Statistical mechanics and Vlasov equation allow for a simplified hamiltonian description of single pass free electron laser saturated dynamics

2 MB·English

by  Andrea Antoniazzi

Statistical mechanics and Vlasov equation allow for a simplified Hamiltonian
description of Single-Pass Free Electron Laser saturated dynamics
Andrea Antoniazzi1,Yves Elskens2,
Duccio Fanelli1,3, Stefano Ruffo1
6 1.Dipartimento di Energetica,
0 Universit`a di Firenze and INFN,
0 via S. Marta, 3, 50139 Firenze, Italy
2 2. Equipe Turbulence Plasma de l’UMR 6633 CNRS–Universit´e de Provence,
n case 321, campus Saint-J´erˆome,
a F-13397 Marseille cedex 13, France
J 3. Department of Cell and Molecular Biology,
7 Karolinska Institute, SE-171 77 Stockholm, Sweden
1
(Dated: February 6, 2008)

]
h A reduced Hamiltonian formulation to reproduce the saturated regime of a Single Pass Free
c Electron Laser, around perfect tuning, is here discussed. Asymptotically, Nm particles are found
e to organize in a dense cluster, that evolves as an individual massive unit. The remaining particles
m fill the surrounding uniform sea, spanning a finite portion of phase space, approximately delimited
- by the average momenta ω+ and ω−. These quantities enter the model as external parameters,
t which can be self-consistently determined within theproposed theoretical framework. To this aim,
a
we make use of a statistical mechanics treatment of the Vlasov equation, that governs the initial
t
s amplification process. Simulations of the reduced dynamics are shown to successfully capture the
.
t oscillating regime observed within theoriginal N-body picture.
a
m
- I. GENERAL BACKGROUND
d
n
o Free-Electron Lasers (FELs) are coherent and tunable radiation sources, which differ from conventional lasers in
c using a relativistic electron beam as their lasing medium, hence the term free-electron.
[
Thephysicalmechanismresponsibleforthelightemissionandamplificationistheinteractionbetweentherelativistic

1 electronbeam, a magnetostatic periodic field generatedin the undulator and an opticalwavecopropagatingwith the
v electrons. Due to the effect of the magnetic field, the electrons are forced to follow sinusoidal trajectories, thus
2 emitting synchrotronradiation. Thisspontaneous emission isthen amplifiedalongthe undulatoruntilthe lasereffect
6
isreached. Amongdifferentschemes,single-passhigh-gainFELsarecurrentlyattractinggrowinginterest,astheyare
3
promising sources of powerful and coherent light in the UV and X ranges. Besides the Self Amplified Spontaneous
1
Emission (SASE) setting [1], seeding schemes may be adopted where a small laser signal is injected at the entrance
0
6 of the undulator and guides the subsequent amplification process [2]. In the following we shall refer to the latter
0 case. Basic features of the system dynamics are successfully captured by a simple one-dimensional Hamiltonian
/ model [15] introduced by Bonifacio and collaboratorsin [3]. Remarkably, this simplified formulation applies to other
t
a physical systems, provided a formal translation of the variables involved is performed. As an example, focus on
m
kinetic plasma turbulence, e.g. the electron beam-plasma instability. When a weak electron beam is injected into
- a thermal plasma, electrostatic modes at the plasma frequency (Langmuir modes) are destabilized. The interaction
d
of the Langmuir waves and the electrons constituting the beam can be studied in the framework of a self-consistent
n
Hamiltonian picture [4], formally equivalent to the one in [3]. In a recent paper [5] we established a bridge between
o
c these two areas of investigation (FEL and plasma), and exploited the connection to derive a reduced Hamiltonian
: model to characterize the saturated dynamics of the laser. According to this scenario, N particles are trapped in
v m
the resonance, i.e. experience a bouncing motion in one of the (periodically repeated) potential wells, and form a
i
X clump that evolves as a single macro-particle localized in space. The remaining particles populate the surrounding
r halo, being almost uniformly distributed in phase space between two sharp boundaries, whose average momentum is
a
labeled ω+ and ω−. The issue of providing a self-consistent estimate for the external parameters Nm, ω+ and ω− is
addressed and solved in this paper.
