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Discrete probability law obtained by assuming Poisson variable parameter is distributed according to half-normal probability law
Numerical integration of Poisson equation for computer simulated plasma
Interpolation of Poisson equation by method adjusting equations to insure existence of discrete solution
Abstract [111]‐Oriented perovskite oxide films exhibit unique interfacial and symmetry breaking effects, which are promising for novel quantum materials as topological insulators and polar metals. However, due to strong polar mismatch and complex structural reconstructions on (111) surfaces/interfaces, it is still challenging to grow high quality [111] perovskite heterostructures, let alone explore the as‐resultant physical properties. Here, the fabrication of ultrathin PbTiO 3 films grown on a SrTiO 3 (111) substrate with atomically defined surfaces, by pulsed laser deposition, is reported. High‐resolution scanning transmission electron microscopy and X‐ray diffraction reveal that the as‐grown [111]PbTiO 3 films are coherent with the substrate and compressively strained along all in‐plane directions. In contrast, the out‐of‐plane lattices are almost unchanged compared with that of bulk PbTiO 3 , resulting in a 4% contraction in unit cell volume and a nearly zero Poisson's ratio. Ferroelectric displacement mapping reveals a monoclinic distortion within the compressed [111]PbTiO 3 , with a polarization larger than 50 µC cm −2 . The present findings, as further corroborated by phase field simulations and first principle calculations, differ significantly from the common [001]‐oriented films. Fabricating oxide films through [111] epitaxy may facilitate the formation of new phase components and exploration of novel physical properties for future electronic nanodevices.
The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier‐Stokes equations needs robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothers, in conjunction with pointwise Jacobi or additive Schwarz methods (ASMs), prove to be an effective smoother. Other polynomial smoothers, however, may provide superior convergence to the multigrid preconditioner. The authors compare the standard Chebyshev polynomial smoothers to both the novel fourth‐kind Chebyshev polynomial smoothers proposed by Lottes as well as smoothers based on the polynomial of best uniform approximation to as proposed by Kraus, Vassilevski, and Zikatanov. At the cost of symmetry, further improvements may be made. For example, a order polynomial smoother on both sides of the V‐cycle may be substituted with an order polynomial smoother on one side at no additional cost. The choice of omitting the postsmoother in favor of higher‐order polynomial presmoothing is advantageous in cases where the multigrid approximation property constant is large. The authors consider a 2D model problem based on finite differences to motivate the choice of polynomial smoother, order, and whether to apply postsmoothing for the target application of high‐order ‐geometric multigrid methods for GPU architectures. Results from both domains demonstrate the substantial improvement of these approaches over the standard Chebyshev polynomial smoother with a symmetric V‐cycle.
Massive scalar fields provide excellent dark matter candidates, whose dynamics are often explored analytically and numerically using nonrelativistic Schrödinger-Poisson (SP) equations in a cosmological context. In this paper, starting from the nonlinear and fully relativistic Klein-Gordon-Einstein (KGE) equations in an expanding universe, we provide a systematic framework for deriving the SP equations, as well as relativistic corrections to them, by integrating out ‘fast modes’ and including nonlinear metric and matter contributions. We provide explicit equations for the leading-order relativistic corrections, which provide insight into deviations from the SP equations as the system approaches the relativistic regime. Upon including the leading-order corrections, our equations are applicable beyond the domain of validity of the SP system, and are simpler to use than the full KGE case in some contexts. As a concrete application, we calculate the mass-radius relationship of solitons in scalar dark matter and accurately capture the deviations of this relationship from the SP system towards the KGE one.
The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L{sup 2} space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ→∞ to the strongly collisional regime τ→0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ-dependent constraint on δ ensuring exponential decay with explicit τ-dependent rates towards the stationary solution. In the strongly collisional limit τ→0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L{sup 2} space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.
This paper introduces an observation-driven (OD) longitudinal integer-valued moving average model of order 1 (INMA(1)) with COM–Poisson innovations under non-stationary moment conditions. This new longitudinal model provides lot of practical flexibility in terms of modeling a wide range of over-, under-dispersion and any mixed level of dispersion. In this set-up, the model parameters of primary interest consist of the regression and dispersion effects while the serial autocorrelation parameters are treated as nuisance. A robust Generalized Quasi-Likelihood approach is formulated to estimate the different set of parameters. The performance of the estimating algorithm is assessed via Monte Carlo experiments under various combination of the serial and dispersion values and is compared with the existing adaptive generalized method of moments. Application to two medical data: epileptic seizures and polyposis counts are also illustrated.
