Aging and nonergodicity beyond the Khinchin theorem

Theory of Relaxation of Quadrupolar Nuclei in Deformable Molecules in Isotropic Solutions. Application to DNA

Purpose. Defining and analysis of the probabilistic and spectral characteristics of random current in regenerative braking mode of DC electric rolling stock are observed in this paper. Methodology. The elements and methods of the probability theory (particularly the theory of stationary and non-stationary processes) and methods of the sampling theory are used for processing of the regenerated current data arrays by PC. Findings. The regenerated current records are obtained from the locomotives and trains in Ukraine railways and trams in Poland. It was established that the current has uninterrupted and the jumping variations in time (especially in trams). For the random current in the regenerative braking mode the functions of mathematical expectation, dispersion and standard deviation are calculated. Histograms, probabilistic characteristics and correlation functions are calculated and plotted down for this current too. It was established that the current of the regenerative braking mode can be considered like the stationary and non-ergodic process. The spectral analysis of these records and “tail part” of the correlation function found weak periodical (or low-frequency) components which are known like an interharmonic. Originality. Firstly, the theory of non-stationary random processes was adapted for the analysis of the recuperated current which has uninterrupted and the jumping variations in time. Secondly, the presence of interharmonics in the stochastic process of regenerated current was defined for the first time. And finally, the patterns of temporal changes of the correlation current function are defined too. This allows to reasonably apply the correlation functions method in the identification of the electric traction system devices. Practical value. The results of probabilistic and statistic analysis of the recuperated current allow to estimate the quality of recovered energy and energy quality indices of electric rolling stock in the regenerative braking mode. But the correlation and spectral analysis is required to determine the electromagnetic compatibility of the electric rolling stock in the regenerative braking mode with the subsystems and the electric traction devices.

The problem of anomalous scaling in magnetohydrodynamics turbulence is considered within the framework of the kinematic approximation, in the presence of a large-scale background magnetic field. The velocity field is Gaussian, delta-correlated in time, and scales with a positive exponent xi. Explicit inertial-range expressions for the magnetic correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy and forcing) anomalous exponents. The complete set of anomalous exponents for the pair correlation function is found nonperturbatively, in any space dimension d, using the zero-mode technique. For higher-order correlation functions, the anomalous exponents are calculated to O(xi) using the renormalization group. The exponents exhibit a hierarchy related to the degree of anisotropy; the leading contributions to the even correlation functions are given by the exponents from the isotropic shell, in agreement with the idea of restored small-scale isotropy. Conversely, the small-scale anisotropy reveals itself in the odd correlation functions: the skewness factor is slowly decreasing going down to small scales and higher odd dimensionless ratios (hyperskewness, etc.) dramatically increase, thus diverging in the r-->0 limit.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

This paper shows how to obtain exact, closed-form expressions for various moments and correlation functions of the solutions of the stochastic, ordinary differential equation d2udz2+β02[1+ηT(z)]u=0,where T(z) is the so-called ``random telegraph'' wave and β02 and η are positive real constants. These moments and correlation functions are calculated by two different methods, one a phase space method and the other a matrix method familiar from optics. It is found that the moments are sums of exponentials. The first-order moments decay exponentially but the second-order moments grow exponentially. The correlation functions are also sums of exponentials and show that the solutions do not form a stationary process. An important application of these results is obtained in the problem of a plane electromagnetic wave normally incident on a randomly stratified dielectric plate. It is shown that, if 𝔖 is the amplitude transmission coefficient of the plate, then 〈1/SS*〉 can be expressed in terms of the second-order moments of the solutions and derivatives of solutions of the stochastic differential equation.

The projective-kinetic method formulated in [1, 2] for a macroscopic description is used to investigate the dynamics of perturbations of a charged subsystem in a strong electric field. A system of macroscopic equations is constructed, and general expressions for the correlation functions (CF) and nonequilibrium kinetic coefficients (NKC) related to them are derived. By the example of the correlation density function and nonequilibrium diffusion coefficient, the CF and NKC asymptotes are examined for a highly nonequilibrium state. It is demonstrated that when the stability changes, the CF density is subject to anomalous discontinuity and the differential conductivity alters its sign in strong fields.

