Inhomogeneous Drift Research Articles

Inhomogeneous affine Volterra processes

We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel K(t,s) and inhomogeneous drift and diffusion coefficients b(s,Xs) and σ(s,Xs). In the case of affine b and σσT we show how the conditional Fourier–Laplace functional can be represented by a solution of an inhomogeneous Riccati–Volterra integral equation. For a kernel of convolution type K(t,s)=K¯(t−s) we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition b and σσT are affine, we prove that the conditional Fourier–Laplace functional is exponential–affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.

On the possibility of track length based Monte-Carlo algorithms for stationary drift-diffusion systems with sources and sinks

The problem of constructing Monte-Carlo solutions of drift-diffusion systems corresponding to Fokker–Planck equations with sources and sinks is revisited. Firstly, a compact formalism is introduced for the specific problem of stationary solutions. This leads to identification of the dwell time as the key quantity to characterize the system and to obtain a proper normalization for statistical analysis of numerical results. Secondly, the question of appropriate track length estimators for drift-diffusion systems is discussed for a 1D model system. It is found that a simple track length estimator can be given only for pure drift motion without diffusion. The stochastic nature of the diffusive part cannot be appropriately described by the path length of simulation particles. Further analysis of the usual situation with inhomogeneous drift and diffusion coefficients leads to an error estimate based on particle trajectories. The result for limits in grid cell size and time step used for the construction of Monte-Carlo trajectories resembles the Courant-Friedrichs-Lewy and von Neumann conditions for explicit methods.

Analytical formulation for the dielectric tensor and field equations of the inhomogeneous drift plasma cylinder in rotating magnetic field

In this procedure, the fundamental electromagnetic equations and fluid equations in a cylindrical coordinate system for a new drift plasma configuration have been analyzed. The system is a long nonhomogeneous drift plasma column, which is imbedded in a uniform transverse magnetic field rotating about the symmetric axis of the system. The elements of the dielectric permittivity tensor are obtained for a pattern propagating in an arbitrary direction, and coupling equations of fields will be derived. It will be observed that the time variable dielectric tensor can be written as non-operational Hermitian and pure spatial operational parts which satisfy the limiting special cases.

Emergence of spatiotemporal dislocation chains in drifting patterns.

One-dimensional patterns subjected to counter-propagative flows or speed jumps exhibit a rich and complex spatiotemporal dynamics, which is characterized by the perpetual emergence of spatiotemporal dislocation chains. Using a universal amplitude equation of drifting patterns, we show that this behavior is a result of a combination of a phase instability and an advection process caused by an inhomogeneous drift force. The emergence of spatiotemporal dislocation chains is verified in numerical simulations on an optical feedback system with a non-uniform intensity pump. Experimentally this phenomenon is also observed in a tilted quasi-one-dimensional fluidized shallow granular bed mechanically driven by a harmonic vertical vibration.

Multi-dimensional, mesoscopic Monte Carlo simulations of inhomogeneous reaction-drift-diffusion systems on graphics-processing units.

For many biological applications, a macroscopic (deterministic) treatment of reaction-drift-diffusion systems is insufficient. Instead, one has to properly handle the stochastic nature of the problem and generate true sample paths of the underlying probability distribution. Unfortunately, stochastic algorithms are computationally expensive and, in most cases, the large number of participating particles renders the relevant parameter regimes inaccessible. In an attempt to address this problem we present a genuine stochastic, multi-dimensional algorithm that solves the inhomogeneous, non-linear, drift-diffusion problem on a mesoscopic level. Our method improves on existing implementations in being multi-dimensional and handling inhomogeneous drift and diffusion. The algorithm is well suited for an implementation on data-parallel hardware architectures such as general-purpose graphics processing units (GPUs). We integrate the method into an operator-splitting approach that decouples chemical reactions from the spatial evolution. We demonstrate the validity and applicability of our algorithm with a comprehensive suite of standard test problems that also serve to quantify the numerical accuracy of the method. We provide a freely available, fully functional GPU implementation. Integration into Inchman, a user-friendly web service, that allows researchers to perform parallel simulations of reaction-drift-diffusion systems on GPU clusters is underway.

The dielectric tensor and field equations in the inhomogeneous cold collisionless magnetized drift plasmas with elliptical cross sections

In this Letter the dielectric permittivity tensor and field equations in multi layer cold collisionless magnetized inhomogeneous drift plasma columns with confocal elliptical cross sections are investigated. It will be shown that the dielectric tensor of each region can be written as non-operational Hermitian and pure spatial operational parts.

