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Results for “FOKKER-PLANCK EQUATION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1966

Cosmic-ray propagation. I.

Charged particle motion in random magnetic field, describing time evolution of particle distribution in pitch angle and position in terms of Fokker- Planck coefficients

INTERPLANETARY MAGNETIC FIELD↗
NASA NTRS1967

Dynamical friction.

Chandrasekhar dynamical friction theory for star in cluster adapted to Fokker-Plank electron in two-component plasma theory and including multiple encounters

STAR CLUSTER↗
NASA NTRS1974

Quasi-linear theory via the cumulant expansion approach

The cumulant expansion technique of Kubo was used to derive an intergro-differential equation for f , the average one particle distribution function for particles being accelerated by electric and magnetic fluctuations of a general nature. For a very restricted class of fluctuations, the f equation degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory for this limited class of fluctuations. For more physically realistic fluctuations, however, quasi-linear theory is at best approximate.

Jones, F. C.↗
NASA NTRS1975

When is quasi-linear theory exact

We use the cumulant expansion technique of Kubo (1962, 1963) to derive an integrodifferential equation for the average one-particle distribution function for particles being accelerated by electric and magnetic fluctuations of a general nature. For a very restricted class of fluctuations, the equation for this function degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory only for this limited class of fluctuations.

Jones, F. C.↗
NASA NTRS1978

Closure of a kinetic model of plasma in strong turbulence by relaxation

A Fokker-Planck kinetic equation for a turbulent plasma is derived by a repeated cascade decomposition. Calculation of the propagator and the kinetic equation determine the transport coefficients (diffusivity and turbulent viscosity) by means of a closure based on a relaxation procedure governing the approach to equilibrium. The k to the minus third power spectral law is obtained, which governs the coupling between the velocity and the electrostatic field fluctuations.

Tchen, C. M.↗
NASA NTRS1980

Costas loop tracking performance in the presence of antenna switching transients

The present paper examines the impact of phase, amplitude and Doppler transients on the tracking performance of a Costas loop, both in the presence and absence of noise; the loop's lock maintenance capability is also examined. Specific considerations involve analyzing second order loop transient behavior in the absence of noise and the presence of the three transients noted above; here, the operational differential equations are formulated and numerically solved. In the context of noise presence, a first order loop is treated in conjunction with amplitude and phase transients; toward this end the time dependent version of the Fokker-Planck partial differential equation is developed and numerically solved. Selected performance curves describe second order loop behavior in the presence of transients, whose values are of varying severity. First order loop behavior in the presence of noise is characterized by the illustration of suitably defined time dependent 'truncated probability density functions' and associated loss of lock probabilities.

Weinberg, A.↗
NASA NTRS1978

Analysis of dynamic system response to product random processes

The response of dynamic systems to the product of two independent Gaussian random processes is developed by use of the Fokker-Planck and associated moment equations. The development is applied to the amplitude modulated process which is used to model atmospheric turbulence in aeronautical applications. The exact solution for the system response is compared with the solution obtained by the quasi-steady approximation which omits the dynamic properties of the random amplitude modulation. The quasi-steady approximation is valid as a limiting case of the exact solution for the dynamic response of linear systems to amplitude modulated processes. In the nonlimiting case the quasi-steady approximation can be invalid for dynamic systems with low damping.

Sidwell, K.↗
NASA NTRS1975

A B-B-G-K-Y framework for fluid turbulence

A kinetic theory for fluid turbulence is developed from the Liouville equation and the associated BBGKY hierarchy. Real and imaginary parts of Fourier coefficients of fluid variables play the roles of particles. Closure is achieved by the assumption of negligible five-coefficient correlation functions and probability distributions of Fourier coefficients are the basic variables of the theory. An additional approximation leads to a closed-moment description similar to the so-called eddy-damped Markovian approximation. A kinetic equation is derived for which conservation laws and an H-theorem can be rigorously established, the H-theorem implying relaxation of the absolute equilibrium of Kraichnan. The equation can be cast in the Fokker-Planck form, and relaxation times estimated from its friction and diffusion coefficients. An undetermined parameter in the theory is the free decay time for triplet correlations. Some attention is given to the inclusion of viscous damping and external driving forces.

Montgomery, D.↗
NASA NTRS1974

Boltzmann equation for the electron gas of a nondegenerate plasma

The collision terms in the Boltzmann equation associated with various processes are derived. For processes having a Fokker-Planck (F-P) limit, the associated F-P operator is derived by means of physical arguments to determine the form of the operator; its multiplying constant is fixed by calculating the total energy exchange rate and comparing with the rate computed by other means. In this manner, the F-P operator is derived for electron-ion scattering, electron-electron scattering in the high-velocity limit, electron-atom elastic scattering, Compton scattering, and the high-velocity limit for inelastic scattering. Other processes considered are bremsstrahlung, radiative-recombination, photoionization, collisional ionization of atoms, and suprathermal-particle ionization of atoms.

Gould, R. J.↗
DOE OSTI2025

Preserving nonlinear constraints in variational flow filtering data assimilation

Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP)—a generic particle flow filtering framework—is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1976

Cosmic-ray transport theory and out-of-the-ecliptic exploration

The reasons for studying cosmic-ray transport theory are summarized and the fundamentally three-dimensional nature of the process is pointed out. It is shown that observations in the solar ecliptic plane cannot unambiguously test transport theories since the solutions to the transport equations depend critically on boundary conditions and variation of parameters such as diffusion tensor out of the ecliptic. Sample calculations (Fokker-Planck coefficient) are shown which illustrate the problem. It is concluded that out-of-the-ecliptic observations are essential to further test transport theory.

Jokipii, J. R.↗
NASA NTRS1981

Analytical description of charged particle transport along arbitrary guiding field configurations

Aspects of charged particle transport theory have been studied by Earl (1973, 1974, 1976) in a number of investigations. The investigations were conducted under the assumption that particles propagate along a DC guiding field with negligible perpendicular transport and that pitch-angle scattering by random fields can be described by the Fokker-Planck formalism. The present study provides a generalized and simplified treatment of focused transport. The first step is taken with respect to a perturbation analysis by which the Boltzmann equation is solved in terms of successively improved approximations to its exact solution. The first approximation is the sum of isotropic pseudodiffusive and anisotropic supercoherent components whose coefficients must satisfy two coupled partial differential equations. The normalization constant K and the pseudodiffusive velocity, which appear in the first-order transport equations, are given by simple integrals over the pitch angle.

Earl, J. A.↗