A parallel, relativistic, three-dimensional particle-in-cell code SPACE has been developed for the simulation of electromagnetic fields, relativistic particle beams, and plasmas. In addition to the standard second-order Particle-in-Cell (PIC) algorithm, SPACE includes efficient novel algorithms to resolve atomic physics processes such as multi-level ionization of plasma atoms, recombination, and electron attachment to dopants in dense neutral gases. SPACE also contains a highly adaptive particle-based method, called Adaptive Particle-in-Cloud (AP-Cloud), for solving the Vlasov-Poisson problems. It eliminates the traditional Cartesian mesh of PIC and replaces it with an adaptive octree data structure. The code's algorithms, structure, capabilities, parallelization strategy, and performance have been discussed. Additionally, typical examples of SPACE applications to accelerator science and engineering problems are described.
The development of an implicit, unconditionally stable, numerical method for solving the Vlasov–Poisson system in one dimension using a phase-space grid is presented. The algorithm uses the Crank–Nicolson discretization scheme and operator splitting allowing for direct solution of the finite difference equations. This method exactly conserves particle number, enstrophy and momentum. A variant of the algorithm which does not use splitting also exactly conserves energy but requires the use of iterative solvers. This algorithm has no dissipation and thus fine-scale variations can lead to oscillations and the production of negative values of the distribution function. We find that overall, the effects of negative values of the distribution function are relatively benign. We consider a variety of test cases that have been used extensively in the literature where numerical results can be compared with analytical solutions or growth rates. We examine higher-order differencing and construct higher-order temporal updates using standard composition methods.
Light-sheet microscopy has revolutionized bioimaging by enabling approximately an order of magnitude reduction in specimen irradiance compared to confocal imaging. Here, we introduce a light-sheet imaging system that enables an additional order of magnitude reduction in specimen irradiance by operating at the Poisson limit. To operate at this limit, we integrated classical illumination with single-photon detection and wavelet-based image reconstruction. This integration enabled brightness quantification and object recognition from fewer than one detected photon per image pixel, corresponding to more than 10-fold lower irradiance levels than modern systems. We demonstrate how such photon-sparse imaging can eradicate photobleaching and enable both dim and bright object imaging, thus, further enhancing the related gains of light-sheet microscopy.
We measured Electron Energy Distribution Functions (EEDFs) from below 200 eV to over 8 keV and spanning five orders-of-magnitude in intensity, produced in a low-power, RF-heated, tandem mirror discharge in the PFRC-II apparatus. The EEDF was obtained from the x-ray energy distribution function (XEDF) using a novel Poisson-regularized spectrum inversion algorithm applied to pulse-height spectra that included both Bremsstrahlung and line emissions. The XEDF was measured using a specially calibrated Amptek Silicon Drift Detector (SDD) pulse-height system with 125 eV FWHM at 5.9 keV. Finally, the algorithm is found to out-perform current leading x-ray inversion algorithms when the error due to counting statistics is high.
Large-amplitude, intermittent fluctuations are ubiquitous in the boundary region of magnetically confined plasmas and lead to detrimental plasma-wall interactions in next-generation, high duty cycle fusion power experiments. Using gas puff imaging data time series from the scrape-off layer in the Alcator C-Mod device, it is demonstrated in this work that the large-amplitude fluctuations can be described as a super-position of pulses with a fixed shape and a constant duration. By applying a new deconvolution algorithm on the data time series with a two-sided exponential pulse function, the arrival times and amplitudes of the pulses can be estimated, and the measurement time series can be reconstructed with high accuracy. The pulse amplitudes are shown to follow an exponential distribution. The waiting times between pulses are uncorrelated, their distribution has an exponential tail, and the number of arrivals is a linear function of time. This demonstrates that pulse arrivals follow a homogeneous Poisson process. Identical statistical properties apply to both ohmic and high confinement mode plasmas, clearly demonstrating the universality of the fluctuation statistics in the boundary region of Alcator C-Mod.