Effects of preexisting voids on electromigration failure of flip chip solder bumps

For the first time, the effects of preexisting voids inside the flip chip solder bump and at the interface of bump/underbump metallization (UBM) are considered on electromigration-induced failures to provide a mechanistic understanding on solder bump electrical reliability. Local peaks of current density due to preexisting voids supported by high operating temperature and temperature gradients and gradients of mechanical stress result in inhomogeneous drift of metal atoms, which leads to the formation of hillocks and more microvoids or large voids in the bump or the bump interfaces. The voids decrease the cross-section of the solder contact, which increases local current density and local resistance. This expected positive feedback cycle can eventually lead to the so-called electromigration failure. In the meantime, the increased volume of the extruded hillocks under electromigration can lead to circuit shorts between neighboring bumps, where the distance becomes closer and closer. A stochastic approach is employed to quantify the effects of various preexisting voids. It is found that preexisting voids at the interface of solder bump/UBM is more prone to growth during electromigration and therefore, a much shorter lifetime of the bump. Voids inside the bump can also speed up the failure of the solder bump and their effects depend on their location, size and total volume fraction. The simulation results can be applied for the reliable design of flip chip solder bumps by modifying the design rule with the consideration of the effects of preexisting voids.

Statistical models of chemical kinetics in liquids

Statistical methods are used increasingly in theoretical chemistry. Applications range from the use of stochastic relaxation techniques in determining minimum energy molecular configurations to dynamical Monte Carlo simulations of molecular motion in liquids. This paper focuses on diffusion-controlled reactions in radiation chemistry. Here the interest is in describing the evolution of isolated clusters, containing a few chemically active particles, resulting from the passage of ionising radiation through a liquid. The subsequent chemistry is determined by the rate at which the particles can encounter each other, pairwise, in the course of random motion. Classically the trajectories of the particles are described as sample paths of a continuous stochastic process, which in many cases can be assumed to be a diffusion. We have devised an approximate theory based on the approximation that pair distances evolve independently. The effect of this geometric distortion has been studied for small systems by simulation. The extension of this work to a wider class of realistic chemical processes poses many problems. Analytical progress is difficult and methods of simulating large spatial systems are not well developed. Problems which remain to be solved relate to the random geometry of the cluster as its constituent particles diffuse, the development of good approximations to first passage time distributions for diffusions with inhomogeneous drift and methods for the analysis of random motion in a liquid on the basis of detailed non-Markovian models of molecular movement.

A class of exact solutions of the Hasegawa-Mima equation

The existence of a new class of drift vortices in a cylindrical plasma pinch is demonstrated starting from an exact solution of the Hasegawa-Mima equation with an inhomogeneous drift velocity. The waves are moving in a circle in the azimuthal direction and represent either a single cyclonic motion or a pair of cyclonic-anticyclonic motions of unequal intensity.

Statistical Models of Chemical Kinetics in Liquids

SUMMARY Statistical methods are used increasingly in theoretical chemistry. Applications range from the use of stochastic relaxation techniques in determining minimum energy molecular configurations to dynamical Monte Carlo simulations of molecular motion in liquids. This paper focuses on diffusion-controlled reactions in radiation chemistry. Here the interest is in describing the evolution of isolated clusters, containing a few chemically active particles, resulting from the passage of ionising radiation through a liquid. The subsequent chemistry is determined by the rate at which these particles can encounter each other, pairwise, in the course of their random motion. Classically the trajectories of the particles are described as sample paths of a continuous stochastic process, which in many cases can be assumed to be a diffusion. We have devised an approximate theory based on the approximation that pair distances evolve independently. The effect of this geometric distortion has been studied for small systems by simulation. The extension of this work to a wider class of realistic chemical processes poses many problems. Analytical progress is difficult and methods of simulating large spatial systems are not well developed. Problems which remain to be solved relate to the random geometry of the cluster as its constituent particles diffuse, the development of good approximations to first passage time distributions for diffusions with inhomogeneous drift and methods for the analysis of random motion in a liquid on the basis of detailed non-Markovian models of molecular movement.

Ionenbewegung unter dem Einflu� von Umladungsst��en

The motion of ions under the influence of an electric field and a density gradient is investigated by taking into account only charge-transfer collisions between the ions and a neutral gas background. The two model cases of constant mean free path and constant mean free time are considered. For weak electric fields the Boltzmann equation can be solved using a special expansion of the velocity distribution. The expansion in Sonine polynomials as given by Chapman and Cowling does not converge. For medium and high electric fields the velocity distribution is calculated by collecting the contributions of the different paths of the ions. The inhomogeneous drift in high electric fields is investigated for a small perturbation by gradients of the electric field, the ion density and of the mean free path.