共查询到20条相似文献，搜索用时 31 毫秒
1.
The Lagrange expansion, which may be used to derive the Fokker-Planck equation, is here used to derive the corresponding expression for the flux of particles subject to a stochastic scattering process. The coefficients which occur in this expression are, in general, not the same as the coefficients which occur in the Fokker-Planck equation itself. In the special case that the particle distribution involves only one independent variable, the particle flux is determined by the familiar Fokker-Planck coefficients. Evaluation of particle flux is of special interest in the study of stochastic acceleration. 相似文献
2.
Simon F. Portegies Zwart Koji Takahashi 《Celestial Mechanics and Dynamical Astronomy》1999,73(1-4):179-186
Recent N-body simulations have shown that there is a serious discrepancy between the results of N-body simulations and the
results of Fokker-Planck simulations for the evolution of globular and rich open clusters under the influence of the galactic
tidal field. In some cases, the lifetime obtained from Fokker-Planck calculations is more than an order of magnitude smaller
than those from N-body simulations. In this paper we show that the principal cause for this discrepancy is an over-simplified
treatment of the tidal field used in previous Fokker-Planck simulations. We performed new Fokker-Planck calculations using
a more appropriate implementation for the boundary condition of the tidal field. The implementation is only possible with
anisotropic Fokker-Planck models, while all previous Fokker-Planck calculations rely on the assumption of isotropy. Our new
Fokker-Planck results agree well with N-body results. Comparison of the two types of simulations gives a better understanding
of the evolution of such clusters.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
3.
On the particle number conservation inherent in the diffusion approach of the Fokker-Planck equation
Jörn E. Kunstmann 《Astrophysics and Space Science》1978,56(1):81-87
Starting from the Fokker-Planck equation we show that, in a strict sense, the usual diffusive transport of charged particles in weak magnetic field fluctuations is not valid until the space-integrated number of particles per velocity interval has reached its final overall constant value and is conserved. Large anisotropies are thus not compatible with diffusion. Diffusion becomes exactly valid after an infinite time, but has reached a good approximation to the real transport before that. The particle-number conservation holds for quite general mean magnetic field configurations, and is not limited to the case of a constant mean field. 相似文献
4.
Jörn E. Kunstmann 《Solar physics》1978,59(2):395-405
The transport of energetic charged particles in fluctuating magnetic fields is analysed by means of the Fokker-Planck equation for the early time, when the diffusive modes are not yet valid. Exact solutions are found for the particle fluxes integrated over space; after these have reached a time-independent final steady state, the usually considered diffusive modes of the velocity-integrated density become valid. The time for that is finite. The analysis is done both for a constant and a divergent mean magnetic field. 相似文献
5.
An analytical theory is developed for the velocity evolution of nonaccreting planetesimal populations, based on the Boltzmann and Fokker-Planck equations. Adapting Shkarofsky's calculation of plasma viscosities, the rate of increase in random velocities due to gravitational encounters between planetesimals of equal mass is found to be one-third to one-half Safronov's result. Comparison with Wetherill's numerical experiments suggests that the Fokker-Planck equation underestimates the effectiveness of encounters and that Safronov's value is approximately correct. For populations of nonuniform sizes, the Fokker-Planck equation indicates an efficient redistribution of energy from the largest bodies to the smaller ones. By conserving angular momentum, the rate of radial spreading of orbits is also derived. 相似文献
6.
Recently an advanced nonlinear diffusion theory for particle transport across the mean magnetic field has been developed. The method used in the derivation of the latter theory is based on the cosmic ray Fokker-Planck equation. In the present article we use the same approach to describe pitch-angle scattering and parallel spatial diffusion nonlinearly. Furthermore, we derive the quasilinear transport theory, the weakly nonlinear theory as well as the Bohm limit as special cases from our more general approach. 相似文献
7.
Bernie D. Shizgal 《Planetary and Space Science》2004,52(10):923-933
Nonequilibrium distributions of space plasmas are often characterized by extended high-energy tails. This paper provides a detailed analysis of the relaxation of such isotropic nonequilibrium plasmas. We consider an energetic charged species dilutely dispersed in a fully ionized plasma, which acts as a heat bath at equilibrium. The minor constituent is referred to as a “test particle” and collisions between test particles are not included. We study the approach to equilibrium with a finite difference method of solution of the Fokker-Planck equation appropriate for collisions between charged particles. The solution of the Fokker-Planck equation is also presented formally as an expansion in the eigenfunctions of the Fokker-Planck operator. The main objective of the paper is the calculation of the energy-dependent relaxation times of the distribution function. A strong energy dependence for these relaxation times is anticipated since, for Coulomb collisions, the Rutherford cross-section varies with relative speed g as g−4. Analogous results for neutral species are presented for comparison in the following paper. 相似文献
8.
