Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.     

Relaxation time for a charge carrier due to its scattering from other charge carriers in superlattices

A calculation of relaxation time for (i) electron–electron scattering in a modulation-doped superlattice of type-I and (ii) electron–electron, hole–hole and electron–hole scattering processes in a compositional superlattice of type-II has been performed, using Fermi's golden rule. As compared to a two-dimensional electron gas system, both intralayer and interlayer interactions, between charge carriers in a superlattice, contribute to relaxation time. It is found that scattering processes at all possible value of momentum transfer contribute to relaxation time, for a given value of temperature and carrier density. We further find interlayer interactions in a superlattice make a significant contribution to relaxation time. Relaxation time is found to decrease on increasing temperature, carrier density and single particle energy, in a superlattice. The computed relaxation time for an electron (hole) in a superlattice enhances on increasing the width of layer consisting of electrons (holes). The electron–hole (hole–electron) scattering process in a type-II superlattice yields maximum contribution to the relaxation time when a hole layer lies exactly in between two consecutive electron layers.     

Quasistationary distributions for autocatalytic reactions

We provide simple conditions for the existence of quasistationary distributions that can be used to describe the long-term behaviour ofopen autocatalytic reaction systems. We illustrate with reference to a particular example that the quasistationary distribution is close to the usual stationary diffusion approximation.     

Williams-watts dielectric relaxation: A fractal time stochastic process

Dielectric relaxation in amorphous materials is treated in a defect-diffusion model where relaxation occurs when a mobile defect, such as a vacancy, reaches a frozen-in dipole. The random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite. When there are many more defects than dipoles, the Williams-Watts decaying fractional exponential relaxation law is derived. The argument of the exponential is related to the number of distinct sites visited by the random walk of the defect. For the same reaction dynamics but with more traps than walkers, an algebraically decaying relaxation is found.     

First passage time and escape time distributions for continuous time random walks

We consider an arbitrary continuous time random walk (ctrw)via unbiased nearest-neighbour jumps on a linear lattice. Solutions are presented for the distributions of the first passage time and the time of escape from a bounded region. A simple relation between the conditional probability function and the first passage time distribution is analysed. So is the structure of the relation between the characteristic functions of the first passage time and escape time distributions. The mean first passage time is shown to diverge for all (unbiased)ctrw’s. The divergence of the mean escape time is related to that of the mean time between jumps. A class ofctrw’s displaying a self-similar clustering behaviour in time is considered. The exponent characterising the divergence of the mean escape time is shown to be (1−H), whereH(0<H<1) is the fractal dimensionality of thectrw.     

Advection and dispersion in time and space

Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly.     

双色噪声驱动光学双稳系统的弛豫时间研究

研究乘性及加性噪声的自关联时间τ₁和τ₂,交叉关联时间τ对光学双稳系统的弛豫时间的影响.弛豫时间T由Mei等所采用的方法得到. 经数值计算, 结果表明: 两噪声间交叉关联时间τ及乘性噪声的自关联时间τ₁延缓系统涨落的消退(T随τ及τ₁的增大而增大);T随加性噪声的自关联时间τ 关键词： 关联噪声 弛豫时间 光学双稳系统     

Relaxation phenomena of polar non-polar liquid mixtures under low and high frequency electric field

Simultaneous calculation of the dipole moment μ_j and the relaxation time τ_j of a certain number of non-spherical rigid aliphatic polar liquid molecules (j) in non-polar solvents (i) under 9.8 GHz electric field is possible from real ε′_ij and imaginary ε″_ij parts of the complex relative permittivity ε*_ij. The low frequency and infinite frequency permittivities ε_0ij and ε_∞ij measured by Purohitet al [1,2] and Srivastava and Srivastava [3] at 25, 35 and 30°C respectively are used to obtain static μ_s. The ratio of the individual slopes of imaginary σ″_ij and real σ′_ij parts of high frequency (hf) complex conductivity σ*_ij with weight fractionsw _jatw _j → 0 and the slopes of σ″_ij— σ′_ij curves for differentw _js [4] are employed to obtain τ_js. The former method is better in comparison to the existing one as it eliminates polar-polar interaction. The hf μ_js in Coulomb metre (C m) when compared with static and reported μs indicate that μ_s s favour the monomer formations which combine to form dimers in the hf electric field. The comparison among μs shows that a part of the molecule is rotating under X-band electric field [5]. The theoretical μ_theos from available bond angles and bond moments of the substituent polar groups attached to the parent molecules differ from the measured μ_js and μs to establish the possible existence of mesomeric, inductive and electromeric effects in polar liquid molecules.     

Toward an Optimal Paramagnetic Relaxation Agent for 31P NMR Spectroscopy

The dynamics of the use of relaxation agents for ³¹P NMR spectroscopy were investigated. It was found that non-lability of the associated ligand is essential, in order for line broadening to be prevented. Thus, chelates were found to be most suitable. In addition, evidence was accumulated that indicates an outer sphere coordination is significant for effective reduction of spin-lattice relaxation times (T₁'s). Finally, the magnitude of the magnetic moment, as pertains to Gd⁺³ and Fe⁺³, was examined and related to effective reduction of T₁'s.     