This paper considers the properties of the tap gain functions in a tapped delay line model of the orbital dipole channel. Specifically, auto-correlation and cross-correlation functions are computed for these tap functions in terms of assumed statistics for the motion of illuminated dipoles. It is demonstrated that when a certain inequality involving the transmitted signal bandwidth is satisfied, these correlation functions become identical to those which would arise from the corresponding tap gain functions of a suitably defined channel consisting of a continuum of motionless scintillating uncorrelated scatterers. In effect, this inequality sets an upper bound to the bandwidth of the transmitted signal, beyond which the orbital dipole channel may not be modeled as such a continuum of uncorrelated scatterers. Simplified expressions for tap gain correlation functions are derived for the case of large signal bandwidths, including the subcase in which the above mentioned inequality is satisfied. In the latter case the tap gain functions are shown to become mutually uncorrelated for all time shifts.

The problem of digital modeling of random processes with given either correlation function or spectral density of the process is considered. These functions of a random process are interconnected by the Wiener–Khinchin theorem. The solution of one function can be used to solve another. The development of a mathematical representation of a stationary random process with a given correlation function based on the Karhunen-Loeve transformation, which is most often used to decorrelate the original process in order to describe it more concisely (data compression problem), has been completed. It is proposed to use the Karhunen–Loev transformation to impart the required correlation properties to the original uncorrelated random process by inverting (converting) this transformation. The form of the required transformation for a discrete (in time) representation of input and output processes of various lengths and methods for ensuring the required modeling accuracy are substantiated. A procedure for obtaining a correlation function from a given spectral density of a simulated random process is presented. An experimental study of the proposed method was carried out in the course of computer simulation in the Mathcad package which simplified the solution of the required computational problems. The initial random process was obtained as a sequence of independent (and, therefore, uncorrelated) random numbers, and the output process, as a result of the transformation, was obtained in the work. The calculated approximate correlation function is compared with the given one and the error variance is determined. The results of modeling random processes with given correlation functions and a homogeneous Markov process with a given transition probability are given as well as an example of the transition from a given spectral density of a random process to its correlation function. The results obtained confirm the effectiveness and feasibility of the developed modeling methods which will allow them to be used in computer research and design of various systems.