Thislong-standingproblemwasfirstpointedoutbyTennysonetal. inthepioneeringwork[6]andrecentlyrevisited
in[4]. AfirstattempttocalculateN ismadein[9]whereasemi-analyticalargumentisproposed. Inthisrespect,the
m
strategy here proposed applies to a large class of phenomena whose dynamics can be modeled within a Hamiltonian
framework [4, 7] displaying the emergence of collective behaviour [8].
The paper is organized as follows. In Section II we introduce the one-dimensional model of a FEL amplifier [3]
and review the derivation of the reduced Hamiltonian [5, 6]. Section III recalls the statistical mechanics approachto
estimate the saturated laser regime. In Sections IV to VI the analytic characterizationof Nm, ω+ and ω− is given in
2
details and the results are then tested numerically in section VII. Finally, in Section VIII we sum up and draw our
conclusions.
II. FROM THE SELF-CONSISTENT N-BODY HAMILTONIAN TO THE REDUCED FORMULATION
Under the hypothesis of one-dimensional motion and monochromatic radiation, the steady state dynamics of a
Single-Pass Free Electron Laser is described by the following set of equations:
dθ
j
= p , (1)
j
dz¯
dp
j = Aeiθj A∗e−iθj , (2)
dz¯ − −
dA 1
= iδA+ e−iθj , (3)
dz¯ N
j
X
where z¯ = 2k ρzγ2/ γ 2 is the rescaled longitudinal coordinate, which plays the role of time. Here, ρ =
[a ω /(4ck )]2/w3/γ rishthi0e so-called Pierce parameter, γ the mean energy of the electrons at the undulator’s
w p w r 0
h i
entrance, k = 2π/λ the wave number of the undulator, ω = (4πe2n/m)1/2 the plasma frequency, c the speed
w w p
of light, n the total electron number density, e and m respectively the charge and mass of one electron. Further,
a = eB /(k mc2), where B is the rms peak undulator field. Here γ = λ (1+a2)/2λ 1/2 is the resonant
w w w w r w w
energy, λ and λ being respectively the period of the undulator and the wavelength of the radiation field. In-
w
(cid:0) (cid:1)
troducing the wavenumber k of the FEL radiation, the two canonically conjugated variables are (θ,p), defined as
θ =(k+k )z 2δρk zγ2/ γ 2 andp=(γ γ )/(ρ γ ). θ correspondstothephaseoftheelectronswithrespectto
w − w r h i0 −h i0 h i0
the ponderomotive wave. The complex amplitude A=A +iA represents the scaled field, transversalto z. Finally,
x y
the detuning parameter is given by δ = ( γ 2 γ2)/(2ργ2), and measures the average relative deviation from the
h i0 − r r
resonance condition.
The above system of equations (N being the number of electrons) can be derived from the Hamiltonian
N p2 I N
j
H = δI+2 sin(θ ϕ), (4)
j
2 − N −
j=1 r j=1
X X
wheretheintensityI andthephaseϕofthewavearegivenbyA= I/Nexp( iϕ). Herethecanonicallyconjugated
−
variables are (p ,θ ) for 1 j N and (I,ϕ). Besides the “energy” H, the total momentum P = p +I is also
j j ≤ ≤ p j j
conserved. By exploiting these conserved quantities, one can recast the FEL equations of motion in the following
P
form for the set of 2N conjugate variables (q ,p ) [10]:
j j
N
1
q˙ = p sinq +δ , (5)
j j l
− √NI
l=1
X
I
p˙ = 2 cosq , (6)
j j
− N
r
where the dot denotes derivation with respect to z¯, and q = θ ϕ mod(2π) is the phase of the jth electron in a
j j
−
proper reference frame. The fixed points of system (5)-(6) are determined by imposing q˙ =p˙ =0 and solving:
j j
N
1
p sinq +δ = 0 , (7)
j l
− √NI
l=1
X
I
2 cosq = 0. (8)
j
N
r
An elliptical fixed point is found for q =q¯=3π/2. The conjugate momentum solves (p¯+δ) P/N p¯+1=0 and
i
−
therefore depends on P/N. We shall return on this issue in the following Sections.