Max K. Wallis 《Planetary and Space Science》1975,23(3):419-430
Interstellar gas streaming through the solar system undergoes both elastic collisions with solar wind ions and destructive, ionizing processes. The Boltzmann equation is set up, with linear Fokker-Planck terms describing the glancing elastic collisions. Solutions combining the dynamical effects of the central force field and the diffusion in velocity space are derived, appropriate to cool gas.For the He component of the streaming gas, if initially at 100 K, the collisional heating dominates inside 2 a.u. upstream and 5 a.u. downstream. A modified formula is given for the density in the downstream wake, as enhanced by gravitational focussing. Calculations of the helium resonant radiation backscatter require substantial modification. 相似文献
9.
Energetic particles, ejected from the Sun during solar flare events, may encounter interplanetary plasma/field conditions, which deviate considerably from the quiet time values used normally to describe the particle propagation. This is due to the presence of a hydromagnetic shock, which is emitted from the Sun at the time of the explosion. In a theoretical blast wave model, which incorporates the interaction with plane polarized Alfvén waves, we have analysed the changes in different terms of the Fokker-Planck equation, which describes energetic particle propagation. In this treatment, the shock influence on energy changes and on the transport coefficients are discussed. 相似文献
10.
The Fokker-Planck equation for small stochastic changes to particles in Kepler orbits has to be formulated in terms of the integrals of motion. We generalize the modelling of proton and electron collisional perturbations to gas particles on trajectories through the solar system in order to include both spatial and velocity diffusion. The general solution is obtained in terms of a 4-dimensional normal distribution. Treatment of the singularity in the Fokker-Planck operator reduces the dimensionality by one. In addition to extending earlier results for anisotropic collisional heating in the thermal approximation, the present formulation gives the changes in density due to the mean repulsive force and to perturbations of trajectories (spatial diffusion). The net diffusion is almost everywhere towards the sun and the density increase is significant in the downstream hydrogen wake, particularly where destructive depletion is strong and gravitational focussing weak. 相似文献
11.
K. G. McClements 《Solar physics》1987,109(2):355-363
The conditions required for the stability of a steady-state electron beam propagating in the solar corona are determined using the quasi-linear theory. The growth rate for electron plasma waves in a magnetized plasma is evaluated, with the electron distribution function being given by an analytic solution of the linearized Fokker-Planck equation. It is shown that, when the gyrofrequency is less than the plasma frequency, the instability has a narrow angular range, with the maximum growth rate occuring along the magnetic field. A stability boundary in parameter space is determined, indicating that electron beams must be highly collimated at injection to be Langmuir unstable at any point in space. The implications of the results for alternative models of hard X-ray bursts are discussed and it is argued that Langmuir instability will not occur on either the trap model or the thermal model. Such models would, therefore, be refuted by the detection of a large flux of plasma microwave radiation associated with hard X-ray emission. 相似文献
12.
We analyze the time variation of microwave spectra and hard X-ray spectra of 1989 March 18, which are obtained from the Solar Array at the Owens Valley Radio Observatory (OVRO) and the Hard X-Ray Burst Spectrometer (HXRBS) on the Solar Maximum Mission (SMM), respectively. From this observation, it is noted that the hard X-ray spectra gradually soften over 50–200 keV on-and-after the maximum phase while the microwaves at 1–15 GHz show neither a change in spectral shape nor as rapid a decay as hard X-rays. This leads to decoupling of hard X-rays from the microwaves in the decay phase away from their good correlation seen in the initial rise phase. To interpret this observation, we adopt a view that microwave-emitting particles and hard X-ray particles are physically separated in an inhomogeneous magnetic loop, but linked via interactions with the Whistler waves generated during flares. From this viewpoint, it is argued that the observed decoupling of microwaves from hard X-rays may be due to the different ability of each source region to maintain high energy electrons in response to the Whistler waves passing through the entire loop. To demonstrate this possibility, we solve a Fokker-Planck equation that describes evolution of electrons interacting with the Whistler waves, taking into account the variation of Fokker-Planck coefficients with physical quantities of the background medium. The numerical Fokker-Planck solutions are then used to calculate microwave spectra and hard X-ray spectra for agreement with observations. Our model results are as follows: in a stronger field region, the energy loss by electron escape due to scattering by the waves is greatly enhanced resulting in steep particle distributions that reproduce the observed hard X-ray spectra. In a region with weaker fields and lower density, this loss term is reduced allowing high energy electrons to survive longer so that microwaves can be emitted there in excess of hard X-rays during the decay phase of the flare. Our results based on spectral fitting of a flare event are discussed in comparison with previous studies of microwaves and hard X-rays based on either temporal or spatial information. 相似文献
13.
C. K. Ng 《Solar physics》1988,114(1):165-179
Two new forms of a simplified Fokker-Planck equation are derived for the transport of low-energy solar energetic particles in an evolving interplanetary magnetic field, carried by a variable radial solar wind. An idealised solution suggests that the invariant anisotropy direction reported by Allum et al. (1974) may be explained within the conventional theoretical framework. The equations may be used to relate studies of solar particle propagation to solar wind transients, and vice versa. 相似文献
14.
