Escape times for rigid Brownian rotators in a bistable potential from the time evolution of the Green function and the characteristic time of the probability evolution

The greatest relaxation time for an assembly of three- dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions (averages of the Legendre polynomials) of the system. This is accomplished by studying the entire time evolution of the Green function (transition probability) by expanding the time dependent distribution as a Fourier series and proceeding to the zero frequency limit of the Laplace transform of that distribution. The procedure is entirely analogous to the calculation of the characteristic time of the probability evolution (the integral of the configuration space probability density function with respect to the position co-ordinate) for a particle undergoing translational diffusion in a potential; a concept originally used by Malakhov and Pankratov (Physica A 229 (1996) 109). This procedure allowed them to obtain exact solutions of the Kramers one-dimensional translational escape rate problem for piecewise parabolic potentials. The solution was accomplished by posing the problem in terms of the appropriate Sturm-Liouville equation which could be solved in terms of the parabolic cylinder functions. The method (as applied to rotational problems and posed in terms of recurrence relations for the decay functions, i.e., the Brinkman approach c.f. Blomberg, Physica A 86 (1977) 49, as opposed to the Sturm-Liouville one) demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function (albeit coupled to all the others in non-linear fashion via the underlying recurrence relation) is governed by a sum of decay functions. The method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non-axially symmetric potentials, where the distribution function is governed by two state variables. (C) 2001 Elsevier Science B.V. All rights reserved.

Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

A variational approach to the relaxation of single domain magnetic particles based on Brown's model

Brown's model for the relaxation of the magnetization of a single domain ferromagnetic particle is considered. This model results in the Fokker-Planck equation of the process. The solution of this equation in the cases of most interest is non- trivial. The probability density of orientations of the magnetization in the Fokker-Planck equation can be expanded in terms of an infinite set of eigenfunctions and their corresponding eigenvalues where these obey a Sturm-Liouville type equation. A variational principle is applied to the solution of this equation in the case of an axially symmetric potential. The first (non-zero) eigenvalue, corresponding to the largest time constant, is considered. From this we obtain two new results. Firstly, an approximate minimising trial function is obtained which allows calculation of a rigorous upper bound. Secondly, a new upper bound formula is derived based on the Euler-Lagrange condition. This leads to very accurate calculation of the eigenvalue but also, interestingly, from this, use of the simplest trial function yields an equivalent result to the correlation time of Coffey et at. and the integral relaxation time of Garanin. (C) 2004 Elsevier B.V. All rights reserved.

Ferroelectricity and isotope effects in hydrogen-bonded KDP crystals

Based on an accurate first principles description of the energetics in H-bonded potassium-dihydrogen-phosphate crystals, we conduct a first study of nuclear quantum effects and of the changes brought about by deuteration. Tunneling is allowed only for clusters involving correlated protons and heavy ion displacements, the main effect of deuteration being a depletion of the proton probability density at the O-H-O bridge center, which in turn weakens its proton-mediated covalent bonding. The ensuing lattice expansion couples self-consistently with the proton off-centering, thus explaining both the giant isotope effect and its close connection with geometrical effects.

Spatially Correlated Multiple-Ananna Channel Capacity Distributions

In this paper, we investigate the capacity of multiple-input multiple-output (MIMO) wireless communication systems over spatially correlated Rayleigh distributed flat fading channels with complex Gaussian additive noise. Specifically, we derive the probability density function of the mutual information between transmitted and received complex signals of MIMO systems. Using this density we derive the closed-form ergodic capacity (mean), delay-limited capacity, capacity variance and outage capacity formulas for spatially correlated channels and then evaluate these formulas numerically. Numerical results show how the channel correlation degrades the capacity of MIMO communication systems. We also show that the density of mutual information of correlated/uncorrelated MIMO systems can be approximated by a Gaussian density with derived mean and variance, even for a finite number of inputs and outputs.

Spectrally efficient cooperative diversity protocol for uplink wireless transmission

A spectrally efficient cooperative protocol for uplink wireless transmission in a centralised communication system is proposed, where each of the N users play the relaying and source roles simultaneously by using superposition (SP) modulation. The probability density function of the mutual information between SP-modulated transmitted and received signals of the cooperative uplink channels is derived. Using the high-signal-to-noise ratio (SNR) approximation of this density function, the outage probability formula of the system as well as its easily computable tight upper and lower bounds are obtained and these formulas are evaluated numerically. Numerical results show that the proposed strategy can achieve around 3 dB performance gain over comparable schemes. Furthermore, the multiplexing and diversity tradeoff formula is derived to illustrate the optimal performance of the proposed protocol, which also confirms that the SP relaying transmission does not cause any loss of data rate. Moreover, performance characterisation in terms of ergodic and outage capacities are studied and numerical results show that the proposed scheme can achieve significantly larger outage capacity than direct transmission, which is similar to other cooperative schemes. The superiority of the proposed strategy is demonstrated by the fact that it can maintain almost the same ergodic capacity as the direct transmission, whereas the ergodic capacity of other cooperative schemes would be much worse.

