Binary N-Step Markov Chain as an Exactly Solvable Model of Long-Range Correlated Systems

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In our model, the conditional probability that the i-th symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The model allows exact analytical treatment. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length L are obtained analytically and numerically. A self-similarity of the studied stochastic process is revealed and the similarity transformation of the chain parameters is presented. The diffusion equation governing the distribution function of the L-words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on L. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.Comment: LaTeX2e, 16 pages, 9 figure

Comment on cond-mat/0107336 ``Critical State Theory for Nonparallel Flux Line Lattices in Type-II Superconductors''

Comment on the paper "Critical State Theory for Nonparallel Flux Line Lattices in Type-II Superconductors" by A. Badia and C. Lopez is presented here.Comment: 1 pag

Memory Functions of the Additive Markov chains: Applications to Complex Dynamic Systems

A new approach to describing correlation properties of complex dynamic systems with long-range memory based on a concept of additive Markov chains (Phys. Rev. E 68, 061107 (2003)) is developed. An equation connecting a memory function of the chain and its correlation function is presented. This equation allows reconstructing the memory function using the correlation function of the system. Thus, we have elaborated a novel method to generate a sequence with prescribed correlation function. Effectiveness and robustness of the proposed method is demonstrated by simple model examples. Memory functions of concrete coarse-grained literary texts are found and their universal power-law behavior at long distances is revealed.Comment: 5 pages, 5 figures, changes of minor nature, 1 figure adde

Continuous stochastic processes with non-local memory

We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into expression for the higher-order transition probability function and stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with prescribed pair correlation function. As illustration, some examples of numerical simulation of the processes with non-local memory are presented.Comment: 7 pages, 2 figure

Rank distributions of words in additive many-step Markov chains and the Zipf law

The binary many-step Markov chain with the step-like memory function is considered as a model for the analysis of rank distributions of words in stochastic symbolic dynamical systems. We prove that the envelope curve for this distribution obeys the power law with the exponent of the order of unity in the case of rather strong persistent correlations. The Zipf law is shown to be valid for the rank distribution of words with lengths about and shorter than the correlation length in the Markov sequence. A self-similarity in the rank distribution with respect to the decimation procedure is observed.Comment: 4pages, 3 figure

Terahertz Transverse-Electric- and Transverse-Magnetic-polarized waves localized on graphene in photonic crystals

We predict the coexistence of both TE- and TM-polarized localized electromagnetic waves that can propagate \emph{in the same frequency range} along a graphene layer inserted in a photonic crystal. In addition, we studied the excitation of these modes by an external wave and have shown that the resonance peaks of the sample transmissivity should be observed due to the excitation of the localized waves, independently of the polarization of the exciting wave. The simplicity of the derived dispersion relations for the localized modes and the possibility to excite waves of both polarizations provide a method for measuring graphene conductivity.Comment: This manuscript was extended to 7 pages and 8 figures and published in PR

Nonlinear Effect of Transport Current on Response of Metals to Electromagnetic Radiation

The nonlinear interaction of DC current flowing in a thin metal film with an external low-frequency AC electromagnetic field is studied theoretically. The nonlinearity is related to the influence of the magnetic field of the DC current and the magnetic field of the wave on the form of electron trajectories. This magnetodynamic mechanism of nonlinearity is the most typical for pure metals at low temperatures. We find that such interaction causes sharp kinks in the temporal dependence of the AC electric field of the wave on surface of the sample. The phenomenon of amplification of the electromagnetic signal on the metal surface is predicted. We also calculate the nonlinear surface impedance and show that it turns out to be imaginary-valued and its modulus decreases drastically with the increase of the wave amplitude.Comment: RevTex, 13 pages, 5 figure

Equivalence of Markov's Symbolic Sequences to Two-Sided Chains

A new object of the probability theory, two-sided chain of events (symbols), is introduced. A theory of multi-steps Markov chains with long-range memory, proposed earlier in Phys. Rev. E 68, 06117 (2003), is developed and used to establish the correspondence between these chains and two-sided ones. The Markov chain is proved to be statistically equivalent to the definite two-sided one and vice versa. The results obtained for the binary chains are generalized to the chains taking on the arbitrary number of states.Comment: 5 page

High-Order Correlation Functions of Binary Multi-Step Markov Chains

Two approaches to studying the correlation functions of the binary Markov sequences are considered. The first of them is based on the study of probability of occurring different ''words'' in the sequence. The other one uses recurrence relations for correlation functions. These methods are applied for two important particular classes of the Markov chains. These classes include the Markov chains with permutative conditional probability functions and the additive Markov chains with the small memory functions. The exciting property of the self-similarity (discovered in Phys. Rev. Lett. 90, 110601 (2003) for the additive Markov chain with the step-wise memory function) is proved to be the intrinsic property of any permutative Markov chain. Applicability of the correlation functions of the additive Markov chains with the small memory functions to calculating the thermodynamic characteristics of the classical Ising spin chain with long-range interaction is discussed.Comment: 9 page

Non-extensive thermodynamics of 1D systems with long-range interaction

A new approach to non-extensive thermodynamical systems with non-additive energy and entropy is proposed. The main idea of the paper is based on the statistical matching of the thermodynamical systems with the additive multi-step Markov chains. This general approach is applied to the Ising spin chain with long-range interaction between its elements. The asymptotical expressions for the energy and entropy of the system are derived for the limiting case of weak interaction. These thermodynamical quantities are found to be non-proportional to the length of the system (number of its particle).Comment: 5 pages, 2 figure