p
3
Foramonokineticinitialbeamwithvelocityresonantwiththewave,equations(1),(2)and(3)predictanexponential
instabilityandalateoscillatingsaturationfortheamplitudeoftheradiationfield. Numericalsimulationsfullyconfirm
this scenario as displayed in fig. 1. In the single particle (q,p) space, a dense core of particles is trapped by the wave
and behaves like a large “macro-particle”, that evolves coherently in the resonance. The distances between these
particles do not grow exponentially fast (as is the case for chaotic motion) but grow at most linearly with time
(for particles trapped in the resonance with different adiabatic invariants, i.e.essentially different action in the single
particle pendulum-like description). This linear-in-time departure of the particles appears in the differential rotation
in fig. 2, while the remaining particles are almost uniformly distributed between two oscillating boundaries. Having
observed the formation of such structures in the phase-space allowed to derive a simplified Hamiltonian model to
characterize the asymptotic evolution of the laser [5, 6]. This reduced formulation consists in only four degrees of
freedom,namelythe wave,the macro-particleandthe twoboundariesdelimitingthe portionofspaceoccupiedbythe
so-called chaotic sea, i.e. the uniform halo surrounding the inner core.
1.5
1.25
2 1
|A|
0.75
0.5
0.25
0
0 20 40 60 80 100
z
FIG. 1: Evolution of the radiation intensity as follows from equations (1), (2) and (3). N = 104 electrons are simulated, for
an initial mono-energetic profile. Here δ=0 and I(0)≃0. Particles are initially uniformly distributed in space.
In[5]wehypothesizedthemacro-particletobeformedbyN individualmassiveunits,andintroducedthevariables
m
(ζ,ξ) to label its position in the phase space.
The N = N N particles of the surrounding halo are treated as a continuum with constant phase space
c m
−
distribution, fsea(θ,p,z¯)=fc, between two boundaries, namely p+(θ,z¯) and p−(θ,z¯) such that:
p± =p0±+p±exp(iθ)+p∗±exp( iθ) , (9)
−
where p0 represents their mean velocity. These assumptions allow to map the original system, after linearizing with
± e e
respect to p±, into [5]:
e ζ¨ = iΦeiζ iΦ∗e−iζ (10)
−
1 i
V˙± = ω±V±+iΦ (11)
2 −2
Φ˙ = i Nc V+−V− +iNme−iζ +iδΦ (12)
2 N ω+ ω− N
−
where [16]
4
A = iΦ (13)
−
p± = V±/2 (14)
p0± = ω± (15)
e
Normalizing the density in the chaotic sea to unitey yields fc = 1/(2π∆ω), where ∆ω := ω+ −ω− represents the
(average)width of the chaotic sea. The above system can be cast in a Hamiltonian form by introducing new actions
I± and their conjugate angles ϕ±:
I ∆ω
V = 4 + e−iϕ+ (16)
+
N
r c
V− = 4I−∆ωeiϕ−. (17)
N
r c
A pictorial representation of the main quantities involved in the analysis is displayed in fig. 3. In addition:
I
Φ=iA= e−i(ϕ+π2). (18)
− N
r
The reduced 4-degrees-of-freedomHamiltonian reads, up to a constant irrelevant to the evolution equations:
ξ2
H4 = δI +ω+I+ ω−I− 2α II+sin(ϕ ϕ+) II−sin(ϕ+ϕ−) 2β√Isin(ϕ ζ), (19)
2N − − − − − − −
m
hp p i
2 2
0 0
-2 -2
-4 a) -4 b)
p 0 1 2 3 4 5 6 0 1 2 3 4 5 6
2 2
0 0
-2 -2
-4 -4
c) d)
0 1 2 3 4 5 6 0 1 2 3 4 5 6
θ
FIG.2: Phasespaceportraits fordifferentpositionalongtheundulator[z¯=a)80,b)81,c)83, d)84]. Thedifferentialrotation
of the macro-particle is clearly displayed. For the parameters choice refer to the caption of Fig. 1.