In this dissertation, we present a detailed theoretical analysis of the spin dynam- ics in small antiferromagnetic systems in view of macroscopic quantum phenom- ena, possible applications in quantum information processing, and transport of the magnetization. Ferromagnetic and antiferromagnetic systems with a size on the nanometer scale such as nanoparticles or magnetic molecular clusters, show intriguing quantum e®ects which are in stark contrast to the behavior of a macroscopic magnetic moment. In recent years, the interest in small magnetic systems has been renewed due to the discovery of possible future technological applications such as data storage or quantum information processing. Fer- romagnetic molecular magnetic systems with a large net spin show incoherent tunneling of the magnetization on a long timescale. This quantum phenomenon is, by now, well established both experimentally and theoretically. Small an- tiferromagnetic systems have so far attracted less attention although quantum e®ects are even more pronounced than in their ferromagnetic counterparts. The main reason for this is that the predicted quantum phenomena such as coherent quantum tunneling of the N¶eel vector are not easily accessible in experiments. On a theoretical level, the description of an antiferromagnetic system is chal- lenging because of the pronounced quantum °uctuations of the spins. Several antiferromagnetic molecular ring molecules have been synthesized to date. The ferric wheels are the most prominent examples. These systems are promising candidates for macroscopic quantum coherence in the form of coherent N¶eel vector tunneling. Although the tunneling rate can be determined from the measurement of thermodynamic properties, a thorough understanding requires theoretical analysis and experimental observation of the spin dynamics. We calculate spin correlation functions using spin coherent state path integrals and ¯nd analytical expressions for the correlation functions of both the N¶eel vector and the total spin. Our results are in good agreement with numerical exact diagonalization for the small systems that are accessible numerically. From the correlation functions, we deduce that the observation of N¶eel vec- tor tunneling requires an experimental probe that couples to a single spin of the antiferromagnetic system only. Both nuclear magnetic resonance and elec- tron spin resonance on doped rings meet this criterion. Nuclear spins coupled only to single electron spins are ideal candidates for local probes because the nuclear spin susceptibiliy exhibits signatures of the coherent electron spin dy- namics. Alternatively, by doping of the ring molecule, an antiferromagnetic system emerges that has uncompensated sublattice spins. The resulting tracer spin would allow one to detect N¶eel vector tunneling with electron spin reso- nance or magnetic susceptibility measurements. Small antiferromagnetic systems with a ¯nite net spin are interesting in view of quantum information processing. Single electron spins are among the most promising candidates for qubits in a solid state system. However, quantum computing is also possible with a wide range of antiferromagnetic clusters which form an e®ective two-state system in the low energy sector. The main advantage of a qubit formed by a spin cluster is that initialization, quantum gate operation, error correction, and readout are possible with techniques applicable to single spins, while the requirements on local control are relaxed. Spin cluster qubits are very insensitive to the details of intracluster exchange interactions and spin placement. Quantum computing is only one of the exciting perspectives in the emerging ¯eld of spintronics in which the spin and charge degrees of freedom of an elec- tron are treated on an equal footing. We analyze transport of magnetization in insulating systems described by a spin Hamiltonian in which the magnetization current is not accompanied by a charge current. The magnetization current through a quasi one-dimensional magnetic wire of ¯nite length suspended be- tween two bulk magnets is determined by the spin conductance which remains ¯nite in the ballistic limit due to contact resistance. Magnetization currents produce an electric ¯eld and hence can be measured directly. For magnetiza- tion transport in an external electric ¯eld, phenomena analogous to the Hall e®ect emerge.

We present a theoretical framework for higher-order correlation functions involving multiple times and multiple points in a classical, many-body system. Such higher-order correlation functions have attracted much interest recently in the context of various forms of multidimensional spectroscopy, and have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses. The theoretical formalism is based upon projection operator techniques that are used to isolate the slow time evolution of dynamical variables by expanding the slowly evolving component of arbitrary variables in an infinite, "multilinear" basis composed of the products of slow variables of the system. Using the formalism, a formally exact mode coupling theory is derived for multiple-point and multiple-time correlation functions. The resulting expressions for higher-order correlation functions are made tractable by applying a rigorous perturbation scheme, called the N-ordering method, which is exact for systems with finite correlation lengths in the thermodynamic limit. The theory is contrasted with standard mode coupling theories in which the noise or fluctuating force appearing in the generalized Langevin equation is assumed to be Gaussian, and it is demonstrated that the non-Gaussian nature of the fluctuating forces leads to important contributions to higher-order correlation functions. Finally, the higher-order correlation functions are evaluated analytically for an ideal gas system for which it is shown that the mode coupling theory is exact.

We discuss the problem of deriving Lévy diffusion, in the form of a Lévy walk, from a densityapproach, namely using a Liouville equation. We address this problem for a case that hasalready been discussed using the method of the continuous time random walk, andconsequently an approach based on trajectory dynamics rather than density timeevolution. We show that the use of the Liouville equation requires the knowledgeof the correlation functions of the fluctuation that generates the Lévy diffusion.We benefit from the results of earlier work proving that the correlation functionis not factorized as in the Poisson case. We show that the Liouville equationgenerates a long-time diffusion whose probability distribution density keeps amemory of the detailed form of the fluctuation correlation function, and not onlyof its long-time inverse power law structure. Although the main result of thispaper rests on an approximate expression for higher-order correlation functions, itbecomes exact in the long-time limit. Thus, we argue that it explains the failure toderive Lévy diffusion from the Liouville equation, thereby supporting the claimthat there exists a conflict between trajectory and density approaches in thiscase.

