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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 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS2003

Theory of Transport of Long Polymer Molecules through Carbon Nanotube Channels

A theory of transport of long chain polymer molecules through carbon nanotube (CNT) channels is developed using Fokker-Planck equation and direct molecular dynamics (MD) simulations. The mean transport or translocation time tau is found to depend on the chemical potential energy, entropy and diffusion coefficient. A power law dependence tau approx. N(sup 2)is found where N is number of monomers in a molecule. For 10(exp 5)-unit long polyethylene molecules, tau is estimated to be approx. 1micro-s. The diffusion coefficient of long polymer molecules inside CNTs, like that of short ones, are found to be few orders of magnitude larger than in ordinary silicate based zeolite systems.

Wei, Chenyu↗
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 NTRS2009

Coupled Particle Transport and Pattern Formation in a Nonlinear Leaky-Box Model

Effects of particle-particle coupling on particle characteristics in nonlinear leaky-box type descriptions of the acceleration and transport of energetic particles in space plasmas are examined in the framework of a simple two-particle model based on the Fokker-Planck equation in momentum space. In this model, the two particles are assumed coupled via a common nonlinear source term. In analogy with a prototypical mathematical system of diffusion-driven instability, this work demonstrates that steady-state patterns with strong dependence on the magnetic turbulence but a rather weak one on the coupled particles attributes can emerge in solutions of a nonlinearly coupled leaky-box model. The insight gained from this simple model may be of wider use and significance to nonlinearly coupled leaky-box type descriptions in general.

Barghouty, A. F.↗
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 NTRS2006

Expectation-Based Control of Noise and Chaos

A proposed approach to control of noise and chaos in dynamic systems would supplement conventional methods. The approach is based on fictitious forces composed of expectations governed by Fokker-Planck or Liouville equations that describe the evolution of the probability densities of the controlled parameters. These forces would be utilized as feedback control forces that would suppress the undesired diffusion of the controlled parameters. Examples of dynamic systems in which the approach is expected to prove beneficial include spacecraft, electronic systems, and coupled lasers.

Zak, Michael↗
DOE OSTI2021

A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. In this work, we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second order convergence in space, first order convergence in time, and it is straightforward to parallelize. Applications are presented to the advection diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
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.↗
NASA NTRS1999

Dynamical Signatures of Living Systems

One of the main challenges in modeling living systems is to distinguish a random walk of physical origin (for instance, Brownian motions) from those of biological origin and that will constitute the starting point of the proposed approach. As conjectured, the biological random walk must be nonlinear. Indeed, any stochastic Markov process can be described by linear Fokker-Planck equation (or its discretized version), only that type of process has been observed in the inanimate world. However, all such processes always converge to a stable (ergodic or periodic) state, i.e., to the states of a lower complexity and high entropy. At the same time, the evolution of living systems directed toward a higher level of complexity if complexity is associated with a number of structural variations. The simplest way to mimic such a tendency is to incorporate a nonlinearity into the random walk; then the probability evolution will attain the features of diffusion equation: the formation and dissipation of shock waves initiated by small shallow wave disturbances. As a result, the evolution never "dies:" it produces new different configurations which are accompanied by an increase or decrease of entropy (the decrease takes place during formation of shock waves, the increase-during their dissipation). In other words, the evolution can be directed "against the second law of thermodynamics" by forming patterns outside of equilibrium in the probability space. Due to that, a specie is not locked up in a certain pattern of behavior: it still can perform a variety of motions, and only the statistics of these motions is constrained by this pattern. It should be emphasized that such a "twist" is based upon the concept of reflection, i.e., the existence of the self-image (adopted from psychology). The model consists of a generator of stochastic processes which represents the motor dynamics in the form of nonlinear random walks, and a simulator of the nonlinear version of the diffusion equation which represents the mental dynamics. It has been demonstrated that coupled mental-motor dynamics can simulate emerging self-organization, prey-predator games, collaboration and competition, "collective brain," etc.

Zak, M.↗
DOE OSTI2021

Transport coefficients for magnetic-field evolution in inviscid magnetohydrodynamics

The magnetized resistivity and electrothermal tensors when substituted into the induction equation lead to electrothermal magnetic field generation, resistive magnetic diffusion, and magnetic field advection due to resistivity gradients, temperature gradients and currents. Here, the advection terms driven by temperature gradient and current have cross-field components (perpendicular to both the magnetic field and the driving term) that depend on significantly modified versions of Braginskii’s transport coefficients. The improved fits to Braginskii’s coefficients given by Epperlein and Haines and Ji and Held give physically incorrect results for cross-field advection at small Hall parameters (product of cyclotron frequency and collision time). The errors in Epperlein and Haines’ fits are particularly severe, giving increasing advection velocities below a Hall parameter of one when they should decrease linearly to zero. Epperlein and Haines’ fits can also give erroneous advection terms due to variations in effective atomic number. The only serious error in Braginskii’s fits is an overestimate in advection due to perpendicular resistivity. New fits for the cross-field advection terms are obtained from a direct numerical solution of the Fokker-Planck equation and Ji and Held’s higher order expansion approach that are continuous functions of the effective atomic number.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Dynamics of heavy quarks in strongly coupled $\mathcal{N}$ = 4 SYM plasma

We calculate the probability distribution P(k) for a heavy quark with velocity v propagating through strongly coupled N = 4 SYM plasma in the ’t Hooft limit (N c → ∞, λ = g 2 N c → ∞) at a temperature T to acquire a momentum k due to interactions with the plasma. This distribution encodes the well-known drag coefficient η D and the transverse and longitudinal momentum diffusion coefficients κ T and κ L . The jet quenching parameter $\hat{q}$ can be extracted from P(k) for v = 1. Going beyond these known Gaussian characteristics of P(k), our calculation determines all of the higher order and mixed moments to leading order in 1/$\sqrt{λ}$ for the first time. These non-Gaussian features of P(k) include qualitatively novel correlations between longitudinal energy loss and transverse momentum broadening at nonzero v. We show that all higher moments scale characteristically with an effective temperature of the boosted plasma in the heavy quark rest frame, and we demonstrate that these non-Gaussian characteristics can be sizable in magnitude and even dominant in physically relevant situations. We use these results to derive a Kolmogorov equation for the evolution of the probability distribution for the total momentum of a heavy quark that propagates through strongly coupled plasma. This evolution equation accounts for all higher order correlations between transverse momentum broadening and longitudinal energy loss, which we have calculated from first principles. It reduces to a Fokker-Planck equation when truncated to only include the effects of η D , κ T and κ L . Remarkably, while heavy quarks do not reach kinetic equilibrium with the plasma if evolved with this Fokker-Planck equation, by showing that the Boltzmann distribution is a static solution of the all-order Kolmogorov equation that we have derived we demonstrate that heavy quarks do reach kinetic equilibrium if evolved with this equation. Our results thus provide a dynamically complete framework for understanding the thermalization of a heavy quark that may be initially far from equilibrium in the strongly coupled N = 4 SYM plasma — as well as new insight into heavy quark transport and equilibration in quark-gluon plasma.

Holography and Hydrodynamics↗