Weak Turbulence and Quasilinear Diffusion for Relativistic Wave-Particle Interactions Via a Markov Approach

We derive weak turbulence and quasilinear models for relativistic charged particle dynamics in pitch-angle and energy space, due to interactions with electromagnetic waves propagating (anti-)parallel to a uniform background magnetic field. We use a Markovian approach that starts from the consideration of single particle motion in a prescribed electromagnetic field. This Markovian approach has a number of benefits, including: 1) the evident self-consistent relationship between a more general weak turbulence theory and the standard resonant diffusion quasilinear theory (as is commonly used in e.g. radiation belt and solar wind modeling); 2) the general nature of the Fokker-Planck equation that can be derived without any prior assumptions regarding its form; 3) the clear dependence of the form of the Fokker-Planck equation and the transport coefficients on given specific timescales. The quasilinear diffusion coefficients that we derive are not new in and of themselves, but this concise derivation and discussion of the weak turbulence and quasilinear theories using the Markovian framework is physically very instructive. The results presented herein form fundamental groundwork for future studies that consider phenomena for which some of the assumptions made in this manuscript may be relaxed.

Dynamics and Merger Rate of Primordial Black Holes in a Cluster

The PBH clusters can be sources of gravitational waves, and the merger rate depends on the spatial distribution of PBHs in the cluster which changes over time. It is well known that gravitational collisional systems experience the core collapse that leads to significant increase of the central density and shrinking of the core. After core collapse, the cluster expands almost self-similarly (i.e., density profile extends in size without changing its shape). These dynamic processes affect the merger rate of PBHs. In this paper, the dynamics of the PBH cluster is considered using the Fokker–Planck equation. We calculate the merger rate of PBHs on cosmic time scales and show that its time dependence has a unique signature. Namely, it grows by about an order of magnitude at the moment of core collapse which depends on the characteristics of the cluster, and then decreases according to the dependence R∝t−1.48. It was obtained for monochromatic and power-law PBH mass distributions with some fixed parameters. Obtained results can be used to test the model of the PBH clusters via observation of gravitational waves at high redshift.

Active Brownian motion with speed fluctuations in arbitrary dimensions: exact calculation of moments and dynamical crossovers

We consider the motion of an active Brownian particle with speed fluctuations in d-dimensions in the presence of both translational and orientational diffusion. We use an Ornstein–Uhlenbeck process for active speed generation. Using a Laplace transform approach, we describe and use a Fokker–Planck equation-based method to evaluate the exact time dependence of all relevant dynamical moments. We present explicit calculations of several such moments and compare our analytical predictions against numerical simulations to demonstrate and analyze the dynamical crossovers, determined by the orientational persistence of activity, speed fluctuation and relaxation. The kurtosis of displacement shows positive and negative deviations from a Gaussian behavior at intermediate times depending on the dominance of speed and orientational fluctuations, respectively.

Одномерное квазилинейное уравнение для описания генерации токов увлечения в плазме токамака геликонами

A quasi-linear equation which allows describing evolution of electron distribution function and generation of non-inductive currents by helicons is obtained. It is shown that in the analysed case the Fokker-Planck equation can be approximated by a one-dimensional equation in the longitudinal electron velocity space with a diffusion coefficient proportional to the helicon power absorbed by electrons due to Landau damping.

Rate of Entropy Production in Stochastic Mechanical Systems

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

Multiphoton Absorption Simulation of Sapphire Substrate under the Action of Femtosecond Laser for Larger Density of Pattern-Related Process Windows

It is essential to develop pattern-related process windows on substrate surface for reducing the dislocation density of wide bandgap semiconductor film growth. For extremely high instantaneous intensity and excellent photon absorption rate, femtosecond lasers are currently being increasingly adopted. However, the mechanism of the femtosecond laser developing pattern-related process windows on the substrate remains to be further revealed. In this paper, a model is established based on the Fokker–Planck equation and the two-temperature model (TTM) equation to simulate the ablation of a sapphire substrate under the action of a femtosecond laser. The transient nonlinear evolutions such as free electron density, absorption coefficient, and electron–lattice temperature are obtained. This paper focuses on simulating the multiphoton absorption of sapphire under femtosecond lasers of different wavelengths. The results show that within the range of 400 to 1030 nm, when the wavelength is large, the number of multiphoton required for ionization is larger, and wider and shallower ablation pits can be obtained. When the wavelength is smaller, the number of multiphoton is smaller, narrower and deeper ablation pits can be obtained. Under the simulation conditions presented in this paper, the minimum ablation pit depth can reach 0.11 μm and the minimum radius can reach 0.6 μm. In the range of 400 to 1030 nm, selecting a laser with a shorter wavelength can achieve pattern-related process windows with a smaller diameter, which is beneficial to increase the density of pattern-related process windows on the substrate surface. The simulation is consistent with existing theories and experimental results, and further reveals the transient nonlinear mechanism of the femtosecond laser developing the pattern-related process windows on the sapphire substrate.

A phenomenological approach to anomalous transport in complex or disordered media

We aim to derive a phenomenological approach to link the theories of anomalous transport governed by fractional calculus and stochastic theory with the conductivity behavior governed by the semi-empirical conductivity formalism involving Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami type conductivity equations. We want to determine the anomalous transport processes in the amorphous semiconductors and insulators by developing a theoretical approach over some mathematical instruments and methods. In this paper, we obtain an analytical expression for the average behavior of conductivity in complex or disordered media via using the fractional-stochastic differential equation, the Fourier-Laplace transform, some natural boundary-initial conditions, and familiar physical relations. We start with the stochastic equation of motion called the Langevin equation, develop its equivalent master equation called Klein-Kramers or Fokker-Planck equation, and consider the time-fractional generalization of the master equation. Once we derive the fractional master equation, then determine the expressions for the mean value of the variables or observables through some calculations and conditions. Finally, we use these expressions in the current density relation to obtain the average conductivity behavior.

Hypothetical Control of Fatal Quarrel Variability

Wars, terrorist attacks, as well as natural catastrophes typically result in a large number of casualties, whose distributions have been shown to belong to the class of Pareto’s inverse power laws (IPLs). The number of deaths resulting from terrorist attacks are herein fit by a double Pareto probability density function (PDF). We use the fractional probability calculus to frame our arguments and to parameterize a hypothetical control process to temper a Lévy process through a collective-induced potential. Thus, the PDF is shown to be a consequence of the complexity of the underlying social network. The analytic steady-state solution to the fractional Fokker-Planck equation (FFPE) is fit to a forty-year fatal quarrel (FQ) dataset.

The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.