Lévy diffusion: the density versus the trajectory approach

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.

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 kinetic equation for a nonuniform plasma is derived from the Liouville equation by a general diagram technique. It describes the evolution of a small spatial inhomogeneity in a plasma whose velocity distribution is far from equilibrium (and hence time-dependent). The equation is valid for short and long times, within the ring approximation. Its explicit form is obtained by the exact closed solution of a singular integral equation. The kinetic equation is non-Markoffian and, contrary to the corresponding homogeneous equation, keeps a trace of this character even in the limit of long times. Only when the velocity distribution and the two-body correlation function reach thermal equilibrium does the equation reduce to a Markoffian limit. The latter is identical with the kinetic equation derived earlier by Guernsey. The treatment of unstable inhomogeneous plasmas is briefly indicated.

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

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.

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.

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.

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.

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.

We have measure the correlation of microwave field and intensity versus polarization of the source and detector in a ensemble of randomly positioned alumina spheres. Unlike correlation functions in space, frequency, and time, for which the corresponding parameter may be unbounded and the field correlation function vanishes asymptotically with increasing parameter shift, the polarization variable is of finite range and the field correlation function vanishes when either the source or detector is rotated by 90 degrees. This facilitates an unambiguous experimental separation of the intensity correlation function into three components, with a form that is independent of the closeness to the localization threshold. The polarization correlation function depends upon the underlying field correlation function in a manner analogous to that found for the intensity correlation function with displacement [1]. The first component of the intensity correlation function is the product of the square of the field correlation function with shift of either source or detector and dominates intensity fluctuations. The field correlation function is simply the cosine of the angular shift of the polarization of the electromagnetic field. The second component is associated with long-range intensity correlation. It is proportional to the sum of the square of the field correlation function with respect to variation of the source and detector. The third component of the polarization correlation function gives rise to universal conductance fluctuations. It is the sum of a multiplicative and additive term involved in the short and long-range contributions to the correlation function and also includes a constant term, which is independent of polarization. Thus, the functional form of the intensity correlation function with polarization is remarkably simple. It is determined exclusively by the cosine of the angular shift of the polarization of the electric field at the source and detector.

A new chiral two-matrix theory for Dirac spectra with imaginary chemical potential

Two-nucleon correlation function in a thermalized dense medium

Dynamics of the one-dimensional transverse Ising model

Violation of the quantum regression theorem and the Leggett–Garg inequality in an exactly solvable model