Discriminating geoelectrical signals measured in seismic and aseismic areas by using Ito models

Geoelectrical signals can be considered as the end-product of complex and collective interactions among system elements. Their dynamical behavior could be different if they are measured in seismic or aseismic areas. Using the Ito equations, which represent a good macroscopic model of phenomena in which microscopic interactions are adequately averaged, we show that geoelectrical data recorded in seismic areas are well discriminated by those measured in aseismic areas. Our findings contribute to a better dynamical characterization of geoelectrical signals.     

One-Way Markov Process Approach to Repeat Times of Large Earthquakes in Faults

One of the uses of Markov Chains is the simulation of the seismic cycle in a fault, i.e. as a renewal model for the repetition of its characteristic earthquakes. This representation is consistent with Reid??s elastic rebound theory. We propose a general one-way Markovian model in which the waiting time distribution, its first moments, coefficient of variation, and functions of error and alarm (related to the predictability of the model) can be obtained analytically. The fact that in any one-way Markov cycle the coefficient of variation of the corresponding distribution of cycle lengths is always lower than one concurs with observations of large earthquakes in seismic faults. The waiting time distribution of one of the limits of this model is the negative binomial distribution; as an application, we use it to fit the Parkfield earthquake series in the San Andreas fault, California.     

Models of the Dynamics of Spatially Separated Broadband Electromagnetic Fields Interacting with Resonant Atoms

The Markov model of spontaneous emission of an atom localized in a spatial region with a broadband electromagnetic field with zero photon density is considered in the conditions of coupling of the electromagnetic field with the broadband field of a neighboring space. The evolution operator of the system and the kinetic equation for the atom are obtained. It is shown that the field coupling constant affects the rate of spontaneous emission of the atom, but is not manifested in the atomic frequency shift. The analytic expression for the radiative decay constant for the atom is found to be analogous in a certain sense to the expression for the decay constant for a singly excited localized ensemble of identical atoms in the conditions when the effect of stabilization of its excited state by the Stark interaction with the vacuum broadband electromagnetic field is manifested. The model is formulated based on quantum stochastic differential equations of the non- Wiener type and the generalized algebra of the Ito differential of quantum random processes.     

The Theory of Individual Based Discrete-Time Processes

A general theory is developed to study individual based models which are discrete in time. We begin by constructing a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are hence intrinsic to the system and can induce qualitative changes to the dynamics predicted from the deterministic map. From the Chapman–Kolmogorov equation for the discrete-time Markov process, we derive the analogues of the Fokker–Planck equation and the Langevin equation, which are routinely employed for continuous time processes. In particular, a stochastic difference equation is derived which accurately reproduces the results found from the Markov chain model. Stochastic corrections to the deterministic map can be quantified by linearizing the fluctuations around the attractor of the map. The proposed scheme is tested on stochastic models which have the logistic and Ricker maps as their deterministic limits.     

Stationary Response of Lotka-Volterra System with Real Noises

A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It o stochastic differential equations. The Ito formula is then carried out to obtain the It o stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged Ito stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.     

Optimal instruments and models for noisy chaos

Analysis of finite, noisy time series data leads to modern statistical inference methods. Here we adapt Bayesian inference for applied symbolic dynamics. We show that reconciling Kolmogorov's maximum-entropy partition with the methods of Bayesian model selection requires the use of two separate optimizations. First, instrument design produces a maximum-entropy symbolic representation of time series data. Second, Bayesian model comparison with a uniform prior selects a minimum-entropy model, with respect to the considered Markov chain orders, of the symbolic data. We illustrate these steps using a binary partition of time series data from the logistic and Henon maps as well as the R?ssler and Lorenz attractors with dynamical noise. In each case we demonstrate the inference of effectively generating partitions and kth-order Markov chain models.     