We describe a new method for solving the linearized 1D Vlasov–Poisson system by using properties of Cauchy-type integrals. Our method remedies critical flaws of the two standard methods, reveals a previously unrecognized Gaussian-in-time-like decay, and can also account for an externally applied electric field. The Landau approximation involves deforming the Bromwich contour around the poles closest to the real axis due to the analytically continued dielectric function, finding the long-time behavior for a stable system: Landau damping. Jackson's generalization encircles all poles while sending the contour to infinity, assuming its contribution vanishes, which is not true in general. This gives incorrect solutions for physically reasonable configurations and can exhibit pathological behavior, of which we show examples. The van Kampen method expresses the solution for a stable equilibrium as a continuous superposition of waves, resulting in an opaque integral. Case's generalization includes unstable systems and predicts a decaying discrete mode for each growing discrete mode, an apparent contradiction to both the Jackson solution and ours. We show, without imposing additional constraints, that the decaying modes are never present in the time evolution due to an exact cancellation with part of the continuum. Our solution is free of integral expressions, is obtained using algebra and Laurent series expansions, does not rely on analytic continuations, and results in a correct asymptotically convergent form in the case of infinite sums. The analysis used can be readily applied in higher-dimensional, electromagnetic systems and also provides a new technique for evaluating certain inverse Laplace transforms.
A plethora of biological, physical, and chemical phenomena involve transport of charged particles (ions). Its continuum-scale description relies on the Poisson–Nernst–Planck (PNP) system, which encapsulates the conservation of mass and charge. The numerical solution of these coupled partial differential equations is challenging and suffers from both the curse of dimensionality and difficulty in efficiently parallelizing. We present a novel particle-based framework to solve the full PNP system by simulating a drift–diffusion process with time- and space-varying drift. We leverage Green’s functions, kernel-independent fast multipole methods, and kernel density estimation to solve the PNP system in a meshless manner, capable of handling discontinuous initial states. The method is embarrassingly parallel, and the computational cost scales linearly with the number of particles and dimension. We use a series of numerical experiments to demonstrate both the method’s convergence with respect to the number of particles and computational cost vis-à-vis a traditional partial differential equation solver.
Kinetic simulations of collisionless plasmas are computationally challenging due to phase-space mixing and filamentation, resulting in fine-scale velocity structures. This study compares three methods developed to reduce artifacts related to limited velocity resolution in Hermite-based Vlasov–Poisson simulations: artificial collisions, filtering, and nonlocal closure approaches. We evaluate each method's performance in approximating the linear kinetic response function and suppressing recurrence in linear and nonlinear regimes. Numerical simulations of Landau damping demonstrate that artificial collisions, particularly higher orders of the Lenard-Bernstein collisional operator, most effectively recover the correct damping rate across a range of wavenumbers. Moreover, Hou-Li filtering and nonlocal closures underdamp high wavenumber modes in linear simulations, and the Lenard-Bernstein collisional operator overdamps low wavenumber modes in both linear and nonlinear simulations. This study demonstrates that hypercollisions offer a robust approach to kinetic simulations, accurately capturing collisionless dynamics with limited velocity resolution.
The Poisson-Boltzmann (PB) theory is widely used to depict ionic systems in a mean-field manner without considering the correlations between ions. A modified theory including ion correlation effect while retaining the simplicity of the original PB theory is desired. Although recent theoretical efforts successfully incorporate ion correlations into the PB equation as the self-energy, the results deviate significantly from our molecular dynamics (MD) simulation results due to the linear form of the self-energy. We herein reinterpret the self-energy term from a physics point of view, on the basis of which we present a new way for incorporating ion correlations into the original PB equation by utilizing the Green’s function with a non-linear form of the self-energy. Justified by our MD simulation results, our modified PB theory with non-linear ion correlations surpasses linearly formed modified PB theories.
Abstract We provide an algorithm and a publicly available code to numerically evolve multicomponent Schrödinger-Poisson (SP) systems with a SO( n ) symmetry, including attractive or repulsive self-interactions in addition to gravity. Focusing on the case where the SP system represents the non-relativistic limit of a massive vector field, non-gravitational self-interactions (in particular spin-spin interactions) introduce complexities related to mass and spin conservation which are not present in purely gravitational systems. We address them with an analytical solution for the `kick' step in the algorithm, where we are able to decouple the multicomponent system completely. Equipped with this analytical solution, the full field evolution is second order accurate, preserves spin and mass to machine precision, and is reversible. Our algorithm allows for an expanding universe relevant for cosmology, and the inclusion of external potentials relevant for laboratory settings.