5
FIG. 3: (q,p) phase space portrait in the deep saturated regime for a monokinetic initial beam (I(0) ≃ 0, pj(0) = 0 and q
uniformly distributed in [0,2π]). The two solid lines result from a numerical fit performed according to the following strategy.
First, the particles located close to the outer boundaries are selected and then the expression p±(q)=ω±∓|V±|sin(q+B±) is
numericallyadjustedtointerpolate theirdistribution. Here, |V±|,B± andω± arefreeparameters. Thenumericsarecompatible
with the simplifying assumption B+ =B− ≃0.
where
N
c
α = (20)
N∆ω
r
N
m
β = (21)
√N
The first four terms represent the kinetic energy of the macro-particle, the oscillation of the wave and the harmonic
contributionsassociatedto the oscillationofthe chaoticsea boundaries. The remaining terms referto the interaction
energy. Total momentum is P = ξ+I +N ω¯. The Hamiltonian (19) allows for a simplified description of the late
c
nonlinear regime of the instability, provided the three parameters ω+, ω− and Nm are given.
To achieve a complete and satisfying theoretical description we need to provide an argument to self-consistently
estimate these coefficients. To this end, we shalluse the analyticalcharacterizationof the asymptotic behaviorof the
laser intensity and beam bunching (a measure of the electrons spatial modulation) obtained in [11] with a statistical
mechanics approach. In the next section these results are shortly reviewed.
III. STATISTICAL THEORY OF SINGLE-PASS FEL SATURATED REGIME
As observed in the previous Section, the process of wave amplification occurs in two steps: an initial exponential
growth followed by a relaxation towards a quasi-stationary state characterized by large oscillations. This regime is
governed by the Vlasov equation, rigorously obtained by performing the continuum limit (N at fixed volume
→ ∞
andenergyper particle)[4, 11,12]onthe discretesystem(1-3). Formally,the followingVlasov-wavesystemis found:
∂f ∂f ∂f
= p +2(A cosθ A sinθ) , (22)
x y
∂z¯ − ∂θ − ∂p
dA
x
= δA + fcosθdθdp , (23)
y
dz¯ −
Z
dA
y
= δA fsinθdθdp . (24)
x
dz¯ −
Z
6
The latter conserves the pseudo-energy per particle
p2
ǫ= f(θ,p)dθdp δ(A2 +A2)+2 (A sinθ+A cosθ)f(θ,p)dθdp (25)
2 − x y x y
Z Z
and the momentum per particle
σ = pf(θ,p)dθdp+(A2 +A2) . (26)
x y
Z
A subsequent slow relaxation towards the Boltzmann equilibrium is observed. This is a typical finite-N effect
and occurs on time-scales much longer than the transit trough the undulator [3, 4, 13]. For our calculations we are
interested in the first saturated state. To estimate analytically the average intensity and bunching parameter in this
regime we exploit the statistical treatment of the Vlasov equation, presented in [11]. In the following, we provide a
shortoutline ofthe strategy. Since the Gibbs ensembles areequivalentfor this model, note thatthe same expressions
are recoveredthrough a canonical calculation [4, 11, 14].
The basic idea is to coarse-grain the microscopic one-particle distribution function f(θ,p,z¯). An entropy is then
associated to the coarse-grained distribution f¯, which essentially counts a number of microscopic configurations.
Neglecting the contribution of the field, since it represents only one degree of freedom within the (N +1) of the
Hamiltonian (4), one assumes
f¯ f¯ f¯ f¯ f¯ f¯
s(f¯)= ln + 1 ln 1 f dθdp ln f dθdp , (27)
0 0
− f f − f − f ≃− f f
Z (cid:20) 0 0 (cid:18) 0(cid:19) (cid:18) 0(cid:19)(cid:21) Z (cid:20) 0 0(cid:21)
where the constant f is related to the initial distribution [17].