We have developed new measurement methods for the dynamic light scattering in a periodically stationary system such as the quasi-elastic light scattering with the sinusoidal (ac) electric field (QELS-SEF). Whereas the usual dynamic light scattering is measured in the stationary system, the scattered light intensity in QELS-SEF obeys a nonstationary stochastic process, more specifically, a periodically stationary one. Hence, the autocorrelation function or the power spectrum in QELS-SEF is a bivariable function of two independent times or frequencies, which cannot be measured by the conventional correlator or spectrum analyzer. We have designed a new kind of spectrum analyzer for QELS-SEF and obtained two-dimensional power spectrum of QELS-SEF by use of an extended Wiener−Khinchin theorem. Furthermore, the data analysis formula for the two-dimensional spectra is calculated to evaluate the complex electrophoretic mobility μ* and the apparent diffusion constant Dapp of polyions from the two-dimensional power spectra. They are in good agreement with the experimental results of QELS-SEF for a polystyrene latex solution.

Consider two interacting sub-systems with nonoverlapping charge distributions. How can the properties of the combined system be expressed in terms of properties of the individual sub-systems alone? This general problem appears in a wide variety of physical, chemical, and biological systems [Joester 2003, Stone 1996, Moore 2001]. In this chapter we provide a prescription for addressing this issue by the computation of (i) response functions and (ii) correlation functions of spontaneous fluctuations of relevant degrees of freedom in the individual sub-systems. The computation of response and correlation functions is greatly simplified by using the density matrix in Liouville space [Mukamel 1995]. Hilbert and Liouville spaces offer very different languages for the description of nonlinear response. Computing dynamical observables in terms of the wavefunction in Hilbert space requires both forward and backward propagations in time. In contrast, the density matrix calculated in Liouville space should only be propagated forward. The choice is between following only the ket, moving it both forward and backward, or following the joint forward dynamics of the ket and the bra. Artificial time variables (Keldysh loops) commonly used in many-body theory [Haug 1996] are connected with the wavefunction. The density matrix which uses the real laboratory time throughout the calculation offers a more intuitive picture. Wavefunction-based theories calculate transition amplitudes, which by themselves are not observable. The density matrix, on the other hand directly calculates physical observables. Moreover, dephasing processes (damping of off-diagonal elements of the density matrix caused by phase fluctuations) which are naturally included into the Liouville space formulation may not be described in terms of the wavefunction. The “causality paradox” of TDDFT [Gross 1996] can be clearly resolved in Liouville space [Mukamel 2005]. In this chapter we present a method for expressing the joint response of two interacting sub-systems in terms of their correlation and response functions. This factorization appears quite naturally in Liouville space. The p order response of the individual systems is a linear combination of 2 distinct (p + 1)-point correlation functions known as Liouville space pathways [Mukamel 1995], which differ by whether the interaction at each time

A simple approximate analytical expression for the direct correlation function of a hard-sphere fluid in D dimensions within the Percus-Yevick equation is proposed. The approximation exactly reproduces the well-known results for D = 1 and D = 3, while for D = 2 it compares well to the numerical results for densities not very close to the freezing density.

A set of unit root tests are applied to test the existence of long-run real interest rate parity among the G-10 countries over the period 1971M1 to 2007M2. Rather than trusting the asymptotic distributions, this article uses simulation techniques to establish the small sample distributions of these tests, conditional on the stationary and nonstationary processes. The empirical results indicate that the tests have stable finite-sample sizes and higher size-adjusted powers such that the two estimated processes can be distinguished from each other. Thus, for six of the nine countries, their series are more likely to come from the estimated Autoregressive (AR) stationary process than from the nonstationary process. Noticeably, the testing results are rather different from those using the asymptotic distributions, in which only three countries support the real interest rate parity.