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Results for “Fokker-Planck diffusion”

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 55 records · Page 3
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NASA NTRS1972

Solar flare particle propagation: Comparision of a new analytic solution with spacecraft measurements

An analytic solution was obtained to the complete Fokker-Planck equation for solar flare particle propagation including the effects of convection, energy-change, corotation, and diffusion. It is assumed that the particles are injected impulsively at a single point in space, and that a boundary exists beyond which the particles are free to escape. Several solar flare particle events were observed with solar and galactic cosmic ray experiment aboard OGO 6. Detailed comparisons of the predictions of the solution with observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times. The solution also yields a time evolution for the vector anisotropy which agrees well with reported observations.

Lupton, J. E.↗
NASA NTRS1979

Comptonization of X-rays by low-temperature electrons

A method is described for calculating the spectrum that results from the Compton scattering of a monochromatic source of X-rays by low-temperature electrons, both for initial-value relaxation problems and for steady-state spatial diffusion problems. The method gives an exact solution of the inital-value problem for evolution of the spectrum in an infinite homogeneous medium if Klein-Nishina corrections to the Thomson cross section are neglected. This, together with approximate solutions for problems in which Klein-Nishina corrections are significant and/or spatial diffusion occurs, shows spectral structure near the original photon wavelength that may be used to infer physical conditions in cosmic X-ray sources. Explicit results, shown for examples of time relaxation in an infinite medium and spatial diffusion through a uniform sphere, are compared with results obtained by Monte Carlo calculations and by solving the appropriate Fokker-Planck equation.

Illarionov, A.↗
NASA NTRS1985

Pitch Angle Distributions of Solar Energetic Particles and the Local Scattering Properties of the Interplanetary Magnetic Field

An approximate solution of the Fokker-Planck equation containing pitch angle scattering and adiabatic focusing is discussed. For modest focusing effects the omnidirectional density obeys an ordinary diffusion equation with a modified diffusion coefficient. The anisotropic part of the distribution function is properly normalized and split into an even and an odd part. The even part is determined by the ratio between the scattering mean free path and the focusing length and by the degree of polarization of the magnetic field fluctuations. The odd part is determined by the deviation of the pitch angle scattering from isotropic scattering. The method supplies a powerful tool to obtain the local characteristics of pitch angle scattering. It is insensitive to long lasting solar injections and to moderate radial variations of the mean free path. The method is applied to solar particle events observed on Helio-1 and -2.

Beeck, J.↗
NASA NTRS1975

Modulation of low-energy cosmic rays

The relation between the diffusion coefficient of cosmic rays in the solar wind and the power spectrum of interplanetary magnetic field fluctuations, established in recent theories, is tested directly for low energy protons (below 80 MeV). In addition, an attempt is made to determine whether the particles are scattered by magnetic field discontinuities or by fluctuations between discontinuities. Predictions of a perturbation solution of the Fokker-Planck equation are compared with observations of the cosmic ray radial gradient. It is found that at energies between 40 and 80 MeV, galactic cosmic ray protons respond to changes in the predicted diffusion coefficients (i.e., the relationship under consideration holds at these low energies). The relation between changes in the proton flux and modulation parameters is best when the contribution of discontinuities is subtracted, which means that scattering is caused by fluctuations between discontinuities. There appears to be no distinct relation between changes in the modulation parameters and changes in the intensity of 20 to 40 MeV protons.

Sari, J. W.↗
NASA NTRS1974

Jupiter's radiation belts

Fluxes of electrons and protons in Jupiter's radiation belts are calculated with the source (radial diffusion inward from the solar wind) and the loss (synchroton radiation). The calculations are tested against the measured radio-wave wavelength distribution, the radio-wave distribution with distance from Jupiter, and the degree of polarization of the radio waves. The Fokker-Planck equation is solved by using the method of Farley and Walt (1971) with the fixed flux at the outer boundary and the zero flux at Jupiter's surface. It is found, in agreement with Brice and McDonough (1973), Jacques and Davis (1973), and Birmingham et al. (1974), that the usual magnetic and electric diffusions, which vary as L to the 10th power and L to the 6th power, respectively, are insufficient to furnish the required electrons and that the diffusion driven by ionospheric winds of Brice and McDonough is strong enough to furnish Jupiter's belts. An additional loss mechanism close to Jupiter is required to remove the electrons, particularly those at low energies.