0
The equilibrium is computed by maximizing this entropy, while imposing the dynamical constraints. This corre-
sponds to solving the constrained variational problem
S(ǫ,σ)= max s(f¯) H(f¯,A ,A )=Nǫ;P(f¯,A ,A )=Nσ; f(θ,p)dθdp=1 , (28)
x y x y
f¯,Ax,Ay(cid:18) (cid:12) Z (cid:19)
(cid:12)
which leads to the equilibrium values (cid:12)
e−β(p2/2+2Asinθ)−λp−µ
f¯ = f (29)
01+e−β(p2/2+2Asinθ)−λp−µ
β
A = A2 +A2 = sin(θ)f¯(θ,p)dθdp, (30)
x y βδ λ
q − Z
where β, λ and µ are the Lagrange multipliers for the energy, momentum and normalization constraints and, in
addition, we have assumed the non-restrictive condition cos(θ ) = 0 [11]. Using then the three equations for the
i
constraints, the statistical equilibrium calculation is reduced to finding the values of the multipliers as functions of
P
energy ǫ and momentum σ. These equations lead directly to the estimates of the equilibrium values for the intensity
I and bunching parameter b= exp(iθ )/N.
i
| |
In the following, we focus on the case of an initially monokinetic beam injected at the wave velocity, while the
P
initial wave intensity is negligible, so that ǫ = 0 and σ = 0. Moreover we let f , which amounts to µ in
0
→ ∞ → ∞
eq. (29). Results are displayed in fig. 4 showing remarkably good agreement between theory and simulations, below
the critical thresholdδ 1.9 that marks the transitionbetween high and low gainregimes. This transition is purely
c
≃
dynamical and cannot be reproduced by the statistical calculation.
Analytically, it turns out that
I (2/(3A3 2δ A))
b= A3 Aδ = 1 | | − | | (31)
| | −| | I (2/(3A3 2δ A))
0
| | − | |
whereI isthe modifiedBesselfunctionofordern. Inparticularforδ =0,onefinds A2 =I/N 0.65andb 0.54.
n
| | ≃ ≃
7
2

g
n
hi Intensity
c
n 1.5
u
B

y,
t
si
n
e 1
t
n
I Bunching Threshold

r
e
s
a
L
0.5
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
δ
FIG. 4: Comparison between theory (solid and long-dashed lines) and simulations (symbols) for a monoenergetic beam with
ǫ=0, σ =0, when varying the detuning δ. The dotted vertical line, δ=δc ≃1.9, represents the transition from the high-gain
to low gain regime [3].
IV. TOWARDS THE ANALYTICAL CHARACTERIZATION OF Nm
Aspreviouslydiscussed,onecanpredictthevalueofthebunchingparameterb,usingtheabovestatisticalmechanics
description. Clearly,thebunchingparameterbdependsonthespatialdistributionoftheparticles. Fromitsdefinition
it immediately follows:
2 2 1/2
N N
1 1
b= cosq + sinq . (32)
i i
 N ! N ! 
i=1 i=1
X X
 
To proceed we can isolate the contribution relative to the macroparticle from that associated to the chaotic sea.
We thus obtain:
2 2 1/2
1 1 1 1
b= cosq + cosq + sinq + sinq . (33)
i i i i
 N i∈macro N i∈sea ! N i∈macro N i∈sea ! 
