Chaotic collisionless evolution in galaxies and charged-particle beams.

Abstract

Both galaxies and charged particle beams can exhibit collisionless evolution on surprisingly short time scales. This can be attributed to the dynamics of chaotic orbits. The chaos is often triggered by resonance caused by time dependence in the bulk potential, which acts almost identically for attractive gravitational forces and repulsive electrostatic forces. The similarity suggests that many physical processes at work in galaxies, although inaccessible to direct controlled experiments, can be tested indirectly via controlled experiments with charged particle beams, such as those envisioned for the University of Maryland electron ring currently nearing completion.

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Chaotic Collisionless Evolution in Galaxies and Charged-
Particle Beams

HENRY E. KANDRUP1, COURTLANDT L. BOHN*2,3, RAMI A. KISHEK4,
PATRICK G. O’SHEA4, MARTIN REISER4, IOANNIS V. SIDERIS2

1Departments of Astronomy and Physics and Institute for Fundamental Theory, University of
Florida, Gainesville, Florida, USA (deceased);
2Department of Physics, Northern Illinois University, DeKalb, Illinois, USA;
3Fermi National Accelerator Laboratory, Batavia, Illinois, USA;
4Institute for Research in Electronics and Applied Physics, University of Maryland, College Park,
Maryland, USA.


ABSTRACT: Both galaxies and charged particle beams can exhibit collisionless evolution on
surprisingly short time scales. This can be attributed to the dynamics of chaotic orbits. The
chaos is often triggered by resonances caused by time dependence in the bulk potential, which
acts almost identically for attractive gravitational forces and repulsive electrostatic forces. The
similarity suggests that many physical processes at work in galaxies, while inaccessible to direct
controlled experiments, can be tested indirectly via controlled experiments with charged-
particle beams such as those envisioned for the University of Maryland Electron Ring currently
nearing completion.

KEYWORDS: Chaos, N-body problem, nonlinear dynamics, collisionless


PREAMBLE

Henry Kandrup and Court Bohn had independently realized that there were important parallels
between the collisionless evolution of charged-particle beams and large stellar systems. Both
desired to pursue this matter explicitly by way of direct experimentation with beams. Also
independently, Martin Reiser obtained funding to build the University of Maryland Electron Ring
(UMER) for the expressed purpose of doing controlled experiments to measure the dynamical
consequences and evolutionary time scales associated with internal Coulomb forces, i.e., space
charge. All of these circumstances led to a strong collaboration. Henry had been eagerly
anticipating the completion of UMER and experiments that the collaboration was planning.
We all endeavored to introduce the notion of an analogy between the dynamics of beams and
galaxies to a broad spectrum of investigators. Before Henry passed away, we had completed a
paper, one that excited Henry immensely, to review the pertinent literature and introduce this idea.
Feedback from referees was generally negative toward publication but positive toward pursuit of
the idea. Loosely translated, the referee reports stated that we have a nice proposal, e.g., to submit
to a funding agency, but we should finish some new experiments prior to journal publication.
The paper has evolved considerably since Henry’s passing, but it retains much of his language,
particularly as concerns galactic dynamics. We, his colleagues, hereby offer this paper as part of
the Symposium that honors Henry. What follows is a version that incorporates all referee
comments and that is edited to mesh with other related Symposium contributions, but that retains
the original flavor and Henry’s unique touch. It would surely have his imprimateur.

