arXiv subjects

J. W. Burby

Publications and source records attributed to J. W. Burby.

At least 19 recordsLinked to original sources

arXiv2026

Diagnosing symplecticity in simulations of high-dimensional Hamiltonian systems

Integrals of the Liouville $1$-form, known as the first Poincar{é} integral invariant, provide a computable figure of merit for monitoring the conservation of symplecticity in the numerical integration of Hamiltonian systems. For smooth loop data, these integrals may be approximated with spectral convergence in the number of sample points, with rates limited by regularity. We devise a numerical integral invariant diagnostic for checking preservation of symplecticity in particle-in-cell (PIC) kinetic plasma simulation codes. As a first application of this diagnostic tool, we check the preservation of symplecticity in symplectic electrostatic particle-in-cell (PIC) methods. Surprisingly, such PIC methods fail to have symplectic time-advance maps if the charge is interpolated to the grid using linear shape functions, as is commonly done in practice. It is found that at least quadratic interpolation is needed to avoid this failure of symplecticity preservation.

physics.plasm-ph↗

arXiv2026

A Structure-Preserving Decorated Particle Method for the Vlasov-Poisson System

We revisit the Scovel-Weinstein framework (Scovel & Weinstein, CPAM 1994) for reducing the Vlasov-Poisson system while preserving its Hamiltonian structure. Standard particle-in-cell (PIC) algorithms approximate the distribution function by macro-particles with position and velocity. In contrast, Scovel-Weinstein decorated particles involve additional shape degrees of freedom, while maintaining a finite-dimensional reduction with Hamiltonian structure inherited from the continuum model. Although the original work established this structure three decades ago, its computational potential has remained largely unexplored. We present a practical implementation of the Scovel-Weinstein model and compare it with a standard PIC algorithm. Numerical experiments demonstrate that macro-particles in standard PIC can be replaced by far fewer decorated particles while retaining comparable accuracy. This decorated particle approach offers a new structure-preserving paradigm for kinetic plasma simulation.

math.NA↗

arXiv2026

Imposing quasineutrality on electrostatic plasmas via the Dirac theory of constraints

We present a method for imposing quasineutrality and, more generally, charge density conservation in the Vlasov-Poisson (VP) and Vlasov-Ampère (VA) systems, which describe electrostatic plasma dynamics, by applying the Dirac theory of constraints. Leveraging the Hamiltonian field formulations of the VP and VA models, we construct generalized Dirac brackets using the Dirac algorithm. The resulting constrained systems enforce charge density conservation, and consequently quasineutrality, given that the initial charge density is zero, through new advection terms in the Vlasov equations involving generalized-force terms, while the electric field is eliminated from the constrained Vlasov dynamics. To verify charge density conservation we conduct one-dimensional numerical experiments using a semi-Lagrangian method, demonstrating that the enforcement of the quasineutrality constraint significantly modifies the dynamics. This approach enables us to identify the forces required to enforce quasineutrality, offering a systematic way to assess the validity of the quasineutral approximation across different kinetic scales.

physics.plasm-ph↗

arXiv2025

Structure-Preserving Neural Ordinary Differential Equations for Stiff Systems

Neural ordinary differential equations (NODEs) are an effective approach for data-driven modeling of dynamical systems arising from simulations and experiments. One of the major shortcomings of NODEs, especially when coupled with explicit integrators, is its long-term stability, which impedes their efficiency and robustness when encountering stiff problems. In this work we present a structure-preserving NODE approach that learns a transformation into a system with a linear and nonlinear split. It is then integrated using an exponential integrator, which is an explicit integrator with stability properties comparable to implicit methods. We demonstrate that our model has advantages in both learning and deployment over standard explicit or even implicit NODE methods. The long-time stability is further enhanced by the Hurwitz matrix decomposition that constrains the spectrum of the linear operator, therefore stabilizing the linearized dynamics. When combined with a Lipschitz-controlled neural network treatment for the nonlinear operator, we show the nonlinear dynamics of the NODE are provably stable near a fixed point in the sense of Lyapunov. For high-dimensional data, we further rely on an autoencoder performing dimensionality reduction and Higham's algorithm for the matrix-free application of the matrix exponential on a vector. We demonstrate the effectiveness of the proposed NODE approach in various examples, including the Robertson chemical reaction problem and the Kuramoto-Sivashinky equation.

math.NA↗

arXiv2025

Exact expressions for nonperturbative guiding center theory in symmetric fields

We apply a recently-developed nonperturbative guiding center formalism to charged particle dynamics in fields with two-parameter continuous symmetry groups. This entails finding exact constants of motion, valid in the nonperturbative regime, that agree with Kruskal's adiabatic invariant series to all orders in the perturbative regime, when the field scale length is large compared with a typical gyroradius. We demonstrate that the nonperturbative guiding center model makes exact predictions in these cases, even though it eliminates the cyclotron timescale, thereby establishing a theoretical baseline for performance of the nonperturbative formalism.

