Browsing by Subject "Boltzmann equation"

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Conservative spectral methods for Fokker-Planck-Landau type equations : simulations, long-time behaviour and error estimates
(2020-05-08) Pennie, Clark Alexander; Martinez Gamba, Irene, 1957-; Chen, Thomas; Morrison, Philip; Ren, Kui; Tsai, Yen-Hsi
The focus of this thesis is to investigate a conservative spectral method for solving Fokker-Planck-Landau type (F. P. L.) equations as a model for plasmas, when coupled to the Vlasov-Poisson equation in the mean-field limit, modelling particle interactions extending from Coulomb to hard sphere potentials. This study will range from numerical examples, that emphasise the strength and accuracy of the method, to a rigorous proof showing that approximations from the numerical scheme converge to analytical solutions. In particular, two sets of novel simulations are included. The first presents benchmark results of decay rates to statistical equilibrium in transport plasma models for Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The other studies the essentially unexplored phenomenon of the plasma sheath for Coulomb interactions, exhibiting the formation of a strong field due to charge separation. These topics will be arranged in three major projects: 1. Entropy decay rates for the conservative spectral scheme modelling Fokker-Planck-Landau type flows in the mean field limit. Benchmark simulations of decay rates to statistical equilibrium are created for F.P.L. equations associated to Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The qualitative decay to the equilibrium Maxwell-Boltzmann distribution is studied in detail through relative entropy for all three types of particle interactions by means of a conservative hybrid spectral and discontinuous Galerkin scheme, adapted from Chenglong Zhang’s thesis in 2014. More precisely, the Coulomb case shows that there is a degenerate spectrum, with a decay rate close to the law of two thirds predicted by upper estimates in work by Strain and Guo in 2006, while the Maxwell type and hard sphere examples both exhibit a spectral gap as predicted by Desvillettes and Villani in 2000. Such decay rate behaviour indicates that the analytical estimates for the Coulomb case is sharp while, still to this date, there is no analytical proof of sharp degenerate spectral behaviour for the F.P.L. operator. Simulations are presented, both for the space-homogeneous case of just particle potential interactions and the space-inhomogeneous case with the mean field coupling through the Poisson equation for total charges in periodic domains. New explicit derivations of spectral collisional weights are presented in the case of Maxwell type and hard sphere interactions and the stability of all three scenarios, including Coulomb interactions, is investigated. 2. Convergence and error estimates for the conservative spectral method for Fokker-Planck-Landau equations. Error estimates are rigorously derived for a semi-discrete version of the conservative spectral method for approximating the space-homogeneous F.P.L. equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in L² spaces globally time, under certain conditions. These estimates, combined with spectral projection control, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates F.P.L. equations associated to any range of potentials. 3. Modelling charge separation with the Landau equation. A model for the plasma sheath is investigated using the space-inhomogeneous linear Landau equation (namely, the F.P.L. equation associated Coulomb interactions), modelling interactions between positive and negative particles. Some theory has been established for the plasma sheath, but this is the first time that an attempt has been made to simulate it with the Landau equation. The particular design of the kinetic model is described and an attempt made to capture physical phenomena associated to separation of charges. Several parameters within the model are varied to try and explain the creation of the sheaths.
A discrete velocity method for the Boltzmann equation with internal energy and stochastic variance reduction
(2015-12) Clarke, Peter Barry; Varghese, Philip L.; Goldstein, David Benjamin; Raja, Laxminarayan; Gamba, Irene; Magin, Thierry
The goal of this work is to develop an accurate and efficient flow solver based upon a discrete velocity description of the Boltzmann equation. Standard particle based methods such as Direct Simulation Monte Carlo (DSMC) have a number of difficulties with complex and transient flows, stochastic noise, trace species, and high level internal energy states. To address these issues, a discrete velocity method (DVM) was developed which models the evolution of a flow through the collisions and motion of variable mass quasi-particles defined as delta functions on a truncated, discrete velocity domain. The work is an extension of a previous method developed for a single, monatomic species solved on a uniformly spaced velocity grid. The collision integral was computed using a variance reduced stochastic model where the deviation from equilibrium was calculated and operated upon. This method produces fast, smooth solutions of near-equilibrium flows. Improvements to the method include additional cross-section models, diffuse boundary conditions, simple realignment of velocity grid lines into non-uniform grids, the capability to handle multiple species (specifically trace species or species with large molecular mass ratios), and both a single valued rotational energy model and a quantized rotational and vibrational model. A variance reduced form is presented for multi-species gases and gases with internal energy in order to maintain the computational benefits of the method. Every advance in the method allows for more complex flow simulations either by extending the available physics or by increasing computational efficiency. Each addition is tested and verified for an accurate implementation through homogeneous simulations where analytic solutions exist, and the efficiency and stochastic noise are inspected for many of the cases. Further simulations are run using a variety of classical one-dimensional flow problems such as normal shock waves and channel flows.