*Voice: (815) 753-6473; fax: (815) 753-8565; clbohn@niu.edu

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I. INTRODUCTION

Many-body systems whose constituents interact via long-range inverse-square-law “Coulomb”
forces, both gravitational and electrostatic, can exhibit macroscopic relaxation and loss of
coherence on time scales much shorter than might be expected on dimensional grounds. This
process moves the system toward a long-lived ‘metaequilibrium’ state, a state that differs from true
thermal equilibrium (which, in the case of galaxies, cannot be accessed dynamically). When a
galaxy has a sizeable gaseous component, the gas will interact with the stellar component and
thereby enhance its relaxation. However, observations and simulations agree that even a relatively
gas-poor (and thus presumably nearly dissipation-free) elliptical galaxy displaced from a
metaequilibrium state as a result of an encounter with another galaxy can readjust itself towards a
new metaequilibrium state within a few hundred million years (i.e., within ~10% of the age of the
galaxy) although the nominal relaxation time tR associated with ‘collisions’ is orders of magnitude
longer than the age of the Universe. And similarly, charged-particle beams, which would be
expected to maintain coherence while traveling some 100 km or more through an accelerator, can
lose coherence and disperse significantly within distances as short as 10 m.
Because collisions would cause relatively slow relaxation, any rapid relaxation must be due to
collisionless, i.e., collective, processes. More specifically, the collective behavior must be
connected with mixing, i.e., the tendency of initially localized clumps of orbits to disperse. Mixing
is much more efficient in a chaotic system than in a system in which the bulk coarse-grained
potential is integrable or near-integrable. An initially localized clump of regular, i.e., non-chaotic,
orbits will typically disperse secularly, i.e., as a power law in time; a clump of chaotic orbits will
instead disperse exponentially.
Allowing for a bulk potential that is strongly chaotic, thereby supporting “chaotic mixing”,
would enable one to understand how a galaxy can ‘relax’ toward a metaequilibrium state on a
comparatively short time scale. Such an understanding is of practical importance in regard to
charged-particle beams. There, rapid collisionless relaxation places strong constraints on
‘emittance compensation’, i.e., processes designed to confine the constituent particles to a compact
volume of phase space, as is required for high-brightness beams.
Theoretical considerations and detailed numerical simulations suggest that, in this setting, the
origin of the chaos that drives the evolution is largely irrelevant. In particular, whether the two-
body forces are attractive or repulsive should not be crucial. What is important is that the long-
range scalings of gravitational and electrostatic forces are identical and that, in both cases, the early
stages of evolution should be driven by long-range, collective interactions (acting ‘globally’) as
opposed to short-range collisional encounters (acting ‘locally’). All that seems to matter is whether
the bulk potential associated with the many-body system admits a large measure of chaotic orbits1.
A complete understanding of these phenomena requires a synthesis of theory, simulations, and
experiments. Performing experiments on self-gravitating systems like galaxies is impossible.
However, controlled experiments can be performed with charged-particle beams, and combining
the results of such experiments with simulations and theory should lead to a clear picture of the role
of chaotic phase mixing in beams. Moreover, as we will exemplify in Sec. II below, the physics
should not depend crucially on whether the force between particles is attractive or repulsive, and
one would thus expect that many results about beams should translate more or less directly into
detailed predictions about the structure and evolution of galaxies. Indeed, one can go one step
further and argue that, in a real sense, carefully constructed experiments involving charged-particle
beams can be used as semi-direct probes of the physics of self-gravitating systems like galaxies.

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II. THE BEAM-GALAXY ANALOGY: THEORETICAL CONSIDERATIONS

That collisional relaxation should be largely irrelevant in many settings involving galaxies and
beams is easily seen. Viewing such effects as an incoherent sum of binary encounters, one
computes, respectively, for galaxies2 and (in gaussian units) for charged particle beams3,4 relaxation
times
3
v
t
R
.
log
~ and
log)(
~
2
3
2
ΛΛ
nq
v
t
n Gm
R
(1)

Here v is a typical speed associated with random motions; G the gravitational constant; m and q the
typical stellar mass and particle charge, respectively; n a characteristic number density, and log
the so-called Coulomb logarithm*, which scales as a positive power of the number of constituent
‘particles’ N.
In either case, assuming the bulk random kinetic and potential energies are comparable in
magnitude implies that tR ~ (N/log )tD, with tD ~ R/v denoting the ‘dynamical time’, a
characteristic orbital time scale defined in terms of the ‘size’ R of the system. For large N
(typically N ~ 109-1012 in realistic, large stellar and particle-beam systems) the relaxation time tR is
clearly orders of magnitude longer than the dynamical time tD; collisional relaxation is slow. By
contrast, mixing of chaotic orbits, i.e., ‘chaotic mixing’, can proceed extremely fast; the e-folding
time associated with the dispersal of an initially localized ‘clump’ of particles, given as the inverse
of the largest positive Lyapunov exponent respective to that clump6, is typically comparable in
magnitude to tD. This is, for example, the case for the systems illustrated in Figs. 2 and 4 discussed
below in Secs. III.A.3 and III.B.3, respectively
Presently there is no known generic algorithm permitting accurate analytic or quasi-analytic
estimates of the largest Lyapunov exponent in three-dimensional bulk potentials. However, recent
work7-9 has shown that, in many cases, an analytic technique developed for systems with many
degrees of freedom10 can be adapted to provide reasonable estimates for lower-dimensional
systems, the breakdown of that approach reflecting typically systems in which autocorrelation
functions for properties of representative orbits have long time tails11. It is therefore relevant to
recall the analytic results for the largest Lyapunov exponent in a three-dimensional time-
independent bulk potential, for this then becomes a quantitative measure of the rate of collisionless
relaxation by way of chaotic mixing:

1)(
3
ξ
L

The auxiliary quantities and are determined from the potential V(x):

1
;
2
κ

wherein the averages are taken over the microcanonical ensemble in the manner
___________________________________
*If one assumes that collisions act as a source of Brownian motion, tR can be related to the time integral of the quantity
N〈F(0)•F(t)〉, where 〈F(0)•F(t)〉 is the autocorrelation function for the test ‘particle’ interacting with a single field
‘particle’.5
.
12
8
33
)(;)(1)()(;
)(
1
)(
2
3 / 1

2
2
πξξ
ξ
+
π
ξξξξκ
ξ
ξχ
+
=



++=
−
≈
TTTL
L

(2)
()
;
2
1
2
2
2
22
VVV
∇−∇=∇=
ξκ
(3)

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[]
[]
∫ ∫
d
∫∫
−
−
≡
EHd
EHAdd
A
),(
),()(
pxpx
pxxpx
δ
δ
. (4)