X X X X
 
Focus on the first two terms of expression (33). We assume the macroparticle to be ideally localized at the
elliptic fixed point solving (7)-(8), i.e. set q = q¯ = 3π/2, for each individual massive unit belonging to the inner
j
agglomeration. Hence, cosq =N cosq¯=0. Asconcernsthesecondcontribution,recallingtheexpressions
i∈macro i m
for the boundaries p−(q) and p+(q) (see caption of fig. 3), one can formally write:
P
8
1 N 1 2π p+(q)
c
cosq cosq dpdq
i
N i∈sea ≃ N 2π(ω+−ω−)Z0 Zp−(q)
X
N 1 2π
c
= (cosq)[ω+ ω−]dq
N 2π(ω+−ω−)Z0 −
N 1 2π
c
[(V+ + V− )sinq]cosqdq =0. (34)
−N 2π(ω+−ω−)Z0 | | | |
The other contributions in eq. (33) can be estimated as follows:
1 N
m
sinq (35)
i
N ≃− N
i∈macro
X
while
1 N 1 2π p+(q)
c
sinq (sinq) dpdq
i
N i∈sea ≃ N 2π(ω+−ω−)Z0 Zp−(q)
X
N 1 2π
c
= sinq(ω+ ω−)dq
N 2π(ω+−ω−)Z0 −
=0
N 1 2π N V
c | (V+{z+ V− )si}n2qdq = c , (36)
−N 2π(ω+−ω−)Z0 | | | | −N 2∆ω
e
where V = V+ + V− . Inserting (34), (35), (36) into (33) yields:
| | | |
e
N N V
m c
b= + (37)
N N 2∆ω
e
and finally
N 2b∆ω V 2(1 b)∆ω
m
= − =1 − (38)
N 2∆ω V − 2∆ω V
− e −
using the relation N =N N . It is worth emphasizing that, neglecting as a first approximationthe amplitudes of
m c
− e e
the sinusoidal boundaries, i.e. setting V− =V+ =0, the previous equation reduces to
N
m
=b , (39)
N
confirming the relevance of the macroparticle picture to the bunching parameter, a physical quantity of paramount
importance for the FEL dynamics. Formula (37) yields bunching parameter values larger than (39), i.e. implies that
theseaalsocontributestoincreasingthebunchingparameter. Thedifferenceincreasesastheseagetsmorepopulated
and the width ω+ ω− is reduced (note that our approximations require V+ + V− <ω+ ω−, see fig. 3).
− | | | | −
V. ESTIMATING THE AVERAGE MOMENTUM OF THE BOUNDARIES ω±
In this section we estimate the unknown quantities ω± by characterizing their functional dependence on Nm. For
this purpose we introduce (see schematic layout of fig. 3):
∆ω
ω± =ω¯ . (40)
± 2
9
∆ω
6 ω
4
2
0
0 100 200 300 400 500
z
FIG. 5: Solid line: ∆ω vs z¯. Dashed line: ω¯ vs z¯. Parameters are set as discussed in the caption of fig. 1.
The problem of estimating ω± is obviously equivalent to providing a self-consistent calculation for ω¯ and ∆ω. The
latter are both monitored as function of time in fig. 5 and shown to be practically constant.
In the following, we shall focus on the case of a system which evolved from an initially monokinetic beam and an
initially infinitesimal wave. It is then convenient to choose the Galilean reference frame moving at the beam initial
velocity. Thistranslatesintothe conditionsǫ=0andσ =0. Thedetuningδ isarbitrarysofar. Itisofprimeinterest
to consider the special case where the beam is injected at the resonant velocity, so that δ = 0. We shall make this
additional assumption in section VII, but the estimates in this section and in the next one do not require it unless
explicitly stated.
Consider the conservation of momentum for the original N-body system (4) and focus on the asymptotic dynam-
ics, which allows one to isolate the contributions respectively associated to the macroparticle and the chaotic sea.