Unveiling Informational Properties of the Chen-Ouillon-Sornette Seismo-Electrical Model

The seismo-electrical coupling is critical to understand the mechanism of geoelectrical precursors to earthquakes. A novel seismo-electrical model, called Chen–Ouillon–Sornette (COS) model, has been developed by combining the Burridge–Knopoff spring-block system with the mechanisms of stress-activated charge carriers (i.e., electrons and holes) and pressure-stimulated currents. Such a model, thus, can simulate fracture-induced electrical signals at a laboratory scale or earthquake-related geoelectrical signals at a geological scale. In this study, by using information measures of time series analysis, we attempt to understand the influence of diverse electrical conditions on the characteristics of the simulated electrical signals with the COS model. We employ the Fisher–Shannon method to investigate the temporal dynamics of the COS model. The result showed that the electrical parameters of the COS model, particularly for the capacitance and inductance, affect the levels of the order/disorder in the electrical time series. Compared to the field observations, we infer that the underground electrical condition has become larger capacitance or smaller inductance in seismogenic processes. Accordingly, this study may provide a better understanding of the mechanical–electrical coupling of the earth’s crust.     

Interplay of electron-electron and electron-phonon interactions in molecular junctions

In this work we consider a current carrying molecular junction with both electron-phonon and electron-electron interactions taken into account. After performing Lang-Firsov transformation and considering Markov approximations in accordance to weak coupling to the electronic leads, we obtain the master equation governing the time evolution of the reduced density matrix of the junction. The steady state of the density matrix can be used to obtain I-V characteristic of the junction in several regimes of strengths of the interactions. Our results indicate that the system can show negative differential conductance (that is, the current decreases by increasing the applied bias voltage) in some regimes as an interplay between the electron-phonon and Coulomb interactions.     

Dynamical modelling of superstatistical complex systems

We show how to construct the optimum superstatistical dynamical model for a given experimentally measured time series. For this purpose we generalise the superstatistics concept and study a Langevin equation with a memory kernel whose parameters fluctuate on a large time scale. We show how to construct a synthetic dynamical model with the same invariant density and correlation function as the experimental data. As a main example we apply our method to velocity time series measured in high-Reynolds-number turbulent Taylor-Couette flow, but the method can be applied to many other complex systems in a similar way.     

Interactions of Lump and Solitons to Generalized (2+1)-Dimensional Ito Systems*

The $(2+1)$-dimensional Ito equation is extended to a general form including some nonintegrable effects via introducing generalized bilinear operators. It is pointed out that the nonintegrable $(2+1)$-dimensional Ito equation contains lump solutions and interaction solutions between lump and stripe solitons. The result shows that the lump soliton will be swallowed or arisen by a stripe soliton in a fixed time. Furthermore, by the interaction between a lump and a paired resonant stripe soliton, the lump will be transformed to an instanton or a rogue wave.     

Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.     

Amplification of entangled light upon reflection from a cavity with nonlinear medium

The kinetic equation for the case of reflection of a broadband entangled light of a nondegenerate parametric source from a cavity is derived with the use of the technique of stochastic differential Ito equations. Statistical properties of the light reflected from a cavity with a nonlinear transparent medium, in which the process of special parametric interaction occurs, are considered on the basis of the obtained equation. Due to the integrals of motion, the entangled light, which is an EPR pair of continuous variables, can be amplified without destruction of the quantum correlation.     

Langevin description of markovian integro-differential master equations

For a given master equation of a discontinuous irreversible Markov process, we present the derivation of stochastically equivalent Langevin equations in which the noise is either multiplicative white generalized Poisson noise or a spectrum of multiplicative white Poisson noise. In order to achieve this goal, we introduce two new stochastic integrals of the Ito type, which provide the corresponding interpretation of the Langevin equations. The relationship with other definitions for stochastic integrals is discussed. The results are elucidated by two examples of integro-master equations describing nonlinear relaxation.     

A Langevin equation for the rates of currency exchange based on the Markov analysis

We propose a method for analyzing the data for the rates of exchange of various currencies versus the U.S. dollar. The method analyzes the return time series of the data as a Markov process, and develops an effective equation which reconstructs it. We find that the Markov time scale, i.e., the time scale over which the data are Markov-correlated, is one day for the majority of the daily exchange rates that we analyze. We derive an effective Langevin equation to describe the fluctuations in the rates. The equation contains two quantities, D⁽¹⁾ and D⁽²⁾, representing the drift and diffusion coefficients, respectively. We demonstrate how the two coefficients are estimated directly from the data, without using any assumptions or models for the underlying stochastic time series that represent the daily rates of exchange of various currencies versus the U.S. dollar.     