Here, E denotes the total particle energy. Upon invoking Poisson’s equation, we see immediately
that the auxiliary quantities are determined from the density distribution. For a galaxy, we have

,
2
; ) 0 (
ρ
G
2 ;4
2
22
ςς
ς
ξςπκρπ
−===∇
GV
(5)

where = (x)/ (0), (x) denoting the mass density. For a beam, we take the external focusing
potential Vf to be quadratic in the coordinates x comoving with the bunch, i.e., Vf(x) = (
wherein = (
Then we have
•x)2/2,
x,
y,
z) corresponds to the focusing strength; the total potential is V = Vf + Vs.
[]
,
)(
2
,)(
2
;4
2
0
2
2
2
0
2
p
0
2
ςωω
ςς
ξςωω
ω
κπρ
−
−
=−=−=∇
p
ps
V
(6)

where
Now, the time scale for chaotic mixing is tm = 1/
in beams versus galaxies becomes apparent: for both classes of systems, the dynamical time is tD ~
-1/2, the auxiliary quantity involves a product of the dispersion in the density profile and the
square of the dynamical time, and f( ) is the same function for both systems. For beams, space
charge is a repulsive collective force that acts to lengthen the dynamical time by weakening the net
focusing force acting on a particle (resulting in what is called the ‘space-charge-depressed period’).
For galaxies no such weakening appears; gravity is strictly attractive.
To do a computational test of this result, one chooses an energy E and integrates a large
number (say, 2,000) tightly localized initial conditions corresponding to an energy very close to E.
These trajectories then spread, and one can calculate moments, such as 〈x2(t)〉, of the corresponding
distribution of trajectories versus time and assess whether they grow exponentially. If they do, then
one can extract the e-folding time and compare it to the analytic estimate. Examples of such
comparisons in galactic and beam systems appear in Fig. 1. The galactic system is a uniform-
density ellipsoid containing a supermassive black hole at its centroid8. The beam system is a
configuration of thermal equilibrium having triaxial symmetry9.
The preceding analytic results follow from a geometric treatment of scleronomous Hamiltonian
systems in the spirit of Pettini and his collaborators10. It does not apply to time-dependent systems,
and thus it is not presently possible to point to an unambiguous analogy between the dynamics of
beams and galaxies involving rheonomous Hamiltonians. A geometric treatment of the latter
would be based on a Finsler metric, i.e., a metric that incorporates velocities and time, but it
becomes unclear how to define an invariant measure to use in place of the microcanonical
ensemble for evaluating phase-space averages, particularly when one considers that resonances
between orbital frequencies and the frequency spectrum of the time-dependent potential come into
play. Nonetheless, a reasonable ansatz is that a successful geometric treatment of rheonomous
systems would result in a connection between beams and galaxies analogous to that of time-
independent systems. The underlying reason is that both systems involve an inverse-square long-
range force, and this force is what drives chaotic mixing.
p0 is the plasma frequency at the bunch centroid and (x) now refers to charge density.
f( )
-1/2. The analogy between chaotic mixing

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Figure 1. (left) Numerical results (diamonds) and analytic (solid line) estimates of the mixing rate for chaotic orbits
evolved in a triaxial galactic potential as a function of black-hole mass MBH and for total particle energy E = 1.0, (b) E
= 0.6, (c) E = 0.4. (right) Numerical (diamonds) and analytic estimates (dashed line) of the mixing rate for chaotic
orbits in a triaxial thermal-equilibrium beam as a function of E. In both figures the unit of
be found in Refs. [8,9] from which these figures are reprinted by permission of the American Physical Society.


III. REGULAR VS. CHAOTIC ORBITS: A TORTURED HISTORY

Chaos has been largely ignored until comparatively recently in both the galactic and
accelerator-dynamics communities. For example, although the famous Hénon-Heiles model12 arose
originally in attempts to understand meridional motions in axisymmetric galaxies, as recently as 15
years ago the potential role of chaos in galaxy structure and evolution was almost completely
neglected (with the exception of a handful of groups in Europe). Only with the advent of high-
resolution photometry, facilitated in part by the Hubble Space Telescope, did many galactic
astronomers begin to recognize that the bulk potentials associated with realistically shaped galaxies
are likely to admit significant measures of chaotic orbits.

A. Galaxies

It has been long recognized that the dominant mechanism for relaxation in galaxies cannot be
‘collisional’. For example, in the 1940s Chandrasekhar2 showed that the relaxation time scale tR on
which binary encounters between individual stars could significantly alter the trajectories of stars in
the Milky Way must be ~1012 years or more. Shorter-time relaxation must somehow involve
collective effects. Two decades later, Lynden-Bell13 proposed a theory of ‘violent relaxation’
which argued, inter alia, that regular (i.e., nonchaotic) phase mixing associated with a time-
dependent potential might explain such collective effects. Substantial evidence for rapid relaxation
accumulated over the next twenty years derived both from numerical simulations of many-body
systems and from the interpretation of observations of galaxies that have been involved in
χ is tD
-1. Further details can