NON-GAUSSIAN STATISTICS OF OIL PRICING TIME-SERIES: A CASE STUDY

The stochastic nature of oil price fluctuations is investigated over a twelve-year period, borrowing feedback from an existing database (USA Energy Information Administration database, available online). We evaluate the scaling exponents of the fluctuations by employing different statistical analysis methods, namely rescaled range analysis (R/S), scale windowed variance analysis (SWV) and the generalized Hurst exponent (GH) method. Relying on the scaling exponents obtained, we apply a rescaling procedure to investigate the complex characteristics of the probability density functions (PDFs) dominating oil price fluctuations. It is found that PDFs exhibit scale invariance, and in fact collapse onto a single curve when increments are measured over microscales (typically less than 30 days). The time evolution of the distributions is well fitted by a Levy-type stable distribution. The relevance of a Levy distribution is made plausible by a simple model of nonlinear transfer. Our results also exhibit a degree of multifractality as the PDFs change and converge toward to a Gaussian distribution at the macroscales.

Causality, apparent 'superluminality' and reshaping in barrier penetration

We consider tunneling of a nonrelativistic particle across a potential barrier. It is shown that the barrier acts as an effective beam splitter which builds up the transmitted pulse from the copies of the initial envelope shifted in the coordinate space backward relative to the free propagation. Although along each pathway causality is explicitly obeyed, in special cases reshaping can result an overall reduction of the initial envelope, accompanied by an arbitrary coordinate shift. In the case of a high barrier the delay amplitude distribution (DAD) mimics a Dirac delta function, the transmission amplitude is superoscillatory for finite momenta and tunneling leads to an accurate advancement of the (reduced) initial envelope by the barrier width. In the case of a wide barrier, initial envelope is accurately translated into the complex coordinate plane. The complex shift, given by the first moment of the DAD, accounts for both the displacement of the maximum of the transmitted probability density and the increase in its velocity. It is argued that analyzing apparent

Anisotropic seismic inversion using a multigrid Monte Carlo approach

We propose a new approach for the inversion of anisotropic P-wave data based on Monte Carlo methods combined with a multigrid approach. Simulated annealing facilitates objective minimization of the functional characterizing the misfit between observed and predicted traveltimes, as controlled by the Thomsen anisotropy parameters (epsilon, delta). Cycling between finer and coarser grids enhances the computational efficiency of the inversion process, thus accelerating the convergence of the solution while acting as a regularization technique of the inverse problem. Multigrid perturbation samples the probability density function without the requirements for the user to adjust tuning parameters. This increases the probability that the preferred global, rather than a poor local, minimum is attained. Undertaking multigrid refinement and Monte Carlo search in parallel produces more robust convergence than does the initially more intuitive approach of completing them sequentially. We demonstrate the usefulness of the new multigrid Monte Carlo (MGMC) scheme by applying it to (a) synthetic, noise-contaminated data reflecting an isotropic subsurface of constant slowness, horizontally layered geologic media and discrete subsurface anomalies; and (b) a crosshole seismic data set acquired by previous authors at the Reskajeage test site in Cornwall, UK. Inverted distributions of slowness (s) and the Thomson anisotropy parameters (epsilon, delta) compare favourably with those obtained previously using a popular matrix-based method. Reconstruction of the Thomsen epsilon parameter is particularly robust compared to that of slowness and the Thomsen delta parameter, even in the face of complex subsurface anomalies. The Thomsen epsilon and delta parameters have enhanced sensitivities to bulk-fabric and fracture-based anisotropies in the TI medium at Reskajeage. Because reconstruction of slowness (s) is intimately linked to that epsilon and delta in the MGMC scheme, inverted images of phase velocity reflect the integrated effects of these two modes of anisotropy. The new MGMC technique thus promises to facilitate rapid inversion of crosshole P-wave data for seismic slownesses and the Thomsen anisotropy parameters, with minimal user input in the inversion process.