physics.plasm-ph↗

arXiv2025

Hamiltonian formulation of the quasineutral Vlasov-Poisson system

Slow manifold reduction and the theory of Poisson-Dirac submanifolds are used to deduce a Hamiltonian formulation for a quasineutral limit of the planar, collisionless, magnetized Vlasov-Poisson system. Motion on the slow manifold models plasma dynamics free of fast Langmuir oscillations. Preservation of quasineutrality requires the bulk plasma flow is incompressible. The electric field is determined by counterbalancing plasma stresses that would otherwise produce compression. The Hamiltonian structure for the quasineutral model synthesizes well-known Poisson brackets for incompressible fluids and collisionless kinetic equations.

math-ph↗

arXiv2025

Poisson-Dirac Submanifolds as a Paradigm for Imposing Constraints in Non-dissipative Plasma Models

We present a generalization of Dirac constraint theory based on the theory of Poisson-Dirac submanifolds. The theory is formulated in a coordinate-free manner while simultaneously relaxing the invertibility condition as seen in standard Dirac constraint theory. We illustrate the the method with two examples: elimination of the electron number density using Gass' Law and ideal MHD as a slow manifold constraint in the ideal two-fluid model.

math-ph↗

arXiv2025

Non-perturbative guiding center model for magnetized plasmas

Perturbative guiding center theory adequately describes the slow drift motion of charged particles in the strongly-magnetized regime characteristic of thermal particle populations in various magnetic fusion devices. However, it breaks down for particles with large enough energy. We report on a data-driven method for learning a non-perturbative guiding center model from full-orbit particle simulation data. We show the data-driven model significantly outperforms traditional asymptotic theory in magnetization regimes appropriate for fusion-born $α$-particles in stellarators, thus opening the door to non-perturbative guiding center calculations.

physics.plasm-ph↗

arXiv2024

Characterization of admissible quasisymmetries

We solve "half" the problem of finding three-dimensional quasisymmetric magnetic fields that do not necessarily satisfy force balance. This involves determining which hidden symmetries are admissible as quasisymmetries, and then showing explicitly how to construct quasisymmetric magnetic fields given an admissible symmetry. The admissibility conditions take the form of a system of overdetermined nonlinear partial differential equations involving second derivatives of the symmetry's infinitesimal generator.

physics.plasm-ph↗

arXiv2023

Spatial dynamics formulation of magnetohydrostatics

We present a formalism for importing techniques from dynamical systems theory in the study of three-dimensional magnetohydrodynamic (MHD) equilibria. By treating toroidal angle as time, we reformulate the equilibrium equations as hydrodynamic equations on the unit disc. They satisfy a variational principle and comprise a Lie-Poisson Hamiltonian system. We use the variational principle to find conservation laws for circulation, vorticity when pressure vanishes, and energy in axisymmetric domains. Combining the Lie-Poisson structure with a construction due to Scovel-Weinstein, we develop a theory of smoothed particle magnetohydrostatics (SPMHS). SPMHS identifies a large class of exact particle-swarm solutions of regularized spatial dynamics equations. The regularization occurs at finer scales than the smallest scales within the physical purview of ideal MHD. Crucially, the SPMHS equations of motion comprise a finite-dimensional Hamiltonian system, which side steps perennial roadblocks to a satisfactory theory of three-dimensional equilibria. In large-aspect-ratio domains, we show the spatial dynamics equations comprise a fast-slow system, where fast dynamics corresponds to the elliptic part of the equilibrium equations and slow dynamics corresponds to the hyperbolic part. Because the fast dynamics is formally normally hyperbolic, Fenichel theory suggests the presence of an exact slow manifold for the spatial dynamics equations that contains all physical solutions. We formally carry out reduction to the slow manifold, obtaining a spatial dynamics formulation for equilibrium Strauss reduced MHD at leading order and a hierarchy of corrections at any desired order in perturbation theory. In large aspect-ratio domains, finding periodic solutions of the slow manifold reduced equations represents a second novel pathway to finding three-dimensional equilibria.

physics.plasm-ph↗

arXiv2023

Variable-moment fluid closures with Hamiltonian structure

Based on ideas due to Scovel-Weinstein, I present a general framework for constructing fluid moment closures of the Vlasov-Poisson system that exactly preserve that system's Hamiltonian structure. Notably, the technique applies in any space dimension and produces closures involving arbitrarily-large finite collections of moments. After selecting a desired collection of moments, the Poisson bracket for the closure is uniquely determined. Therefore data-driven fluid closures can be constructed by adjusting the closure Hamiltonian for compatibility with kinetic simulations.