Emergent phenomena in an interacting Bose gas
(2022-08) Hott, Michael Erich; Chen, Thomas (Ph. D. in mechanical engineering and Ph. D. in mathematical physics); Pavlović, Nataša; Seiringer, Robert; Gamba, Irene M
In this thesis, I intend to study the quantum fluctuation dynamics in a Bose gas on a torus Λ = (L𝕋)³ that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a Bose-Einstein condensate (BEC) with density N surrounded by thermal fluctuations with density 1, we assume that the system is described by a mean-field Hamiltonian. We extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics. Using a Fock-space approach, we provide explicit error bounds. It is known that the BEC and the HFB fluctuations both evolve at microscopic time scales t ∼ 1. Given a quasifree initial state, we determine the time evolution of the centered correlation functions ⟨a⟩, ⟨aa⟩ − ⟨a⟩², ⟨a⁺a⟩ − |⟨a⟩|² at mesoscopic time scales t ∼ λ⁻², where 0 < λ ≪ 1 denotes the size of the HFB interaction. For large but finite N, we consider both the case of fixed system size L ∼ 1, and the case L ∼ λ⁻²⁻. In the case L ∼ 1, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in ℚ; this phenomenon is reminiscent of the Talbot effect. For the case L ∼ λ⁻²⁻, we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have λ ∼ (log log N / log N)[superscript α], for different values of α > 0.
Global Lp solutions of the Boltzmann equation with an angle-potential concentrated collision kernel and convergence to a Landau solution
(2017-05-05) Akopian, Sona; Martínez Gamba, Irene, 1957-; Vasseur, Alexis F; Caffarelli, Luis A; Pavlovic, Natasa; Chen, Thomas; Ren, Kui; Morrison, Phil J
We solve the Cauchy problem associated to the space homogeneous Boltzmann equation with an angle-potential singular concentration modeling the collision kernel, proposed in 2013 by Bobylev and Potapenko. The potential under consideration ranges from Coulomb to hard spheres cases, however, the motivation of such a collision kernel is to treat the (extreme) case of Coulomb potentials, on which this particular form of collision operator is well defined. We show that the scaled angle-potential singular concentration in a grazing collisions limit makes the Boltzmann operator converge in the sense of distributions to the Landau operator acting on the Boltzmann solutions, and also that solutions of this type of Boltzmann equation converge to solutions of the Landau equation that conserve mass, momentum and energy.
Higher order extensions of the Boltzmann equation
(2020-08-04) Ampatzoglou, Ioakeim; Pavlović, Nataša; Gamba, Irene M.; Caffarelli, Luis; Chen, Thomas; Vasseur, Alexis; Morrison, Philip
This dissertation investigates extensions of the Boltzmann equation to higher order interactions and consists of two parts, which are submitted separately for publication, see [6, 4]. In the first part of the dissertation, we present a rigorous derivation of a novel kinetic equation, which we call ternary Boltzmann equation, describing the limiting behavior of a classical system of particles with three particle instantaneous interactions. Derivation of such an equation required development of new conceptual and geometrical ideas to treat interactions among three particles and their evolution in time. We also show that a symmetrized version of the ternary Boltzmann equation has the same conservation laws and entropy production properties as the classical binary operator. The superposition of this ternary equation with the classical Boltzmann equation, which we call the binary-ternary Boltzmann equation, could be understood as a step towards modeling a dense gas in non-equilibrium, since both binary and ternary interactions between particles are taken into account. In the second part of the dissertation, we show global well-posedness near vacuum for the binary-ternary Boltzmann equation for monoatomic gases with a wide range of hard and soft potentials. Well-posedness of the ternary equation for these potentials follows as a special case. This is the first global well-posedness result for the binary-ternary Boltzmann equation and for the ternary Boltzmann equation. To prove global well-posedness, we implement a Kaniel-Shinbrot iteration and related works to the ternary correction of the Boltzmann equation to approximate the solution of the nonlinear equation by monotone sequences of supersolutions and subsolutions which converge, for small initial data, to the global in time solution of the binary-ternary equation. This analysis required establishing new convolution type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution.
Mittag-Leffler moments and weighted L∞ estimates for solutions to the Boltzmann equation for hard potentials without cutoff
(2016-05) Tasković, Maja; Martínez Gamba, Irene, 1957-; Pavlović, Nataša; Caffarelli, Luis A.; Chen, Thomas; Figalli, Alessio; Morrison, Philip J.; Vasseur, Alexis F.
