Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe invariance of the diffusion coefficient with the iterative operations of charged particles transport equation
62
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The Spatial Parallel Diffusion Coefficient (SPDC) is one of the important quantities describing energetic charged particle transport. There are three different definitions for the SPDC, i.e., the Displacement Variance definition $kappa_{zz}^{DV}=lim_{trightarrow t_{infty}}dsigma^2/(2dt)$, the Ficks Law definition $kappa_{zz}^{FL}=J/X$ with $X=partial{F}/partial{z}$, and the TGK formula definition $kappa_{zz}^{TGK}=int_0^{infty}dt langle v_z(t)v_z(0) rangle$. For constant mean magnetic field, the three different definitions of the SPDC give the same result. However, for focusing field it is demonstrated that the results of the different definitions are not the same. In this paper, from the Fokker-Planck equation we find that different methods, e.g., the general Fourier expansion and perturbation theory, can give the different Equations of the Isotropic Distribution Function (EIDFs). But it is shown that one EIDF can be transformed into another by some Derivative Iterative Operations (DIOs). If one definition of the SPDC is invariant for the DIOs, it is clear that the definition is also an invariance for different EIDFs, therewith it is an invariant quantity for the different Derivation Methods of EIDF (DMEs). For the focusing field we suggest that the TGK definition $kappa_{zz}^{TGK}$ is only the approximate formula, and the Ficks Law definition $kappa_{zz}^{FL}$ is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition $kappa_{zz}^{DV}$ is the invariant quantity to the kinds of the DIOs. Therefore, for spatially varying field the displacement variance definition $kappa_{zz}^{DV}$, rather than the Ficks law definition $kappa_{zz}^{FL}$ and TGK formula definition $kappa_{zz}^{TGK}$, is the most appropriate definition of the SPDCs.
rate research
Read More
It is very important to understand stochastic diffusion of energetic charged particles in non-uniform background magnetic field in plasmas of astrophysics and fusion devices. Using different methods considering along-field adiabatic focusing effect, various authors derived parallel diffusion coefficient $kappa_parallel$ and its correction $T$ to $kappa_{parallel 0}$, where $kappa_{parallel 0}$ is the parallel diffusion coefficient without adiabatic focusing effect. In this paper, using the improved perturbation method developed by He & Schlickeiser and iteration process, we obtain a new correction $T$ to $kappa_{parallel 0}$. Furthermore, by employing the isotropic pitch-angle scattering model $D_{mumu}=D(1-mu^2)$, we find that $T$ has the different sign as that of $T$. In this paper the spatial perpendicular diffusion coefficient $kappa_bot$ with the adiabatic focusing effect is also obtained.
We establish an integral variational principle for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions, for arbitrary reaction terms. This principle allows to obtain in a simple way the dependence of the speed on the Stefan constant. As an application a generalized Zeldovich-Frank-Kamenetskii lower bound for the speed, valid for monostable and combustion reaction terms, is given.
We present the results of the first Charged-Particle Transport Coefficient Code Comparison Workshop, which was held in Albuquerque, NM October 4-6, 2016. In this first workshop, scientists from eight institutions and four countries gathered to compare calculations of transport coefficients including thermal and electrical conduction, electron-ion coupling, inter-ion diffusion, ion viscosity, and charged particle stopping powers. Here, we give general background on Coulomb coupling and computational expense, review where some transport coefficients appear in hydrodynamic equations, and present the submitted data. Large variations are found when either the relevant Coulomb coupling parameter is large or computational expense causes difficulties. Understanding the general accuracy and uncertainty associated with such transport coefficients is important for quantifying errors in hydrodynamic simulations of inertial confinement fusion and high-energy density experiments.
Lie group methods are applied to the time-dependent, monoenergetic neutron diffusion equation in materials with spatial and time dependence. To accomplish this objective, the underlying 2nd order partial differential equation (PDE) is recast as an exterior differential system so as to leverage the isovector symmetry analysis approach. Some of the advantages of this method as compared to traditional symmetry analysis approaches are revealed through its use in the context of a 2nd order PDE. In this context, various material properties appearing in the mathematical model (e.g., a diffusion coefficient and macroscopic cross section data) are left as arbitrary functions of space and time. The symmetry analysis that follows is restricted to a search for translation and scaling symmetries; consequently the Lie derivative yields specific material conditions that must be satisfied in order to maintain the presence of these important similarity transformations. The principal outcome of this work is thus the determination of analytic material property functions that enable the presence of various translation and scaling symmetries within the time- dependent, monoenergetic neutron diffusion equation. The results of this exercise encapsulate and generalize many existing results already appearing in the literature. While the results contained in this work are primarily useful as phenomenological guides pertaining to the symmetry behavior of the neutron diffusion equation under certain assumptions, they may eventually be useful in the construction of exact solutions to the underlying mathematical model. The results of this work are also useful as a starting point or framework for future symmetry analysis studies pertaining to the neutron transport equation and its many surrogates.
We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable convection-diffusion-reaction systems. Some representative examples of exactly solvable systems are presented. We also describe how an equivalent convection-diffusion-reaction system can be constructed which admits the same similarity solution of another convection-diffusion-reaction system.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
