Coulomb’05 High intensity beam dynamics
September 12 - 16, 2005 – Senigallia (AN), Italy
From long-range interactions
to collective behaviour
and from hamiltonian chaos
to stochastic models
Yves Elskens
umr6633 CNRS — univ. Provence
Marseille
http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=IP464
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1. Effective dyn., collective deg. freedom
2. Kinetic concepts
3. Vlasov
4. Limitations, extensions : macroparticle,
granularity (N<), entropy production...
• 5. Boltzmann, Landau, Balescu-Lenard
• 6. Quasilinear limit : transport
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1. Long range yields collective
degrees of freedom
• Ex. mollified Coulomb (Fourier truncated) :
H(q,p) = i pi2/(2m)
- k n i,j kn-2 cos kn.(qi-qj)
dt2 qj = (1/m) n En(qj)
En(x) = - k j kn-1 sin kn.(x-qj)
 r,n Ar,n(t) sin (kn.x - wr,nt)
with envelopes A varying slowly
Antoni, Elskens & Sandoz, Phys. Rev. E 57 (1998) 5347
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1 wave and 1 particle
• Integrable system
• Locality in velocity : p-wj/kj 2 ~ 4 kj Ij1/2
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Beam-plasma paradigm
Underlying plasma
electrostatic modes
(Langmuir, Bohm-Gross)
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M waves and N particles
• Effective lagrangian
• Effective hamiltonian
H(p, q, I, f)
= i pi2/2 + j wj Ij - i,j kj Ij1/2 cos (kjqi-fj)
coupling type mean field (global), 2 species
constants : H, P = i pi + j kj Ij
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Effective hamiltonian
• Dynamical reduction to an effective
lagrangian and hamiltonian (“good chaos”
vs quasi-constants of motion) N0 >> M + N1
Ex. : N0 particles, Coulomb
 M modes (collective, principal)
+ N1 particles (resonant or test)
Effective dynamics & thermodynamics
Elskens & Escande, Microscopic dynamics of plasmas and chaos
(IoP, 2002)
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2. micro- < ... < macroscopic :
Kinetic concepts
• Phase space for the dynamics : R6N
Instantaneous state : x = ((q1,p1), ...,
(qN,pN))
Probability distribution : f(N)(x,t) dNx
Realization : f(N)(y,t) = Pj=1N d(yj-xj(t))
Evolution (Liouville) : df/dt = -[H,f]
tf + Sj (pj/m).fSenigallia,
/qjSeptember
+ Sj 2005
Fj(x).f /pj = 0
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Kinetic concepts
• Observations : m space (Boltzmann) R6
Instantaneous state : {(q1,p1), ..., (qN,pN)}
Marginal distribution :
f(1)(q1,p1,t) dq1dp1
= .. f(N)(q1,p1,t) Pj=2N dqjdp1
... symmetrized :
f(1s)(q,p,t) = N-1 Sj f(1)(qj,pj,t)
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Kinetic concepts
• Realization : f(1s)(y,t) = N-1 Sj=1N d(yj-xj(t))
Evolution (BBGKY) :
tf(1) + (p/m).qf(1) + F(q,p).pf(1) = 0
with F(q,p) = F[f (N)] = ...
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Kinetic concepts
• Fluid moments :
n(q,t) = N  f(1s)(q,p,t) dp
n u(q,t) = N  (p/m) f(1s)(q,p,t) dp
...
• Conservation laws by integration and
closure
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Kinetic concepts
• Weak coupling : molecular independence
approximation
f(N)(q,p,t)  Pj f(1)(qj,pj,t)
... coherent with Liouville ? No !
... supported by dynamical chaos ?
... good approximation ?