In this thesis we study analytic properties of solutions to the spatially homogeneous Boltzmann equation for collision kernels corresponding to hard potentials without the angular cutoff assumption, i.e. the angular part of the kernel is non-integrable with prescribed singularity rate. We study behavior in time of such solutions for large velocities i.e. their tails. We do this in two settings - L¹ and L∞. In the L¹ setting, we study Mittag-Leffler moments of solutions of the Cauchy problem under consideration. These moments, obtained by integrating the solution against a Mittag-Leffler function, are a generalization of exponential moments since Mittag-Leffler functions asymptotically behave like exponential functions. Mittag-Leffler moments can be also represented as infinite sums of renormalized polynomial moments. However, instead of considering renormaliztion by integer factorials that would lead to classical exponential moments, we renormalize by Gamma functions with non-integer arguments. By analyzing the convergence of partial sums sequences of these infinite sums, we prove the propagation and generation in time of Mittag-Leffler moments. In the case of propagation, orders of these moments depend on the singularity rate of the angular collision kernel. In the case of generation, the orders depend on the potential rate of the kernel. The proof uses a subtle combination of angular averaging and angular singularity cancellation, to show that partial sums satisfy an ordinary differential inequality with a negative term of the highest order while controlling all positive terms, whose solutions are uniformly bounded in time and number of terms. These techniques apply to both generation and propagation of Mittag-Leffler moments, with some variations depending on the case. In the L∞ setting, we prove that solutions to the Boltzmann equation that satisfy propagation in time of weightedL¹ bounds also satisfy propagation in time of weighted L∞ bounds. To emphasize that the propagation in time of weighted L∞ bounds relies on the propagation in time of weighted L¹ bounds, we express our main result using certain general weights. Consequently we apply the main result to cases of exponential and Mittag-Leffler weights, for which propagation in time of weighted L¹ bounds holds. Hence we obtain propagation in time of exponentially or Mittag-Leffler weighted L∞ bounds on the solution.
Modeling reactive rarefied systems using a novel quasi-particle Boltzmann solver
(2020-12-04) Poondla, Yasvanth Kumar; Varghese, Philip L.; Goldstein, David Benjamin, doctor of aeronautics; Raja, Laxminarayan; Liechty, Derek; Moore, Christopher
The goal of this work is to build up the capability of Quasi-Particle Simulation (QuiPS), a novel flow solver, such that it can adequately model the rarefied portion of an atmospheric reentry trajectory. Direct Simulation Monte Carlo (DSMC) is the conventional solver for such conditions, but struggles to resolve transient flows, trace species, and high level internal energy states due to stochastic noise. Quasi-Particle Simulation (QuiPS) is a novel Boltzmann solver that describes a system with a discretized, truncated velocity distribution function. The resulting fixed-velocity, variable weight quasi-particles enable smooth variation of macroscopic properties. The distribution function description enables use of a variance reduced collision model, greatly minimizing expense near equilibrium. Improvements made to the method in this work include parallelization of the collision integral routine, modification of the velocity space definition to improve performance and resolution of the distribution function, and the addition of a neutral chemistry model. Chemistry's dependence on the tail of a distribution function necessitates accurate resolution of said tail, a computationally challenging proposition. The effects of these additions are verified and studied through a number of 0D calculations, including simulations for which analytic solutions exist and model simulations intended to capture relevant physics present in more complicated problems. The explicit representation of internal distributions in QuiPS reveals some of the flaws in existing physics models. Variance reduction, a key feature of QuiPS can greatly reduce expense of multi-dimensional calculations, but is only cheaper when the gas composition is near chemical equilibrium.
The Landau limit of the Boltzmann equation and fast particle motion in a tokamak
(2021-08-12) Szczekutowicz, Anna A.; Martínez Gamba, Irene, 1957-; Arbogast, Todd; Breizman, Boris; Engquist, Bjorn; Haack, Jeffery R
I propose to study two different problems related to recent developments on kinetic theories. The first is a collisional model for Coulomb interactions, which presents an alternative to the classical Landau equation. The second project relates to particle motion in a tokamak. For the former, I derive and examine a new, modified Landau operator derived from an associated Boltzmann equation with a specific differential cross section. The improved cross section has a stronger physical meaning, with an angular cut-off condition that is dependent on the square of the relative particle speed. This differs significantly from the cut-off used in the classical formal derivation, where the cut-off is determined by a small constant. The investigation of the new models arising from the improved cross section encourages the creation of new numerical and analytical approaches. The new, modified Landau operator with velocity-dependent Coulomb logarithm derived in this work, resulting from using a velocity-dependent cutoff angular cross section for the Boltzmann operator, yields several improvements. The new, modified Landau operator will also be a better approximation of the Boltzmann operator by making fewer assumptions on the underlying two-body interaction of particles. The resulting modified Landau operator provides a different relaxation rate to equilibrium, in particular changing the rate at which the bulk and tail relax to equilibrium compared to the previous constant CL model. This change in relaxation behavior is likely to impact the related hydrodynamic transport coefficients of the model. Furthermore, the improved velocity dependent CL can be incorporated into many other existing simulation tools without considerably more effort. Additionally, Strain and Guo's analytical error estimates were supported by the numerical findings. For the latter, I introduce a mass and energy conservative Discontinuous Galerkin (DG) numerical scheme for a modified Vlasov-Maxwell model for collisionless transport. This scheme models follows the Littlejohn Lagrangian for the guiding center motion of particles moving in a periodic structure under the influence of a magnetic field, such as a tokamak. Modeled after the work of Li et al., this new system converts the classical Vlasov flow for weak electric fields, which is modeled in six dimensions for space, momenta, and time, into a straight line Hamiltonian form modeled in four dimensions of poloidal and toroidal space and momenta reference frame. The Discontinuous Galerkin scheme employs special, enhanced basis functions that were created to ensure mass and energy conservation through the application of the Hamiltonian in the new coordinates. Furthermore, L²-stability is shown for the scheme.