A time series analysis and a non-homogeneous Poisson model with multiple change-points applied to acoustic data

High levels of the so-called community noise may produce hazardous effect on the health of a population exposed to them for large periods of time. Hence, the study of the behaviour of those noise measurements is very important. In this work we analyse that in terms of the probability of exceeding a given threshold level a certain number of times in a time interval of interest. Since the datasets considered contain missing measurements, we use a time series model to estimate the missing values and complete the datasets. Once the data is complete, we use a non-homogeneous Poisson model with multiple change-points to estimate the probability of interest. Estimation of the parameters of the models are made using the usual time series methodology as well as the Bayesian point of view via Markov chain Monte Carlo algorithms. The models are applied to data obtained from two measuring sites in Messina, Italy.     

New approach to the quantum tunneling process: Characteristic times for transmission and reflection

In [1] we have demonstrated that scattering of a quantum particle on a one-dimensional potential barrier should be considered as a combined process involving two alternative elementary transmission and reflection processes. For symmetric potential barriers, we have found solutions of the Schrödinger equation which describe the transmission and reflection processes in all stages of scattering. The present work studies time aspects of both processes. The local and asymptotic group tunneling times, dwell time, and Larmor tunneling time are determined for each process. Among these time characteristics, the group tunneling times should be considered as auxiliary. As to the dwell and Larmor tunneling times, they are the best estimates (of the expected values) of times the quantum particle in stationary and localized nonstationary states dwells in the barrier region. Moreover, the Larmor time is simply the dwell time averaged over the corresponding ensemble of particles. This characteristic can be measured experimentally and hence the suggested model of scattering can be verified.     

The weak coupling limit as a quantum functional central limit

We show that, in the weak coupling limit, the laser model process converges weakly in the sense of the matrix elements to a quantum diffusion whose equation is explicitly obtained. We prove convergence, in the same sense, of the Heisenberg evolution of an observable of the system to the solution of a quantum Langevin equation. As a corollary of this result, via the quantum Feynman-Kac technique, one can recover previous results on the quantum master equation for reduced evolutions of open systems. When applied to some particular model (e.g. the free Boson gas) our results allow to interpret the Lamb shift as an Ito correction term and to express the pumping rates in terms of quantities related to the original Hamiltonian model.     

Markov Properties of Electrical Discharge Current Fluctuations in Plasma

Using the Markovian method, we study the stochastic nature of electrical discharge current fluctuations in the Helium plasma. Sinusoidal trends are extracted from the data set by the Fourier-Detrended Fluctuation analysis and consequently cleaned data is retrieved. We determine the Markov time scale of the detrended data set by using likelihood analysis. We also estimate the Kramers-Moyal’s coefficients of the discharge current fluctuations and derive the corresponding Fokker-Planck equation. In addition, the obtained Langevin equation enables us to reconstruct discharge time series with similar statistical properties compared with the observed in the experiment. We also provide an exact decomposition of temporal correlation function by using Kramers-Moyal’s coefficients. We show that for the stationary time series, the two point temporal correlation function has an exponential decaying behavior with a characteristic correlation time scale. Our results confirm that, there is no definite relation between correlation and Markov time scales. However both of them behave as monotonic increasing function of discharge current intensity. Finally to complete our analysis, the multifractal behavior of reconstructed time series using its Keramers-Moyal’s coefficients and original data set are investigated. Extended self similarity analysis demonstrates that fluctuations in our experimental setup deviates from Kolmogorov (K41) theory for fully developed turbulence regime.     

A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation

Bird's direct simulation Monte Carlo method for the Boltzmann equation is considered. The limit (as the number of particles tends to infinity) of the random empirical measures associated with the Bird algorithm is shown to be a deterministic measure-valued function satisfying an equation close (in a certain sense) to the Boltzmann equation. A Markov jump process is introduced, which is related to Bird's collision simulation procedure via a random time transformation. Convergence is established for the Markov process and the random time transformation. These results, together with some general properties concerning the convergence of random measures, make it possible to characterize the limiting behavior of the Bird algorithm.