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The Slowing-Down Distribution of Energetic Nuclei in a Fully Ionized Gas

Description: The steady-state energy distribution is obtained for a group of initially monoenergetic charged particles slowing down in a filly ionized gas. The distribution function is obtained analytically by solving the FokkerPlanck equation in energy space. With reference to thermonuclear reactions, the results obtained suggest that in some situations the contribution of the slowingdown distribution to the reaction rate may be significant. (auth)
Date: May 1, 1958
Creator: Kidder, R. E.
Partner: UNT Libraries Government Documents Department

Deterministic Brownian Motion

Description: The goal of this thesis is to contribute to the ambitious program of the foundation of developing statistical physics using chaos. We build a deterministic model of Brownian motion and provide a microscpoic derivation of the Fokker-Planck equation. Since the Brownian motion of a particle is the result of the competing processes of diffusion and dissipation, we create a model where both diffusion and dissipation originate from the same deterministic mechanism - the deterministic interaction of that particle with its environment. We show that standard diffusion which is the basis of the Fokker-Planck equation rests on the Central Limit Theorem, and, consequently, on the possibility of deriving it from a deterministic process with a quickly decaying correlation function. The sensitive dependence on initial conditions, one of the defining properties of chaos insures this rapid decay. We carefully address the problem of deriving dissipation from the interaction of a particle with a fully deterministic nonlinear bath, that we term the booster. We show that the solution of this problem essentially rests on the linear response of a booster to an external perturbation. This raises a long-standing problem concerned with Kubo's Linear Response Theory and the strong criticism against it by van Kampen. Kubo's theory is based on a perturbation treatment of the Liouville equation, which, in turn, is expected to be totally equivalent to a first-order perturbation treatment of single trajectories. Since the boosters are chaotic, and chaos is essential to generate diffusion, the single trajectories are highly unstable and do not respond linearly to weak external perturbation. We adopt chaotic maps as boosters of a Brownian particle, and therefore address the problem of the response of a chaotic booster to an external perturbation. We notice that a fully chaotic map is characterized by an invariant measure which is a continuous ...
Date: August 1993
Creator: Trefán, György
Partner: UNT Libraries


Description: A theoretical treatment of the loss of plasmas from the ends of magnetic mirror machines, based on the FokkerPlanck equation is given. The plasma losses are described in terms of the evolution in time of distribution functions. Some supplementary calculations based on these distribution functions were made. These include charge-exchangereaction rates and fusion-reaction rates for the DD and DT reactions. (auth)
Date: April 1, 1960
Creator: Roberts, J.E. & Carr, M.L.
Partner: UNT Libraries Government Documents Department

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

Description: The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.
Date: January 1, 2008
Creator: Densmore, Jeffery D; Warsa, James S; Lowrie, Robert B & Morel, Jim E
Partner: UNT Libraries Government Documents Department

Transition form collisional to kinetic reconnection in large-scale plasmas

Description: Using first-principles fully kinetic simulations with a Fokker-Planck collision operator, it is demonstrated that Sweet-Parker reconnection layers are unstable to a chain of plasmoids (secondary islands) for Lundquist numbers beyond S >{approx} 1000. The instability is increasingly violent at higher Lundquist number, both in terms of the number of plasmoids produced and the super-Alfvenic growth rate. A dramatic enhancement in the reconnection rate is observed when the half-thickness of the current sheet between two plasmoids approaches the ion inertial length. During this transition, the reconnection electric field rapidly exceeds the runaway limit, resulting in the formation of electron-scale current layers that are unstable to the continual formation of new plasmoids.
Date: January 1, 2009
Creator: Daughton, William S; Roytershteyn, Vadim S; Albright, Brian J; Yin, Lin; Bowers, Kevin J & Karimabadi, Homa
Partner: UNT Libraries Government Documents Department

Stochastic Cooling with Schottky Band Overlap

Description: Optimal use of stochastic cooling is essential to maximize the antiproton stacking rate for Tevatron Run II. Good understanding and characterization of the cooling is important for the optimization. The paper is devoted to derivation of the Fokker-Planck equations justified in the case of near or full Schottky base overlap for both longitudinal and transverse coolings.
Date: December 1, 2005
Creator: Lebedev, Valeri
Partner: UNT Libraries Government Documents Department