physics.plasm-ph↗

arXiv2023

Isodrastic Magnetic fields for suppressing transitions in guiding-centre motion

In a magnetic field, transitions between classes of guiding-centre motion can lead to cross-field diffusion and escape. We say a magnetic field is isodrastic if guiding centres make no transitions between classes of motion. Therefore, this is an important ideal for enhancing confinement. First, we present a weak formulation, based on the longitudinal adiabatic invariant, generalising omnigenity. To demonstrate that isodrasticity is strictly more general than omnigenity, we construct weakly isodrastic mirror fields that are not omnigenous. Then we present a strong formulation that is exact for guiding-centre motion. We develop a first-order treatment of the strong version via a Melnikov function and show that it recovers the weak version. The theory provides quantification of deviations from isodrasticity that can be used as objective functions in optimal design. The theory is illustrated on some simple examples.

physics.plasm-ph↗

arXiv2022

Mean field theory for intense light-matter interactions in high energy density plasmas

We present a generalization of Vlasov-Maxwell kinetic theory that accounts for intense electromagnetic fields. A strongly-radiating, possibly optically-thick plasma is decomposed into fragments, each comprising a charged particle together with its self-generated electromagnetic field. Assuming weak inter-fragment correlations, but strong intra-fragment correlations, a mean-field evolution equation for the single-fragment distribution functional is derived. We also identify the equation's Hamiltonian formulation. By incorporating strong correlations between a charged particle and the field it generates, the new model captures the effects of strong radiation reaction non-perturbatively. The fragment kinetic formalism offers an attractive approach to modeling exotic light-matter interactions such as nonlinear and multiple Compton scattering.

physics.plasm-ph↗

arXiv2021

Nearly-periodic maps and geometric integration of noncanonical Hamiltonian systems

M. Kruskal showed that each continuous-time nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. When the nearly-periodic system is also Hamiltonian, Noether's theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analoue of Kruskal's theory. Nearly-periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a $U(1)$-action. When the limiting rotation is non-resonant, these maps admit formal $U(1)$ symmetries to all orders in perturbation theory. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, we prove that the formal $U(1)$ symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether's theorem. When the unperturbed $U(1)$-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

math.DS↗

arXiv2021

Normal stability of slow manifolds in nearly-periodic Hamiltonian systems

M. Kruskal showed that each nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly-invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly-periodic Hamiltonian systems on barely symplectic manifolds -- manifolds equipped with closed, non-degenerate $2$-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivatives of the well-known adiabatic invariant. We use these results to investigate the problem of embedding guiding center dynamics of a magnetized charged particle as a slow manifold in a nearly-periodic system. We prove that one previous embedding, and two new embeddings enjoy long-term normal stability, and thereby strengthen the theoretical justification for these models.

math.DS↗

arXiv2021

Improved accuracy in degenerate variational integrators for guiding center and magnetic field line flow

First-order accurate degenerate variational integration (DVI) was introduced in C. L. Ellison et. al, Phys. Plasmas 25, 052502 (2018) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introducing second order accurate DVI schemes, both with and without non-uniform time stepping. We show that it is not in general possible to construct a second order scheme with a preserved two-form by composing a first order scheme with its adjoint, and discuss the conditions under which such a composition is possible. We build two classes of second order accurate DVI schemes. We test these second order schemes numerically on two systems having noncanonical variables, namely the magnetic field line and guiding center systems. Variational integration for Hamiltonian systems with nonuniform time steps, in terms of an extended phase space Hamiltonian, is generalized to noncanonical variables. It is shown that preservation of proper degeneracy leads to single-step methods without parasitic modes, i.e. to non-uniform time step DVIs. This extension applies to second order accurate as well as first order schemes, and can be applied to adapt the time stepping to an error estimate.

physics.plasm-ph↗

arXiv2020

Fast neural Poincaré maps for toroidal magnetic fields

Poincaré maps for toroidal magnetic fields are routinely employed to study gross confinement properties in devices built to contain hot plasmas. In most practical applications, evaluating a Poincaré map requires numerical integration of a magnetic field line, a process that can be slow and that cannot be easily accelerated using parallel computations. We show that a novel neural network architecture, the HénonNet, is capable of accurately learning realistic Poincaré maps from observations of a conventional field-line-following algorithm. After training, such learned Poincaré maps evaluate much faster than the field-line integration method. Moreover, the HénonNet architecture exactly reproduces the primary physics constraint imposed on field-line Poincaré maps: flux preservation. This structure-preserving property is the consequence of each layer in a HénonNet being a symplectic map. We demonstrate empirically that a HénonNet can learn to mock the confinement properties of a large magnetic island by using coiled hyperbolic invariant manifolds to produce a sticky chaotic region at the desired island location. This suggests a novel approach to designing magnetic fields with good confinement properties that may be more flexible than ensuring confinement using KAM tori.

physics.plasm-ph↗

arXiv2020

Generalized Grad-Shafranov equation for non-axisymmetric MHD equilibria

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad-Shafranov equation, which is a single elliptic PDE in two space dimensions. By revealing a hidden symmetry, we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad-Shafranov equation. In contrast to solutions of the classical Grad-Shafranov equation, solutions of he generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad-Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

physics.plasm-ph↗