RUNAWAY PHENOMENA
General Considerations
Runaway phenomena occur in ionized gases and plasmas when the collisional
mechanisms are not strong enough to stop particles to being accelerated to
arbitrary large velocities by the electric field.
The time evolution of an ensemble of charged particles subject
to a constant electric field and colliding with a
(charged or neutral) background medium, can happen according to
one of two possible scenarios. If collisions are effective
enough in removing the kinetic energy gained by the particles from
the field, the distribution
function and the macroscopic (or average) quantities that
characterize the ensemble relax towards their asymptotic
steady-state values; if collisions are not sufficiently effective,
then the energy gain of the
ensemble continues indefinitely, particles are accelerated to
arbitrarily large velocities and an asymptotic steady-state
does not exist. This latter phenomenon is called ``runaway".
The asymptotic behaviour of the average velocity of the
ensemble v(t) characterizes the two
situations and
is often used as a definition of a runaway regime and a
non-runaway regime. In the runaway regime
v(t) diverges as t goes to infinity,
while in the non-runaway regime v(t) tends to a constant value
as t goes to infinity.
The mathematical description of runaway phenomena usually
involves the Boltzmann equation, with the scattering kernel
or model collision operator suitable for the physical system
which is being studied. The Fokker-Planck equation is also
used very often, especially in plasma physics.
Results
In (5), we have carried out a mathematical and numerical
study of the solutions of the linear, one-dimensional, Boltzmann
equation in runaway regime and we have addressed the problem of the
approach to the asymptotic state, in which the average velocity
diverges with time. We have used a BGK collision
operator with a collision frequency which is constant inside a
compact support in velocity space and vanishes outside. We
find that the time evolution of the
distribution function and the average velocity occurs on two different
time scales. On a short time scale, collisions cause
the distribution function to relax towards a
stationary profile and the average velocity to saturate at a constant value,
showing the formation of a plateau; a small runaway flux is also present,
indicating the leak of particles towards higher velocities.
The
stationary profile can be considered as the asymptotic steady-state of an
auxiliary problem, in which the size of the support of the collision
frequency tends to infinity. By using techniques of functional analysis and semigroup theory,
we have obtained a rigorous result about the behaviour of the
solution during the approach to the quasi steady-state and
its persistence in time. If the collision frequency function
is sufficiently close to a constant over a large interval
of the velocity space, then for some time the solution and the
average velocity follow closely the solution and the average velocity
of the model with constant collision frequency. The proof
is quite general and is not restricted to BGK
collision kernels.
On a longer time scale, the runaway flux depopulates the bulk of the
distribution, which deviates more and more from the stationary shape;
eventually, most particles lie outside of the support of the collision
frequency and the solution simply propagates along the characteristics
of the collisionless system. Asymptotically in time, a travelling wave
pattern is formed and the average velocity tends to infinity.
In (4), the presence of the two time scales has been exploited to find
approximate expressions for the solution of the Boltzmann equation in
runaway regime, by using a Multiple-Time-Scale expansion. In (1) and (2),
the problem has been addressed by Laplace Transform methods.
-
L. Demeio, Laplace transform approach to runaway phenomena, Internal
Report N. 1/1998, Dipartimento di Matematica ``V. Volterra", Universita'
degli Studi di Ancona.
-
L. Demeio and G. Frosali,
Approximate solutions of kinetic equations in runaway regime,
Proc. IX Int. Conf. on Waves and Stability in Cont.
Media, Bari (Italy), October 6-11, 1997, Suppl. Rend. Cir. Mat. Palermo
Serie II, N. 57, 211-216 (1998).
-
L. Demeio and G. Frosali,
Asymptotic analysis of kinetic equations in runaway regime, Atti
del XIII Congresso Nazionale AIMETA, Siena, 29/9-3/10/1997, vol. I, 95-100,
Ed. ETS
-
L. Demeio Multiple time scale analysis of runaway phenomena, Transport
Theory and Statistical Physics, 27 (3-4), 333 (1998).
-
J. Banasiak, L. Demeio Quasi steady-state solutions of kinetic equations
in runaway regime, Transport Theory and Statistical Physics 28 (1),
1-29 (1999).