Variational correction to the FERMI beam solution

Description: We consider the time-independent, monoenergetic searchlight problem for a purely scattering, homogeneous slab with a pencil beam of nuclear particles impinging upon one surface. The scattering process is assumed sufficiently peaked in the forward direction so that the Fokker-Planck differential scattering operator can be used. Further, the slab is assumed sufficiently thin so that backscattering is negligibly small. Generally, this problem is approximated by the classic Fermi solution. A number of modifications of Fermi theory, aiming at improved accuracy, have been proposed. Here, we show that the classic Fermi solution (or any approximate solution) can I be improved via a variational formalism.
Date: October 1, 1996
Creator: Su, Bingjing & Pomraning, G.C.
Partner: UNT Libraries Government Documents Department

Discrete ordinates transport methods for problems with highly forward-peaked scattering

Description: The author examines the solutions of the discrete ordinates (S{sub N}) method for problems with highly forward-peaked scattering kernels. He derives conditions necessary to obtain reasonable solutions in a certain forward-peaked limit, the Fokker-Planck (FP) limit. He also analyzes the acceleration of the iterative solution of such problems and offer improvements to it. He extends the analytic Fokker-Planck limit analysis to the S{sub N} equations. This analysis shows that in this asymptotic limit the S{sub N} solution satisfies a pseudospectral discretization of the FP equation, provided that the scattering term is handled in a certain way (which he describes) and that the analytic transport solution satisfies an analytic FP equation. Similar analyses of various spatially discretized S{sub N} equations reveal that they too produce solutions that satisfy discrete FP equations, given the same provisions. Numerical results agree with these theoretical predictions. He defines a multidimensional angular multigrid (ANMG) method to accelerate the iterative solution of highly forward-peaked problems. The analyses show that a straightforward application of this scheme is subject to high-frequency instabilities. However, by applying a diffusive filter to the ANMG corrections he is able to stabilize this method. Fourier analyses of model problems show that the resulting method is effective at accelerating the convergence rate when the scattering is forward-peaked. The numerical results demonstrate that these analyses are good predictors of the actual performance of the ANMG method.
Date: April 1, 1998
Creator: Pautz, S.D.
Partner: UNT Libraries Government Documents Department

Numerical Methods for Stochastic Partial Differential Equations

Description: This is the final report of a Laboratory Directed Research and Development (LDRD) project at the Los Alamos National laboratory (LANL). The objectives of this proposal were (1) the development of methods for understanding and control of spacetime discretization errors in nonlinear stochastic partial differential equations, and (2) the development of new and improved practical numerical methods for the solutions of these equations. The authors have succeeded in establishing two methods for error control: the functional Fokker-Planck equation for calculating the time discretization error and the transfer integral method for calculating the spatial discretization error. In addition they have developed a new second-order stochastic algorithm for multiplicative noise applicable to the case of colored noises, and which requires only a single random sequence generation per time step. All of these results have been verified via high-resolution numerical simulations and have been successfully applied to physical test cases. They have also made substantial progress on a longstanding problem in the dynamics of unstable fluid interfaces in porous media. This work has lead to highly accurate quasi-analytic solutions of idealized versions of this problem. These may be of use in benchmarking numerical solutions of the full stochastic PDEs that govern real-world problems.
Date: July 8, 1999
Creator: Sharp, D.H.; Habib, S. & Mineev, M.B.
Partner: UNT Libraries Government Documents Department

Application of scaling properties of the Vlasov and the Fokker-Planck equations to improved macroparticle models

Description: Numerical simulations of cooling processes over minutes or hours of real time are usually carried out using direct solution of the Fokker-Planck equation. However, by using scaling rules derived from that equation, it is possible to use macroparticle representations of the beam distribution. Besides having applications for cooling alone, the macroparticle approach allows combining the cooling process with other dynamical processes which are represented by area-preserving maps. A time-scaling rule derived from the Vlasov equation can be used to adjust the time step of a map-based dynamics calculation to one more suitable for combining with a macroparticle Fokker-Planck calculation. The time scaling for the Vlasov equation is also useful for substantially more rapid calculations when a macroparticle model of a conservative multiparticle system requires a large number of macroparticles to faithfully produce the collective potential or when the model must simulate a long time period.
Date: July 12, 2001
Creator: MacLachlan, James A.
Partner: UNT Libraries Government Documents Department

Fokker-Planck Modelling of Delayed Loss of Charged Fusion Products in TFTR.