Averaging over the number of particles yields:
N N
c m
p + p +J =σ (41)
sea macro 0
N N
where p stands for the average momentum of the chaotic sea and the subscript ′0′ labels the initial condition.
sea
To simplify the calculations, we introduced the rescaled intensity J = I/N. As already observed in [5, 6], the
macroparticle rotates in phase space. This rotation is directly coupled to the oscillations displayed by the laser
intensity. Averaging over a bounce period z , one formally gets:
rot
N N
c m
p + p + J =0 (42)
sea macro
N h i N h i h i
where stands for the time average. Focus now on p . Since particles are uniformly filling the chaotic sea, one
sea
h·i h i
can use the approximationoutlined before eq. (9) (see also caption of fig.3):
1 z¯0+zrot 1 p+(q)
psea = pfsea(θ,p,z¯)dθdp dz¯= pdpdθ (ω++ω−)/2=ω¯ . (43)
h i zrot Zz¯0 (cid:18)ZZ (cid:19) 2π(ω+−ω−)Z Zp−(q) ≃
As alreadyoutlined inthe preceding discussion,we assume that the macroparticleoscillatesaroundthe fixed point
andthereforeeachindividualelementconstituting themacroparticleverifiesthe condition q =q¯. Inaddition, from
j
h i
equation (7):
1 1
p :=p¯= sinq + sinq δ. (44)
macro i i
h i *N√J i∈macro + *N√J i∈sea +−
X X
10
To proceed in the analysis, we approximate the right hand side in equation (44) as:
1 1 1 N 1
m
sinq sin q = sinq¯= (45)
i i
*N√J i∈mXacro +≃ N hJii∈mXacro h i N hJii∈mXacro − N hJi
consistentlywiththeargumentafter(33). Tpheaboverelationisderivpedbyperformingalinearizaption(seeAppendix),
validated numerically and supported a posteriori by the correctness of the results. The contribution of the chaotic
sea reads:
1 1 N 1 2π p+(q)
c
sinq sinq sinq dpdq
i j
*N√J i∈Xsea +≃ N hJijX∈sea ≃ N hJi2π(∆ω)Z0 Zp−(q)
p p
N 1 2π 2π N V
= c  (ω+ ω−)sinqdq (V+ + V− )sin2qdq= c . (46)
N J 2π∆ω − − | | | | −N J 2∆ω
h i Z0 Z0  h i e
p  =0  p
 
Merging equations (44), (45)|and (46) a{nzd recallin}g (37) and (31), one obtains
N 1 N V 1 b
m c
p δ = δ = J . (47)
macro
h i≃− N J − N 2∆ω J − − J − −h i
h i e h i h i
Thusthemacroparticlemovesontheaverapgeatthesameveplocityasthecepnterofthechaoticsea. Insertingequations
(43) and (47) into (42) and solving for ω¯, we find,
N N b
m
ω¯ = J + +δ = J . (48)
Nc "−h i N J !# −h i
h i
To getan expressionfor ∆ω, we consider the energy conspervation for the originalN-body model (4). By averaging
over one complete rotation of the macroparticle,we write:
1 N p2 1 N
h i=1 ii +2 √J sinq =δ J . (49)
i
N 2 N h i
P * i=1 +
X
We then bring into evidence the contributions associated to the massive agglomerate and to the particles of the
surrounding halo, for both the kinetic and the interaction terms:
N p2 N p2 1 1
mh macroi + ch seai + 2√J sinq + 2√J sinq =δ J . (50)
i i
N 2 N 2 * N i∈macro + * N i∈sea + h i
X X
Hereafter we make use of p2 p 2, which in turn amounts to assume small oscillations around the
h macroi ≃ h macroi
mean p , consistently with (44) which neglects such oscillations. The kinetic energy associated to the uniform
macro
h i
sea can be estimated as follows:
hp2seai = 1 dθ ω+ p2dp= 1ω+3 −ω−3 = 1 3ω¯2+ ∆ω2 . (51)
2 2π(ω+−ω−)Z Zω− 2 6ω+−ω− 6(cid:18) 4 (cid:19)
Inthisestimateweassumedtheparticlestobedistributeduniformlyinarectangularbox,disregardingthesinusoidal
shape of the boundaries. The modulation of the outer frontiers results in higher order corrections which can be
neglected.
The interaction term follows directly from (44) and (47). Inserting (51) in (50), one finds:
N
∆ω2 = 36 J 2 24δ J . (52)
N h i − h i
c
(cid:2) (cid:3)