Electromagnetic radiation of charged particles in stochastic motion
Eur. Phys. J. C
Electromagnetic radiation of charged particles in stochastic motion
Tiberiu Harko 1 2
Gabriela Mocanu 0
0 Astronomical Institute of the Romanian Academy , 19 Cires ̧ilor Street, Cluj-Napoca 400487 , Romania
1 Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
2 Department of Physics, Babes-Bolyai University , Kogalniceanu Street, Cluj-Napoca 400084 , Romania
The study of the Brownian motion of a charged particle in electric and magnetic fields has many important applications in plasma and heavy ions physics, as well as in astrophysics. In the present paper we consider the electromagnetic radiation properties of a charged non-relativistic particle in the presence of electric and magnetic fields, of an exterior non-electromagnetic potential, and of a friction and stochastic force, respectively. We describe the motion of the charged particle by a Langevin and generalized Langevin type stochastic differential equation. We investigate in detail the cases of the Brownian motion with or without memory in a constant electric field, in the presence of an external harmonic potential, and of a constant magnetic field. In all cases the corresponding Langevin equations are solved numerically, and a full description of the spectrum of the emitted radiation and of the physical properties of the motion is obtained. The power spectral density of the emitted power is also obtained for each case, and, for all considered oscillating systems, it shows the presence of peaks, corresponding to certain intervals of the frequency.
1 Introduction
The stochastic motion of particles in different physical
systems, and under the influence of various forces, is a
fundamental area of research in plasma physics, astronomy,
condensed matter physics, and biology [
1,2
]. In particular, the
motion and the radiation of charged particles play a key role
in the understanding of the nuclear fusion processes in the
tokamak plasmas. In the presence of an external electric field,
in a fully ionized plasma, electrons with energies higher than
certain critical values of the energy are continuously
accelerated at very high energies. These electrons are called runaway
electrons [3].
The study of runaway electrons, and in particular of their
radiation, represents a field of great importance in different
areas of research such as astronomy, accelerators, or nuclear
fusion [
4–6
].
An important physical problem in plasma and fusion
physics is the anomalous transport in a magnetically confined
plasma in a region where the magnetic surfaces are destroyed
[
7
]. One of the basic methods in the study of the anomalous
transport is based on the analogy between the transport
problem and the random walk or Brownian motion theory [
7,8
].
For this case, the starting point is the equation of motion of
a charged test particle, feeling the action of a magnetic field
and of interparticle collisions. The latter are represented by a
random force and the equation of motion becomes a
stochastic differential equation, the Langevin equation [
7–11
]
d2r
dt 2 = F r (t ), r˙(t ), t + η(t ),
(1)
where F [r (t ), r˙(t ), t ] is the systematic (“average”) force
acting on the article, and η(t ) is a random force
modeling the effects of the interparticle collisions. The
anomalous transport in plasmas is usually attributed to the
magnetic fluctuations in a very strong “basic” magnetic field
Bo, which undergoes small fluctuations in a perpendicular
direction.
The Langevin equation Eq. (1) gives a correct physical
and statistical description of the random Brownian motion
only in the large time limit. This requires that the
considered time intervals must be large enough as compared to the
characteristic relaxation time of the velocity autocorrelation
function [
11
]. The description of the dynamics of a
homogeneous system without restriction on a time scale can be
realized by generalizing the Langevin equation. This
generalization implies the introduction of a systematic force term
with an integral kernel, which substitutes the simple friction
term [
12
]. From a physical point of view the convolution
term describes the memory, or the retardation effects. From
an astrophysical point of view such a term can be used to
model the stochastic oscillations of the accretion disks [
13
]
in the presence of colored noise [
14,15
].
The study of the Brownian motion of charged particles
in magnetic fields based on the Langevin equations with the
dynamic friction proportional to the particle velocity was
initiated in [
16
], where the diffusion of ions in plasma across
the magnetic field, with the stochasticity arising from the
fluctuations of the electric field was considered. In this work
the mean square displacement was found in the limit of long
times. These studies were further extended in [
17,18
], where
it was shown that for a special symmetry of the dynamical
friction matrix for Larmor periods of the ord (...truncated)