Limit probability distributions for an infinite-order phase transition model

The limiting probability distributions for the one-dimensional inhomogeneous spin system considered in a previous paper, which exhibits an infinite-order phase transition, are computed. It turns out that below the critical temperature or in the presence of an external magnetic field, the spins are completely polarized.     

高能不等核碰撞和弛豫时间的相对论动力论方程描述

本文用相对论动力论方程描述高能重离子碰撞时空演化,并用它分析在200A GeV的¹⁶O束流和³²S束流下,于快度中心区的末态粒子快度分布,确定了不同系统的弛豫时间.     

Solutions for a non-Markovian diffusion equation

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.     

Langevin equation for a free particle driven by power law type of noises

We study a generalized Langevin equation for a free particle driven by N internal noises. The mean square displacement and velocity autocorrelation function are derived in case of a mixture of Dirac delta, power law and Mittag-Leffler noises. Additionally, a frictional memory kernel of distributed order is considered. The long time limit and short time limit are analyzed, and the dominant contributions of noises on particle dynamics is discussed. Various different diffusive behaviors (subdiffusion, superdiffusion, normal diffusion, ultraslow diffusion) are obtained. The considered problem may be used in the theory of anomalous diffusion in complex environment.     

Anomalous diffraction approximation method for retrieval of spherical and spheroidal particle size distributions in total light scattering

Using total light scattering technique to measure the particle size distribution has advantages of simplicity in measurement principle and convenience in the optical arrangement. However, the calculation of extinction efficiency based on Mie theory for a spherical particle is expensive in both time and resources. Thus, a simple but accurate approximation formula for the exact extinction efficiency may be very desirable. The accuracy and limitations of using the anomalous diffraction approximation (ADA) method for calculating the extinction efficiency of a spherical particle are investigated. Meanwhile, the monomodal and bimodal particle size distributions of spherical particles are retrieved using the genetic algorithm in the dependent model. Furthermore, the spheroidal model in the retrieval of non-spherical particle size distribution is also discussed, which verifies the non-sphericity has a significant effect on the retrieval of particle size distribution compared with the assumed homogeneous isotropic sphere. Both numerical computer simulations and experimental results illustrate that the ADA can be successfully applied to retrieve the particle size distributions for spherical and spheroidal particles with high stability even in the presence of random noise. The method has advantages of simplicity, rapidity, and suitability for in-line particle size measurement.     

Using relaxation time approximation for collisions in laser-cluster interaction simulation

Numerical experiments were performed for cluster dynamics on a one-dimensional cluster model consisting of hydrogen atoms exposed to intense laser pulses. The algorithm deployed the relativistic equations for solving motions of both protons and electrons, respectively. In contrary to traditionally and extensively used method which treats collisions within a jumbo cluster in a statistical way, we introduced a phenomenological relaxation time parameter to cope collisions among particles in a cluster, thereby profoundly reducing computational workload while still attaining the essential physics. Positive ion’s maximum kinetic energy and maximum kinetic energy saturation with laser intensity were explored, which were found to be in fairly good accordance with experimental observations, corroborating the effectiveness of our method and inferring that this treatment is useful in numerical experiments on light-cluster interaction.     

Structure-specific magnetic field inhomogeneities and its effect on the correlation time

We describe the relationship between the correlation time and microscopic spatial inhomogeneities in the static magnetic field. The theory takes into account diffusion of nuclear spins in the inhomogeneous field created by magnetized objects. A simple general expression for the correlation time is obtained. It is shown that the correlation time is dependent on a characteristic length, the diffusion coefficient of surrounding medium, the permeability of the surface and the volume fraction of the magnetized objects. For specific geometries (spheres and cylinders), exact analytical expressions for the correlation time are given. The theory can be applied to contrast agents (magnetically labeled cells), capillary network, BOLD effect and so forth.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.     

The influence of fractality on the time evolution of the diffusion process

In the literature, the deviations from standard behaviors of the solutions of the kinetic equation and the analogous diffusion equation are put forward by investigations which are carried out in the frame of fractional mathematics and nonextensive physics. On the other hand, the physical origins of the order of derivative namely α in fractional mathematics and the entropy index q in nonextensive physics are a topic of interest in scientific media. In this study, the solutions of the diffusion equation which have been obtained in the framework of fractional mathematics and nonextensive physics are revised. The diffusion equation is solved by the cumulative diminuation/growth method which has been developed by two of the present authors and physical nature of the parameters α and q are enlightened in connection with fractality of space and the memory effect. It has been emphasized that the mathematical basis of deviations from standard behavior in the distribution functions could be established by fractional mathematics where as the physical mechanism could be revealed using the cumulative diminuation/growth method.     

Modeling of Self-Diffusion and Relaxation Time NMR in Multi-Compartment Systems

The theory of pulsed field gradient (pfg) NMR applied to molecules in cellular systems which contain different subcellular compartments separated by permeable membranes, acting as diffusion barriers, has been extended. A numerical model of restricted diffusion and magnetization relaxation behavior in pfg-CPMG NMR experiments, based on the Fick's second law of diffusion, is presented. This model is applicable to a wide range of systems and allows the exploration of temporal and spatial behavior of the magnetization with and without the influence of gradient pulses. Results of the numerical experiments show their correspondence to the previously observed ones and demonstrate the importance of the inclusion of the time domain data in analyzing diffusion measurements.