Boltzmann-Fokker-Planck Equation Research Articles

High-order diamond differencing schemes for the Boltzmann Fokker–Planck equation in 1D and 2D Cartesian geometries

This paper presents the implementation of high-order diamond differencing schemes, specifically the DD1 and DD2 schemes which are 4th- and 6th-order accurate respectively, to handle the spatial discretization of the Boltzmann Fokker–Planck equation in 1D and 2D Cartesian geometries. While being as computationally expensive as linear/quadratic discontinuous Galerkin schemes, the DD1/DD2 schemes are respectively an order more accurate. The energy deposition calculations presented in this work show that, besides providing correction to the oscillations and a reduced propensity to yield negative fluxes than the classical diamond difference closure relation, they tend to perform better than discontinuous Galerkin schemes, even in regions with abrupt variations of the solution due to rapid energy loss, to interface implying high-Z variations and to domain borders.

SU‐E‐T‐22: A Deterministic Solver of the Boltzmann‐Fokker‐Planck Equation for Dose Calculation

Purpose:The Boltzmann‐Fokker‐Planck equation (BFPE) accurately models the migration of photons/charged particles in tissues. While the Monte Carlo (MC) method is popular for solving BFPE in a statistical manner, we aim to develop a deterministic BFPE solver based on various state‐of‐art numerical acceleration techniques for rapid and accurate dose calculation.Methods:Our BFPE solver is based on the structured grid that is maximally parallelizable, with the discretization in energy, angle and space, and its cross section coefficients are derived or directly imported from the Geant4 database. The physical processes that are taken into account are Compton scattering, photoelectric effect, pair production for photons, and elastic scattering, ionization and bremsstrahlung for charged particles.While the spatial discretization is based on the diamond scheme, the angular discretization synergizes finite element method (FEM) and spherical harmonics (SH). Thus, SH is used to globally expand the scattering kernel and FFM is used to locally discretize the angular sphere. As a Result, this hybrid method (FEM‐SH) is both accurate in dealing with forward‐peaking scattering via FEM, and efficient for multi‐energy‐group computation via SH. In addition, FEM‐SH enables the analytical integration in energy variable of delta scattering kernel for elastic scattering with reduced truncation error from the numerical integration based on the classic SH‐based multi‐energy‐group method.Results:The accuracy of the proposed BFPE solver was benchmarked against Geant4 for photon dose calculation. In particular, FEM‐SH had improved accuracy compared to FEM, while both were within 2% of the results obtained with Geant4.Conclusion:A deterministic solver of the Boltzmann‐Fokker‐Planck equation is developed for dose calculation, and benchmarked against Geant4.Xiang Hong and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500).

About an H-theorem for systems with non-conservative interactions

We present some arguments in favor of an H-theorem for a generalization of the Boltzmann equation including non-conservative interactions and a linear Fokker–Planck-like thermostatting term. Such a non-linear equation describing the evolution of the single particle probability Pi(t) of being in state i at time t is a suitable model for granular gases and is referred to here as the Boltzmann–Fokker–Planck (BFP) equation. The conjectured H-functional, which appears to be non-increasing, is HC(t) = ∑iPi(t)lnPi(t)/Πi with Πi = limt→∞Pi(t), in analogy with the H-functional of Markov processes. The extension to continuous states is straightforward. A simple proof can be given for the elastic BFP equation. A semi-analytical proof is also offered for the BFP equation for so-called inelastic Maxwell molecules. Other evidence is obtained by solving particular BFP cases through numerical integration or through ‘particle schemes’ such as the direct simulation Monte Carlo.

Modeling of high energy ion implantation based on splitting of the Boltzmann transport equation

For the straight-ahead form of the Boltzmann transport equation, describing high energy heavy ion transport in matter, the splitting in the Boltzmann–Fokker–Planck equation is considered. The splitting consists of decomposition of differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. An approximation of the scattering integral with singular differential cross-section by the Fokker–Planck operator of second order, including the energy straggling term, is considered. For this form of the Fokker–Planck equation boundary conditions, preserving the number of particles in flux, and the energy dissipated in target are suggested. An effective algorithm for numerical integration of the Boltzmann–Fokker–Planck equation, which preserves the number of particles in flux, and the energy deposited in target is developed. The convergence of the implantation profile, nuclear and electronic energy deposition, calculated from the Boltzmann–Fokker–Planck equation, to the respective exact distributions, calculated from the Monte-Carlo method, was investigated for various values of the splitting parameter for the different ion-target mass ratios and for a large energy interval. For the universal potential, it is shown the existence of an optimal value of splitting parameter, at which solutions of the Boltzmann–Fokker–Planck equation accurately approximate solutions of the exact equation with minimal computational efforts.

High energy ion range and deposited energy calculation using the Boltzmann–Fokker–Planck splitting of the Boltzmann transport equation

We consider the splitting of the straight-ahead Boltzmann transport equation in the Boltzmann–Fokker–Planck equation, decomposing the differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. The convergence of implantation profile, nuclear and electronic energy depositions, calculated from the Boltzmann–Fokker–Planck equation, to the respective exact distributions, calculated from Monte-Carlo method, was exanimate in a large-energy interval for various values of splitting parameter and for different ion-target mass relations. It is shown that for the universal potential there exists an optimal value of splitting parameter, for which range and deposited energy distributions, calculated from the Boltzmann–Fokker–Planck equation, accurately approximate the exact distributions and which minimizes the computational expenses.

An analytical solution to the Boltzmann–Fokker–Planck equation for multi-component non-homogeneous plasmas

The previously obtained analytical solution to the Boltzmann equation for nonhomogeneous plasmas with relative large temperature and density gradients is generalized in the following sense. (i) The relatively simple Bhatnagar, Gross and Krook collision operator is replaced by the Fokker-Planck operator expressed in terms of relaxation rates (slowing down, energy exchange, etc.). (ii) The simple Lorentzian plasma model is replaced by a multi-component plasma model with realistic masses and temperatures.

Photon Transport Studies in Media with Strongly Anisotropic Scattering with the Boltzmann-Fokker-Planck Equation

The Boltzmann-Fokker-Planck (BFP) equation has been applied to treat the photon transport in highly anisotropic scattering media. Some benchmark cases relating to haze and cloud media have been investigated. It is found that, in general, the number of Legendre moments required for the BFP equation is a factor of --5 lower compared to that for the standard (discrete ordinates S/sub n/) approach for solving the Boltzmann equation for the comparable precision on the integral quantities like the albedo and transmission factor. But BFP is much more economical if one seeks precision on angular fluxes.

Investigations of Nuclear Elastic Scattering Effects by Means of the Boltzmann-Fokker-Planck Equation

Nuclear elastic scattering (NES) has recently been recognized as an important slowing down mechanism for fast ions injected into a background plasma. The present paper proposes an improved method to include this effect into slowing down calculations. This method consists of calculating multigroup cross-section data including the transfer matrices up to a desired degree of Legendre expansion and in supplying the data to a Boltzmann-Fokker-Planck (BFP) equation solved by a discrete ordinates scheme. The physical model of the BFP equation and the accuracy of the numerical method guarantee a good representation of NES.