A. G. Nikoghossian 《Astrophysics》2007,50(2):175-186
The formation of spectral lines in a turbulent atmosphere with a spatially correlated velocity field is examined. A new approach
for solving this problem is proposed which is not based on the Fokker-Planck formalism. The invariant imbedding method makes
it possible to reduce the problem of finding the mean radiant intensity observed in a line to solving a system of differential
equations. This possibility is based on determining the mean intensity of the radiation emerging from the medium for a fixed
value of the turbulent velocity at its boundary. A separate integral equation is derived for this quantity. The dependence
of the line profile, integrated intensity, and width on the mean correlation length and the average value of the hydrodynamic
velocity is studied. It is shown that the transition from a microturbulent regime to a macroturbulent regime occurs within
a comparatively narrow range of variation in the correlation length. The proposed method yields a solution to the problem
for a family of inhomogeneous atmospheres with different optical thicknesses, which makes it easy to determine the radiation
field inside the turbulent medium. This approach can be generalized in various ways, in particular, it can be applied without
significant changes to the case where the correlation length depends on position within the atmosphere.
__________
Translated from Astrofizika, Vol. 50, No. 2, pp. 219–232 (May 2007). 相似文献
15.
基于在^3He丰富事件中，高能^3He和重离子具有相似的幂律谱分布这一观测结果，通过数值求解Fokker-Planck方程，探讨经阿尔芬波湍动速后的离子分布随时间的演化特征。计算结果表明：加速源区的等离子体密度和阿尔芬波湍动能量密度对粒子能谱分布起主要作用，如果取加速源区等离子体密度n=(0.1-1)10^10cm^-3、磁场强度B=50-100Gs、湍动能量密度为0.4-2ergs cm^-3，则在1秒左右的时间内，湍动阿尔芬波能够将^3He和重离子加速到10MeV/nucleon量级，能谱指数为2.0-3.5。理论计算与观测结果基本一致。 相似文献
16.
B. R. Frieden 《Astrophysics and Space Science》1996,244(1-2):387-391
Fisher informationI is a classical concept that originates in estimation theory. Through the Cramer-Rao inequality, it defines the smallest possible error in the estimation of a parameter in the presence of noise obeying a given probability law. More recently, Fisher information has been incorporated within a variational principle for forming the laws of physics (Schrödinger wave equation, Dirac equation, etc.). The premise is thatdI/dt0, witht the time, so that, at equilibrium,I=min. The premise has recently been proven for any process obeying a Fokker-Planck differential equation. Hence, Fisher information provides a new measure of the passage of time. All errors of estimation increase, on average, with time. 相似文献
17.
由分析可知，在光学厚的介质中麦氏分布的极端相对论电子在磁场中能产生同步加速辐射，其辐射谱为瑞利—金斯谱．基于Fokker-Planck方程，这种波场对快电子的加速将产生幂律分布的电子能谱．考虑到日冕活动区的物理条件，在太阳耀斑中观测到的10Mev左右的电子能谱很可能就是由同步加速辐射加速快电子产生的． 相似文献
18.
M. Hénon 《Astrophysics and Space Science》1971,13(2):284-299
19.
Heat transport in the Sun is describable by a Fokker-Planck, or diffusion, transfer equation. A study of the general character of the solutions of the transfer equation shows that the inhibition of convective transport beneath the photosphere produces a photospheric dark ring surrounded by a bright ring, or at best, a dark area surrounded by a bright ring. The mean temperature beneath the sunspot is unavoidably above normal, so that the enhanced gas pressure would disperse, rather than concentrate, the magnetic field. Hence we conclude that the inhibition of convection cannot be the cause of a sunspot.We suggest, instead, that a sunspot is a region of enhanced, rather than inhibited, energy transport and emissivity. The magnetic field of the sunspot causes a dynamical overstability in the outer thousand km of the convective zone, generating copious fluxes of hydromagnetic waves, which propagate rapidly out of the region along the magnetic field. We suggest that this heat engine is so efficient as to convert at least three fourths of the heat flux into waves. Solutions of the heat transport equation in the presence of a heat sink automatically resemble the observed sunspot, including a dark interior, a sharp transition at the edge of the umbra, and an extended grey area around the outside, the penumbra. The mean temperature is reduced, causing the observed concentration of the magnetic field.The enhanced radiation is in the form of hydromagnetic waves, which do not appear in ordinary photographs, but which light up the sky over the sunspot in a manner conspicuous in any UV or X-ray picture. In this respect, then, a sunspot is effectively a hole in the Sun, extending down to temperatures of 2 × 104 K or more.This work was supported in part by the National Aeronautics and Space Administration under Grant NGL 14-001-001. 相似文献
20.
In this paper we consider a random motion of magnetic bright points (MBP) associated with magnetic fields at the solar photosphere.
The MBP transport in the short time range [0–20 minutes] has a subdiffusive character as the magnetic flux tends to accumulate
at sinks of the flow field. Such a behavior can be rigorously described in the framework of a continuous time random walk
leading to the fractional Fokker-Planck dynamics. This formalism, applied for the analysis of the solar subdiffusion of magnetic
fields, generalizes the Leighton’s model.
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