scholarly journals UBER v1.0: a universal kinetic equation solver for radiation belts

2021 ◽  
Vol 14 (9) ◽  
pp. 5825-5842
Author(s):  
Liheng Zheng ◽  
Lunjin Chen ◽  
Anthony A. Chan ◽  
Peng Wang ◽  
Zhiyang Xia ◽  
...  
Keyword(s):  
Radiation Belt ◽  
Variance Reduction ◽  
Wide Spectrum ◽  
Kinetic Equations ◽  
Adaptive Algorithms ◽  
Planck Equation ◽  
Particle Interactions ◽  
Coordinate Systems ◽  
Computational Accuracy ◽  
Numerical Solver

Abstract. Recent proceedings in radiation belt studies have proposed new requirements for numerical methods to solve the kinetic equations involved. In this article, we present a numerical solver that can solve the general form of the radiation belt Fokker–Planck equation and Boltzmann equation in arbitrarily provided coordinate systems and with user-specified boundary geometry, boundary conditions, and equation terms. The solver is based upon the mathematical theory of stochastic differential equations, whose computational accuracy and efficiency are greatly enhanced by specially designed adaptive algorithms and a variance reduction technique. The versatility and robustness of the solver are exhibited in four example problems. The solver applies to a wide spectrum of radiation belt modeling problems, including the ones featuring non-diffusive particle transport such as that arising from nonlinear wave–particle interactions.

scholarly journals UBER v1.0: A universal kinetic equation solver for radiation belts

2021 ◽  
Author(s):  
Liheng Zheng ◽  
Lunjin Chen ◽  
Anthony A. Chan ◽  
Peng Wang ◽  
Zhiyang Xia ◽  
...  
Keyword(s):  
Radiation Belt ◽  
Variance Reduction ◽  
Wide Spectrum ◽  
Kinetic Equations ◽  
Adaptive Algorithms ◽  
Planck Equation ◽  
Particle Interactions ◽  
Coordinate Systems ◽  
Computational Accuracy ◽  
Numerical Solver

Abstract. Recent proceedings in the radiation belt studies have proposed new requirements for numerical methods to solve the kinetic equations involved. In this article, we present a numerical solver that can solve the general form of radiation belt Fokker-Planck equation and Boltzmann equation in arbitrarily provided coordinate systems, and with user-specified boundary geometry, boundary conditions, and equation terms. The solver is based upon the mathematical theory of stochastic differential equations, whose computational accuracy and efficiency are greatly enhanced by specially designed adaptive algorithms and variance reduction technique. The versatility and robustness of the solver is exhibited in three example problems. The solver applies to a wide spectrum of radiation belt modeling problems, including the ones featuring non-diffusive particle transport such as that arises from nonlinear wave-particle interactions.

Stochastic Particle Dynamics

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Brownian Motion ◽  
Kinetic Theory ◽  
Stochastic Dynamics ◽  
Kinetic Equations ◽  
Planck Equation ◽  
Central Place ◽  
Conclusive Evidence ◽  
Dense Fluids ◽  
Complementary Role ◽  
Collisional Interactions

Dense fluids and liquids molecules are in constant interaction; hence, they do not fit into the Boltzmann’s picture of a clearcut separation between free-streaming and collisional interactions. Since the interactions are soft and do not involve large scattering angles, an effective way of describing dense fluids is to formulate stochastic models of particle motion, as pioneered by Einstein’s theory of Brownian motion and later extended by Paul Langevin. Besides its practical value for the study of the kinetic theory of dense fluids, Brownian motion bears a central place in the historical development of kinetic theory. Among others, it provided conclusive evidence in favor of the atomistic theory of matter. This chapter introduces the basic notions of stochastic dynamics and its connection with other important kinetic equations, primarily the Fokker–Planck equation, which bear a complementary role to the Boltzmann equation in the kinetic theory of dense fluids.

Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution

2020 ◽  
Vol 30 (04) ◽  
pp. 685-725 ◽  
Author(s):  
Giulia Furioli ◽  
Ada Pulvirenti ◽  
Elide Terraneo ◽  
Giuseppe Toscani
Keyword(s):  
Steady State ◽  
Wealth Distribution ◽  
Kinetic Equations ◽  
Planck Equation ◽  
Logarithmic Sobolev Inequality ◽  
One Dimensional ◽  
Distribution Of Wealth ◽  
Multi Agent ◽  
Fokker Planck Equations ◽  
Fokker Planck

We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

scholarly journals Verification of SpacePy's radial diffusion radiation belt model

2011 ◽  
Vol 4 (3) ◽  
pp. 2165-2197 ◽  
Author(s):  
D. T. Welling ◽  
J. Koller ◽  
E. Camporeale
Keyword(s):  
Radiation Belt ◽  
Source Term ◽  
Model Verification ◽  
Radiation Environment ◽  
Source Terms ◽  
Time Step ◽  
Manufactured Solutions ◽  
Quantitative Verification ◽  
Numerical Solver ◽  
Selection Of

Abstract. Model verification, or the process of ensuring that the prescribed equations are properly solved, is a necessary step in code development. Careful, quantitative verification guides users when selecting grid resolution and time step and gives confidence to code developers that existing code is properly instituted. This work introduces the RadBelt radiation belt model, a new, open-source version of the Dynamic Radiation Environment Assimilation Model (DREAM) and uses the Method of Manufactured Solutions (MMS) to quantitatively verify it. Order of convergence is investigated for a plethora of code configurations and source terms. The ability to apply many different diffusion coefficients, including time constant and time varying, is thoroughly investigated. The model passes all of the tests, demonstrating correct implementation of the numerical solver. The importance of DLL and source term dynamics on the selection of time step and grid size is also explored. Finally, an alternative method to apply the source term is examined to illustrate additional considerations required when non-linear source terms are used.

PLASMA WAKES DRIVEN BY NEUTRINOS, PHOTONS AND ELECTRON BEAMS

2007 ◽  
Vol 21 (03n04) ◽  
pp. 343-350
Author(s):  
R. BINGHAM ◽  
L. O. SILVA ◽  
J. T. MENDONCA ◽  
P. K. SHUKLA ◽  
W. B. MORI ◽  
...  
Keyword(s):  
Single Particle ◽  
Electron Beams ◽  
Kinetic Equations ◽  
Particle Interactions ◽  
Photon Beams ◽  
Dense Plasmas ◽  
New Class ◽  
Collective Plasma ◽  
Collective Interactions ◽  
Electron And Photon

There is considerable interest in the propagation dynamics of intense electron and photon neutrino beams in a background dispersive medium such as dense plasmas, particularly in the search for a mechanism to explain the dynamics of type II supernovae. Neutrino interactions with matter are usually considered as single particle interactions. All the single particle mechanisms describing the dynamical properties of neutrino's in matter are analogous with the processes involving single electron interactions with a medium such as Compton scattering, and Cerenkov radiation etc. However, it is well known that beams of electrons moving through a plasma give rise to a new class of processes known as collective interactions such as two stream instabilities which result in either the absorption or generation of plasma waves. Intense photon beams also drive collective interactions such as modulational type instabilities. In both cases relativistic electron beams of electrons and photon beams can drive plasma wakefields in plasmas. Employing the relativistic kinetic equations for neutrinos interacting with dense plasmas via the weak force we explore collective plasma streaming instabilities driven by Neutrino electron and photon beams and demonstrate that all three types of particles can drive wakefields.

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