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.

The role of Mittag-Leffler functions in anomalous relaxation

The inertia-corrected Debye model of rotational Brownian motion of polar molecules was generalized by Coffey et al. [Phys. Rev. E, 65, 32 102 (2002)] to describe fractional dynamics and anomalous rotational diffusion. The linear- response theory of the normalized complex susceptibility was given in terms of a Laplace transform and as a function of frequency. The angular-velocity correlation function was parametrized via fractal Mittag-Leffler functions. Here we apply the latter method and complex-contour integral- representation methods to determine the original time-dependent amplitude as an inverse Laplace transform using both analytical and numerical approaches, as appropriate. (C) 2004 Elsevier B.V. All rights reserved.

Microscopic models for dielectric relaxation in disordered systems

It is shown how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended to yield the empirical Havriliak-Negami (HN) equation of anomalous dielectric relaxation from a microscopic model based on a kinetic equation just as in the Debye model. This kinetic equation is obtained by means of a generalization of the noninertial Fokker-Planck equation of conventional Brownian motion (generally known as the Smoluchowski equation) to fractional kinetics governed by the HN relaxation mechanism. For the simple case of noninteracting dipoles it may be solved by Fourier transform techniques to yield the Green function and the complex dielectric susceptibility corresponding to the HN anomalous relaxation mechanism.

Polarization Relaxation in an Ionic Liquid Confined between Electrified Walls

The response of a room temperature molten salt to an external electric field when it is confined to a nanoslit is studied by molecular dynamics simulations. The fluid is confined between two parallel and oppositely charged walls, emulating two electrified solid-liquid interfaces. Attention is focused on structural, electrostatic, and dynamical properties, which are compared with those of the nonpolarized fluid. It is found that the relaxation of the electrostatic potential, after switching the electric field off, occurs in two stages. A first, subpicosecond process accounts for 80% of the decay and is followed by a second subdiffusive process with a time constant of 8 ps. Diffusion is not involved in the relaxation, which is mostly driven by small anion translations. The relaxation of the polarization in the confined system is discussed in terms of the spectrum of charge density fluctuations in the bulk.

Relaxation processes in room temperature ionic liquids: The case of 1-butyl-3-methyl imidazolium hexafluorophosphate

A detailed investigation on the nature of the relaxation processes occurring in a typical room temperature ionic liquid (RTIL), namely, 1-butyl-3-methyl imidazolium hexafluorophosphate ([bmim][PF6]), is reported. The study was conducted using both elastic and inelastic neutron scattering over a wide temperature range from 10 to 400 K, accessing the dynamic features of both the liquid and glassy amorphous states. In this study, the inelastic fixed energy scan technique has been applied for the first time to this class of materials. Using this technique, the existence of two relaxation processes below the glass transition and a further diffusive process occurring above the glass-liquid transition are observed. The low temperature processes are associated with methyl group rotation and butyl chain relaxation in the glassy state and have been modeled in terms of two Debye-like, Arrhenius activated processes. The high temperature process has been modeled in terms of a Kohlraush-Williams-Watts relaxation, with a distinct Vogel-Fulcher-Tamman temperature dependence. These results provide novel information that will be useful in rationalizing the observed structural and dynamical behavior of RTILs in the amorphous state.

Temperature Dependence of the Primary Relaxation in 1-Hexyl-3-methylimidazolium bis{(trifluoromethyl)sulfonyl}imide

We present results from complementary characterizations of the primary relaxation rate of a room temperature ionic liquid (RTIL), 1-hexyl-3-methylimidazolium bis{(trifluoromethyl)sulfonyl} imide, [C(6)mim][Tf2N], over a wide temperature range. This extensive data set is successfully merged with existing literature data for conductivity, viscosity, and NMR diffusion coefficients thus providing, for the case of RTILs, a unique description of the primary process relaxation map over more than 12 decades in relaxation rate and between 185 and 430 K. This unique data set allows a detailed characterization of the VTF parameters for the primary process, that are: B = 890 K, T-0 = 155.2 K, leading to a fragility index m = 71, corresponding to an intermediate fragility. For the first time neutron spin echo data from a fully deuteriated sample of RTIL at the two main interference peaks, Q = 0.76 and 1.4 angstrom(-1) are presented. At high temperature (T > 250 K), the collective structural relaxation rate follows the viscosity behavior; however at lower temperatures it deviates from the viscosity behavior, indicating the existence of a faster process.

Review of constitutive models for describing the time dependent behaviour of soft clays

Different classes of constitutive models have been proposed to capture the time-dependent behaviour of soft soil (creep, stress relaxation, rate dependency). This paper critically reviews many of the models developed based on understanding of the time dependent stress-strain-stress rate-strain rate behaviour of soils and viscoplasticity in terms of their strengths and weaknesses. Some discussion is also made on the numerical implementation aspects of these models. Typical findings from numerical analyses of geotechnical structures constructed on soft soils are also discussed. The general elastic viscoplastic (EVP) models can roughly be divided into two categories: models based on the concept of overstress and models based on non-stationary flow surface theory. Although general in structure, both categories have their own strengths and shortcomings. This review indicates that EVP analysis is yet to be vastly used by the geotechnical engineers, apparently due to the mathematical complication involved in the formulation of the constitutive models, unconvincing benefit in terms of the accuracy of performance prediction, requirement of additional soil parameter(s), difficulties in determining them, and the necessity of excessive computing resources and time. © 2013 Taylor & Francis.

A multiconfigurational time-dependent Hartree-Fock method for excited electronic states. II. Coulomb interaction effects in single conjugated polymer chains

Conjugated polymers have attracted considerable attention in the last few decades due to their potential for optoelectronic applications. A key step that needs optimisation is charge carrier separation following photoexcitation. To understand better the dynamics of the exciton prior to charge separation, we have performed simulations of the formation and dynamics of localised excitations in single conjugated polymer strands. We use a nonadiabatic molecular dynamics method which allows for the coupled evolution of the nuclear degrees of freedom and of multiconfigurational electronic wavefunctions. We show the relaxation of electron-hole pairs to form excitons and oppositely charged polaron pairs and discuss the modifications to the relaxation process predicted by the inclusion of the Coulomb interaction between the carriers. The issue of charge photogeneration in conjugated polymers in dilute solution is also addressed. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3600404]

Hybrid time dependent/independent solution for the He i line ratio temperature and density diagnostic for a thermal helium beam with applications in the scrape-off layer-edge regions in tokamaks

Spectroscopic studies of line emission intensities and ratios offer an attractive option in the\r\ndevelopment of non-invasive plasma diagnostics. Evaluating ratios of selected He I line\r\nemission profiles from the singlet and triplet neutral helium spin systems allows for simultaneous\r\nmeasurement of electron density (ne) and temperature (Te) profiles. Typically, this powerful\r\ndiagnostic tool is limited by the relatively long relaxation times of the 3S metastable term of helium\r\nthat populates the triplet spin system, and on which electron temperature sensitive lines are based.\r\nBy developing a time dependent analytical solution, we model the time evolution of the two spin\r\nsystems. We present a hybrid time dependent/independent line ratio solution that improves the\r\nrange of application of this diagnostic technique in the scrape-off layer (SOL) and edge plasma\r\nregions when comparing it against the current equilibrium line ratio helium model used at\r\nTEXTOR.