Distribution of the Demmel Condition Number of Wishart Matrices

This paper studies the Demmel condition number of Wishart matrices, a quantity which has numerous applications to wireless communications, such as adaptive switching between beamforming and diversity coding, link adaptation, and spectrum sensing. For complex Wishart matrices, we give an exact analytical expression for the probability density function (p.d.f.) of the Demmel condition number, and also derive simplified expressions for the high tail regime. These results indicate that the condition of complex Wishart matrices is dominantly decided by the difference between the matrix dimension and degree of freedom (DoF), i.e., the probability of drawing a highly ill conditioned matrix decreases considerably when the difference between the matrix dimension and DoF increases. We further investigate real Wishart matrices, and derive new expressions for the p.d.f. of the smallest eigenvalue, when the difference between the matrix dimension and DoF is odd. Based on these results, we succeed to obtain an exact p.d.f. expression for the Demmel condition number, and simplified expressions for the high tail regime.

Hyperfine interaction in the ground state of the negatively charged nitrogen vacancy center in diamond

The N-14, N-15, and C-13 hyperfine interactions in the ground state of the negatively charged nitrogen vacancy (NV-) center have been investigated using electron-paramagnetic-resonance spectroscopy. The previously published parameters for the N-14 hyperfine interaction do not produce a satisfactory fit to the experimental NV- electron-paramagnetic-resonance data. The small anisotropic component of the NV- hyperfine interaction can be explained from dipolar interaction between the nitrogen nucleus and the unpaired-electron probability density localized on the three carbon atoms neighboring the vacancy. Optical spin polarization of the NV- ground state was used to enhance the electron-paramagnetic-resonance sensitivity enabling detailed study of the hyperfine interaction with C-13 neighbors. The data confirmed the identification of three equivalent carbon nearest neighbors but indicated the next largest C-13 interaction is with six, rather than as previously assumed three, equivalent neighboring carbon atoms.

Nakagami-m Phase Model: Further Results and Validation

This paper contributes to and expands on the Nakagami-m phase model. It derives exact, closed-form expressions for both the phase cumulative distribution function and its inverse. In addition, empirical first- and second-order statistics obtained from measurements conducted in a body-area network scenario were used to fit the phase probability density function, the phase cumulative distribution function, and the phase crossing rate expressions. Remarkably, the unlikely shapes of the phase statistics, as predicted by the theoretical formulations, are actually encountered in practice.

Ergodic Capacity of Cognitive TAS/GSC Relaying in Nakagami-m Fading Channels

We examine the impact of transmit antenna selection with receive generalized selection combining (TAS/GSC) for cognitive decode-and-forward (DF) relaying in Nakagami-m fading channels. We select a single transmit antenna at the secondary transmitter which maximizes the receive signal-to-noise ratio (SNR) and combine a subset of receive antennas with the largest SNRs at the secondary receiver. In an effort to assess the performance, we first derive the probability density function and cumulative distribution function of the end-to-end SNR using the moment generating function. We then derive new exact closed-form expression for the ergodic capacity. More importantly, by deriving the asymptotic expression for the high SNR approximation of the ergodic capacity, we gather deep insights into the high SNR slope and the power offset. Our results show that the high SNR slope is 1/2 under the proportional interference power constraint. Under the fixed interference power constraint, the high SNR slope is zero.

Improving signal reliability for indoor off–body communications using spatial diversity at the base station

In this paper, we investigate the potential improvement in signal reliability for indoor off-body communications when using spatial diversity at the base station. In particular, we utilize two hypothetical indoor base stations operating at 5.8 GHz each featuring four antennas which are spaced at either half- or one-wavelength apart. Three on-body locations are considered along with four types of user movement. The cross-correlation between the received signal envelopes observed at each base station antenna element was calculated and found to be always less than 0.5. Selection, maximal ratio, and equal gain combining of the received signal has shown that the greatest improvement is obtained when the user is mobile, with a maximum diversity gain of 11.34 dB achievable when using a four branch receiver. To model the fading envelope obtained at the output of the virtual combiners, we use diversity specific, theoretical probability density functions for multi-branch receivers operating in Nakagami-m fading channels. It is shown that these equations provide an excellent fit to the measured channel data.

Shadowed fading in body–to–body communications channels in an outdoor environment at 2.45 GHz

In this paper a number of outdoor body-to-body communications channels at 2.45 GHz which are deemed to be susceptible to shadowed fading are analyzed. The newlyproposed shadowed K-Il model is used to characterize thesechannels. Its probability density function is shown to provide an improved fit to the distribution of the signal fading compared to established models such as lognormal, Nakagamiand Rice.