Stansberry, K. G.↗
NASA NTRS1974

The Fokker-Planck coefficient for pitch-angle scattering of cosmic rays

For the case of homogeneous, isotropic magnetic field fluctuations, it is shown that most theories which are based on the quasi-linear and adiabatic approximations yield the same integral for the Fokker-Planck coefficient for the pitch-angle scattering of cosmic rays. For example, despite apparent differences, the theories due to Jokipii and to Klimas and Sandri yield the same integral. It is also shown, however, that this integral in most cases has been evaluated incorrectly in the past. For small pitch angles, the errors in previous evaluations are fortuitously of minor importance. For large pitch angles, however, these errors become more significant; and for pitch angles of 90 deg, the actual Fokker-Planck coefficient contains a delta function which has been overlooked in the past. The implications of these corrections on the possibility of relating cosmic-ray diffusion coefficients to observed properties of the interplanetary magnetic field are discussed.

Fisk, L. A.↗
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 NTRS1975

Cosmic ray positron and negatron spectra between 20 and 800 MeV measured in 1974

A balloon-borne spark chamber magnetic spectrometer was used to measure separate spectra of positrons and negatrons in two flights during summer, 1974. The total electron flux is about 0.3 m(-2) s(-1) sr(-1) MeV(-1) between 70 and 800 MeV, and increases toward lower energies. The positron spectrum decreases sharply toward lower energies from a value of about 0.08 m(-2) s(-1) sr(-1) MeV(-1) at 650 MeV, and only upper limits are obtained for positrons below 200 MeV. At energies above 180 MeV, the spherically symmetric Fokker-Planck equation provides reasonable fits to both the positron and total electron data. At energies below 180 MeV the data are consistent with a continuation of the same diffusion coefficient and local source of negatrons, or a change in the diffusion coefficient to a constant value.

Hartman, R. C.↗
NASA NTRS1974

Mirroring within the Fokker-Planck formulation of cosmic ray pitch angle scattering in homogeneous magnetic turbulence

The Fokker-Planck coefficient for pitch angle scattering, appropriate for cosmic rays in homogeneous, stationary, magnetic turbulence, is computed from first principles. No assumptions are made concerning any special statistical symmetries the random field may have. This result can be used to compute the parallel diffusion coefficient for high energy cosmic rays moving in strong turbulence, or low energy cosmic rays moving in weak turbulence. Becuase of the generality of the magnetic turbulence which is allowed in this calculation, special interplanetary magnetic field features such as discontinuities, or particular wave modes, can be included rigorously. The reduction of this results to previously available expressions for the pitch angle scattering coefficient in random field models with special symmetries is discussed. The general existance of a Dirac delta function in the pitch angle scattering coefficient is demonstrated. It is proved that this delta function is the Fokker-Planck prediction for pitch angle scattering due to mirroring in the magnetic field.

Goldstein, M. L.↗
NASA NTRS1987

The effect of neutrino transport on the collapse of iron stellar cores

A multigroup flux-limited diffusion approximation to neutrino transport, correct to first order in material velocities, is described. The role of neutrino-electron scattering in determining core deleptonization is studied using a modified Fokker-Planck approximation in which adjacent neutrino energy groups are coupled. The moderate deleptonizaton obtained during infall restricts homologous core masses to less than 0.8 solar mass. With 0.6-0.8 solar mass of outer core to traverse, nuclear dissociation alone stalls the shock waves produced at core bounce. In addition, neutrino energy losses remove at least another 4 x 10 to the 51st ergs. In no case is matter ejected promptly as a direct result of the initial shock wave.

Myra, E. S.↗
NASA NTRS1988

Analytical solutions of a class of multidimensional Fokker-Planck equations

Analytical representations of the solutions of Kolmogorov's forward (i.e., Fokker-Planck) and backward partial differential equations are obtained by employing Girsanov's (1960) transformation. The equations considered are restricted to those with the drift vector being the gradient of a function and the diffusion matrix being nonsingular and dependent upon time only.

Zhang, Weijian↗
NASA NTRS1992

Tidal evolution of globular clusters. I - Method

Tidal evolution of globular clusters is regulated by both Galactic tidal effects and internal relaxation processes. In order to investigate the tidal evolution of globular clusters, a numerical scheme which utilizes a Fokker-Planck approach as well as direct numerical integration of the restricted three-body problem is developed. In the inner regions of the cluster, stellar orbits are mapped with the cluster's gravitational potential and orbit-averaged diffusion coefficients. In the outer regions, the Galactic tidal field is explicitly included in the direct orbital integration. This method is presented here with some tests on King-Michie models.