Description: The results of a Fokker-Planck simulation of the ripple-induced loss of charged fusion products in the Tokamak Fusion Test Reactor (TFTR) are presented. It is shown that the main features of the measured "delayed loss" of partially thermalized fusion products, such as the differences between deuterium-deuterium and deuterium-tritium discharges, the plasma current and major radius dependencies, etc., are in satisfactory agreement with the classical collisional ripple transport mechanism. The inclusion of the inward shift of the vacuum flux surfaces turns out to be necessary for an adequate and consistent explanation of the origin of the partially thermalized fusion product loss to the bottom of TFTR.
Date: August 1, 1998
Creator: Edenstrasser, J.W.; Goloborod'ko, V.Ya.; Reznik, S.N.; Yavorskij, V.A. & Zweben, S.
Partner: UNT Libraries Government Documents Department

Bunch heating by coherent synchrotron radiation

Description: The authors discuss here effects which define the steady-state rms energy spread of a microbunch in a storage ring. It is implied that the longitudinal microwave instability is controlled by low {alpha} lattice. In this case the coherent synchrotron radiation, if exists, may be the main factor defining the bunch temperature. Another effect comes from the fact that a nonlinear momentum compaction of such lattices makes Haissinskii equation not applicable, and the coherent synchrotron radiation may effect not only bunch lengthening but the energy spread as well.
Date: October 1, 1995
Creator: Heifets, S.A. & Zolotorev, M.
Partner: UNT Libraries Government Documents Department

Fully kinetic simulations of magnetic reconnction in semi-collisional plasmas

Description: The influence of Coulomb collisions on the dynamics of magnetic reconnection is examined using fully kinetic simulations with a Monte-Carlo treatment of the Fokker-Planck collision operator. This powerful first-principles approach offers a bridge between kinetic and fluid regimes, which may prove useful for understanding the applicability of various fluid models. In order to lay the necessary groundwork, the collision algorithm is first carefully bench marked for a homogeneous plasma against theoretical predictions for beam-plasma interactions and electrical resistivity. Next, the collisional decay of a current layer is examined as a function of guide field, allowing direct comparisons with transport theory for the parallel and perpendicular resistivity as well as the thermoelectric force. Finally, the transition between collisional and collision less reconnection is examined in neutral sheet geometry. For modest Lundquist numbers S {approx}< 1000, a distinct transition is observed when the thickness of the Sweet-Parker layers falls below the ion inertia length {delta}{sub sp} {approx}< d,. At higher Lundquist number, deviations from the Sweet-Parker scaling are observed due to the growth of plasmoids (secondary-islands) within the elongated resistive layer. In certain cases, this instability leads to the onset of fast reconnection sooner than expected from {delta}{sub sp} {approx} d, condition. After the transition to fast reconnection, elongated electron current layers are formed which are unstable to the formation of new plasmoids. The structure and time-dependence of the electron diffusion region in these semi-collisional regimes is profoundly different than reported in two-fluid simulations.
Date: January 1, 2009
Creator: Daughton, William S; Roytershteyn, Vadim S; Albright, Brian J; Yin, Lin; Bowers, Kevin J & Karimabadi, Homa
Partner: UNT Libraries Government Documents Department

Quantum plasma dynamics: Distorted atomic reaction rates for high temperature plasmas. Final report, September 15, 1991--September 14, 1994

Description: The effect of plasma perturbers on ionic systems inside a plasma is complex, time-dependent, and stochastic. It affects not only the population of the various excited and charge states of the ions, but also the life time of these states. The broadening and population of the states inside plasmas must be treated simultaneously in a coherent way. The purpose of this project was to develop a systematic theoretical procedure to treat the effects of plasma perturbers on the atomic reaction rates, in a way consistent with the rate equation approach to plasma modeling. As explained in the following, we have completed the program set out originally. Further works needed are also pointed out. We first formulated a theoretical procedure to treat this problem in a simple form such that the theory may be applied to realistic cases. The crucial features of the conventional approaches for plasma modeling and diagnostics, the pressure broadening theory (PBT) and the rate equations approach (REQ), are adopted in treating the level distortions and population determination. The PBT is effective in treating the distortion, while the REQ approach is suitable in treating multiple cascade effects on the level population. In order to combine both these theories, however, careful examination of the assumptions which have gone into the PBT and REQ formulations had to be carried out, in terms of the distorted state dressing and coarse grain averaging. The various relaxation times involved in the perturber collisions, ionic decay, and averaging time intervals were compared. The resulting theory separates the plasma perturbations into two parts, slow and fast relative to the typical atomic relaxation time involved for a particular state of the target ion. The slow part is to be included in an effective plasma potential, and is to be used in distorting the ionic states.
Date: September 25, 1996
Creator: Hahn, Yukap
Partner: UNT Libraries Government Documents Department

Multidimensional electron-photon transport with standard discrete ordinates codes

