrations. The following observations were made from transverse field μSR experiments: (i) Loading with hydrogen has a strong impact on muon mobility in Nb, and shifts the motional narrowing region from 60 to above 170 K; (ii) in β‐NbHx, the inverse muon correlation times τ−1μ are of the same order as the hydrogen jump rates τ−1H. However, the activation energies are distinctly different (EHa≂230 meV, Eμa≂140–150 meV). Moreover, the concentration dependence of τ−1μ exhibits strong negative deviations from a (1−c) behavior; (iii) the H/D isotope effects as measured by μ+ diffusion in β‐NbH(D)x is significantly nonclassical. While the activation energy for τ−1μ is slightly larger in the deuteride, the ratio of jump rates is significantly larger tha...

Random walk on a random walk

The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found.

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

Abstract In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Phys. J. B Special Issue [http://epjb.epj.org/open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on] containing articles which show incredible possibilities of the CTRWs.

A Model for Interevent Times with Long Tails and Multifractality in Human Communications: An Application to Financial Trading

Social, technological, and economic time series are divided by events which are usually assumed to be random, albeit with some hierarchical structure. It is well known that the interevent statistics observed in these contexts differs from the Poissonian profile by being long-tailed distributed with resting and active periods interwoven. Understanding mechanisms generating consistent statistics has therefore become a central issue. The approach we present is taken from the continuous-time random-walk formalism and represents an analytical alternative to models of nontrivial priority that have been recently proposed. Our analysis also goes one step further by looking at the multifractal structure of the interevent times of human decisions. We here analyze the intertransaction time intervals of several financial markets. We observe that empirical data describe a subtle multifractal behavior. Our model explains this structure by taking the pausing-time density in the form of a superstatistics where the integral kernel quantifies the heterogeneous nature of the executed tasks. A stretched exponential kernel provides a multifractal profile valid for a certain limited range. A suggested heuristic analytical profile is capable of covering a broader region.

Influence of a Uniform Driving Force on Tracer Diffusion in a One-Dimensional Hard-Core Lattice Gas

The influence of a uniform driving force on tracer diffusion is investigated for a one-dimensional lattice gas where particles jump stochastically to unoccupied neighboring sites. A new, simple calculation is presented for the diffusion coefficient of a tracer particle with respect to its average drift, obtained recently by rigorous methods by De Masi and Ferrari. A theoretical expression describing the tracer particle mean square displacement approximately for all times is derived and found to be in excellent agreement with the results of Monte Carlo simulations.