Oh, K. S.↗
DOE OSTI2024

Ensemble variational Fokker-Planck methods for data assimilation

Particle flow filters solve Bayesian inference problems by smoothly transforming a set of particles into samples from the posterior distribution. Particles move in state space under the flow of an McKean-Vlasov-Itˆo process. This work introduces the Variational Fokker-Planck (VFP) framework for data assimilation, a general approach that includes previously known particle flow filters as special cases. The McKean-Vlasov-Itˆo process that transforms particles is defined via an optimal drift that depends on the selected diffusion term. It is established that the underlying probability density - sampled by the ensemble of particles - converges to the Bayesian posterior probability density. For a finite number of particles the optimal drift contains a regularization term that nudges particles toward becoming independent random variables. Based on this analysis, we derive computationally-feasible approximate regularization approaches that penalize the mutual information between pairs of particles, and avoid particle collapse. Moreover, the diffusion plays a role akin to a particle rejuvenation approach that aims to alleviate particle collapse. The VFP framework is very flexible. Different assumptions on prior and intermediate probability distributions can be used to implement the optimal drift, and localization and covariance shrinkage can be applied to alleviate the curse of dimensionality. A robust implicit-explicit method is discussed for the efficient integration of stiff McKean- Vlasov-Itˆo processes. Here, the effectiveness of the VFP framework is demonstrated on three progressively more challenging test problems, namely the Lorenz ’63, Lorenz ’96 and the quasi-geostrophic equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1975

A dynamic model for the time evolution of the modulated cosmic ray spectrum

A recently developed model predicts an energy dependent phase lag in the modulated cosmic ray density U(t) given by U(t) approximately equal to US (t - tau) where US is the solution to the Fokker-Planck equation under time independent conditions and tau is the average time spent by particles inside the modulating region. The delay times tau are functions of modulating parameters R (the radius of the modulating cavity), V (the solar wind velocity), and K (the effective average diffusion-coefficient which is a function of energy). This model is applied to predict the time evolution of the modulated cosmic ray proton spectrum over a simulated solar cycle. A modulation produced mostly by varying R over the solar cycle is less consistent with the observations.

Ogallagher, J. J.↗
NASA NTRS1992

Kinetic description of electron beams in the solar chromosphere

We formulate the relativistic Fokker-Plank equation for a beam of accelerated electrons interacting with a partially ionized plasma. In our derivation we conserved those terms contributing to velocity diffusion and found that this effect cannot be neglected a priori. We compute the terms accounting for elastic and inelastic collisions with neutral hydrogen and helium. Collisions with neutral hydrogen are found to be dominant throughout the chromosphere, except at the uppermost layers close to the transition region. As an application, we compute the loss of energy and momentum for a power-law beam impinging on the solar chromosphere, for a particular case in which the Fokker-Planck equation can be integrated analytically. We find that most of the beam energy is deposited in a relatively thin region of the chromosphere, a result which is largely insensitive to the theoretical method employed to compute the energy deposition rate.

Gomez, Daniel O.↗
NASA NTRS1995

Stochastic charging of dust grains in planetary rings: Diffusion rates and their effects on Lorentz resonances

Dust grains in planetary rings acquire stochastically fluctuating electric charges as they orbit through any corotating magnetospheric plasma. Here we investigate the nature of this stochastic charging and calculate its effect on the Lorentz resonance (LR). First we model grain charging as a Markov process, where the transition probabilities are identified as the ensemble-averaged charging fluxes due to plasma pickup and photoemission. We determine the distribution function P(t;N), giving the probability that a grain has N excess charges at time t. The autocorrelation function tau(sub q) for the strochastic charge process can be approximated by a Fokker-Planck treatment of the evolution equations for P(t; N). We calculate the mean square response to the stochastic fluctuations in the Lorentz force. We find that transport in phase space is very small compared to the resonant increase in amplitudes due to the mean charge, over the timescale that the oscillator is resonantly pumped up. Therefore the stochastic charge variations cannot break the resonant interaction; locally, the Lorentz resonance is a robust mechanism for the shaping of etheral dust ring systems. Slightly stronger bounds on plasma parameters are required when we consider the longer transit times between Lorentz resonances.

Schaffer, L.↗