Description: A method is described for generating electron cross sections that are compatible with standard discrete ordinates codes without modification. There are many advantages of using an established discrete ordinates solver, e.g. immediately available adjoint capability. Coupled electron-photon transport capability is needed for many applications, including the modeling of the response of electronics components to space and man-made radiation environments. The cross sections have been successfully used in the DORT, TWODANT and TORT discrete ordinates codes. The cross sections are shown to provide accurate and efficient solutions to certain multidimensional electronphoton transport problems.
Date: December 31, 1995
Creator: Drumm, C.R.
Partner: UNT Libraries Government Documents Department

Adjoint electron-photon transport Monte Carlo calculations with ITS

Description: A general adjoint coupled electron-photon Monte Carlo code for solving the Boltzmann-Fokker-Planck equation has recently been created. It is a modified version of ITS 3.0, a coupled electronphoton Monte Carlo code that has world-wide distribution. The applicability of the new code to radiation-interaction problems of the type found in space environments is demonstrated.
Date: February 1, 1995
Creator: Lorence, L.J.; Kensek, R.P.; Halbleib, J.A. & Morel, J.E.
Partner: UNT Libraries Government Documents Department

Multidimensional electron-photon transport with standard discrete ordinates codes

Description: A method is described for generating electron cross sections that are comparable with standard discrete ordinates codes without modification. There are many advantages of using an established discrete ordinates solver, e.g. immediately available adjoint capability. Coupled electron-photon transport capability is needed for many applications, including the modeling of the response of electronics components to space and man-made radiation environments. The cross sections have been successfully used in the DORT, TWODANT and TORT discrete ordinates codes. The cross sections are shown to provide accurate and efficient solutions to certain multidimensional electron-photon transport problems. The key to the method is a simultaneous solution of the continuous-slowing-down (CSD) portion and elastic-scattering portion of the scattering source by the Goudsmit-Saunderson theory. The resulting multigroup-Legendre cross sections are much smaller than the true scattering cross sections that they represent. Under certain conditions, the cross sections are guaranteed positive and converge with a low-order Legendre expansion.
Date: April 1, 1997
Creator: Drumm, C.R.
Partner: UNT Libraries Government Documents Department

Ionization cooling and muon dynamics

Description: Muon colliders potential to provide a probe for fundamental particle physics is very interesting. To obtain the needed collider luminosity, the phase space volume must be greatly reduced within the muon life time. The Ionization cooling is the preferred method used to compress the phase space and reduce the emittance to obtain high luminosity muon beams. The authors note that, the ionization losses results not only in damping, but also heating. They discuss methods used including moments methods, Focker Plank Equation, and Multi Particle Codes. In addition they show how a simple analysis permits us to estimate the most part of the optimal system parameters, such as optimal damping rates, length of the system and energy.
Date: January 1, 1998
Creator: Parsa, Z.
Partner: UNT Libraries Government Documents Department

Determination of the Electron Cyclotron Current Drive Profile

Description: Evaluation of the profile of non-inductive current density driven by absorption of electron cyclotron waves (ECCD) using time evolution of the poloidal flux indicated a broader profile than predicted by theory. To determine the nature of this broadening, a 1-1/2 D transport calculation of current density evolution was used to generate the signals which the DIII-D motional Stark effect (MSE) diagnostic would measure in the event that the current density evolution followed the neoclassical Ohm's law with the theoretical ECCD profile. Comparison with the measured MSE data indicates the experimental data is consistent with the ECCD profile predicted by theory. The simulations yield a lower limit on the magnitude of the ECCD which is at or above the value found in Fokker-Planck calculations of the ECCD including quasilinear and parallel electric field effects.
Date: November 1, 1999
Creator: Luce, T.C.; Petty, C.C.; Schuster, D.I. & Makowski, M.A.
Partner: UNT Libraries Government Documents Department

Modification of the Current Profile in DIII-D by Off-Axis Electron Cyclotron Current Drive

Description: Localized non-inductive currents due to electron cyclotron wave absorption have been measured on the DIII-D tokamak. Clear evidence of the non-inductive currents is seen on the internal magnetic field measurements by motional Stark effect spectroscopy. The magnitude and location of the non-inductive current is evaluated by comparing the total and Ohmic current profiles of discharges with and without electron cyclotron wave power. The measured current agrees with Fokker-Planck calculations near the magnetic axis, but exceeds the predicted value as the location of the current drive is moved to the half radius.
Date: July 1, 1999
Creator: Luce, T.C.; Lin-Liu, Y.R.; Harvey, R.W.; Giruzzi, G.; Lohr, J.M.; Petty, C.C. et al.
Partner: UNT Libraries Government Documents Department