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3. Vlasov
• Coupling of mean field type :
F1(q,p) = F1[f (N)] = N-1 Sj=2N F1j(qj-q1)
and for N :
F1(q,p)   F1j(q’-q1) f (1s) (q’,p’) dq’dp’
if the force is smooth enough (not pure
Coulomb – OK if mollified)
then : Vlasov
Spohn, Large scale dynamics of interacting particles (Springer, 1991)
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Vlasov
• Estimates for separation of solutions
 f(1s)(y,t) - g(1s)(y,t) 
<  f(1s)(y,0) - g(1s)(y,0) elt 
l : majorant for Liapunov exponent in R6N
idea : test particles
norm .  weak enough for Dirac
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Vlasov
• Ex. : g(1s)(y,0) “smooth”
f(1s)(y,0) = N-1 Sj=1N d(yj-xj(0))
 f(1s)(y,0) - g(1s)(y,0) < c N-1/2
• limN limt  limt limN
Firpo, Doveil, Elskens, Bertrand, Poleni & Guyomarc'h, Phys. Rev. E
64 (2001) 026407
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4. M waves and N particles
• Effective hamiltonian
mean field, 2 species
H(p, q, I, f)
= i pi2/2 + j wj Ij - i,j kj Ij1/2 cos (kjqi-fj)
• for M fixed, N : Vlasov
• M=1 : free electron laser, CARL, ...
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4.1. Cold beam instability
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Cold beam instability
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Cold beam instability
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Cold beam instability
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4.2. Instability and damping
Warm beam : gL = c df/dv
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Warm beam instability
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Warm beam instability
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Warm beam instability
N2 : gLt = 200
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Warm beam instability
N2 : gLt = 200
particles initially in range
0.99 < v < 1.00
1.03 < v < 1.04
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Vlasov
• Casimir invariants
dt f(1s)(q,p,t) = 0
 dt  F[f(1s)(q,p,t)] dq dp = 0 (if exists)
conserve all entropies !
• Trend to equilibrium ? No hamiltonian
attractor !... but weak convergence
 g(q,p) f(1s)(q,p,t) dq dp (for any g)
via filamentation
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Warm beam instability
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4.3. Thermalization (M=1)
Dynamics : non-linear regimes (trapping)
Canonical ensemble : phase transition
Firpo & Elskens, Phys. Rev. Lett. 84 (2000) 3318
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Thermalization (M>>1)
Y. Elskens & N. Majeri (2005)
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4.4. Chaos & entropy production
• Chaos : Liapunov exponents > 0
l1 = sup limt ln dx(t) / dx(0) 
l1+l2 = sup limt ln da12(t) / da12(0) 
da12(t) = dx1(t)  dx2(t)
...
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Chaos & entropy production
• Hamilton
 Poincaré-Cartan : dt Sj=13N dpj  dqj = 0
 symmetric spectrum l6N-j = -lj
 Liouville : dt Pj=13N dpjdqj = 0
 sum Sj=16N lj = 0
 no attractor !
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Chaos & entropy production
• Dynamical complexity : entropy production
per time unit
dSmacro/dt < kB hKS ~ kB Sj lj+
Arnold & Avez, Problèmes ergodiques de la mécanique classique
(Gauthier-Villars, 1967)
Pesin, Russ. Math. Surveys 32 n°4 (1977) 55
Elskens, Physica A 143 (1987) 1
Dorfman, An introduction to chaos in nonequilibrium statistical
mechanics (Cambridge, 1999)
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5. Kinetic approach :
Boltzmann and variations
• Forces with short range (collisions), dilution
Boltzmann Ansatz :
tf(1s) + (p/m).qf(1s) + Fext(q,p).pf(1s)
= Q[f(2s)]
(BBGKY)
 Q[f(1s)  f(1s)] (non-local in p)
=   (f+(1s) f*+(1s) - f(1s) f*(1s)) b(w, p*-p) dw dp*
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Boltzmann
• Valid with probability 1 in Grad limit :
N , Nr2 = cst
for 0 < t < tfree/5
or for expansion in vacuum...
 longer time ? open problem !
Spohn, Large scale dynamics of interacting particles (Springer, 1991)
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Boltzmann
• Entropy :
n sBoltzmann(q,t)
= - kB  f(1s)(q,p,t) ln (f(1s)(q,p,t)/f0) dp
• H theorem : dsBoltzmann/dt > 0
and = iff f(1s) locally maxwellian ; then
sBoltzmann[f(1s)] = smicrocan[n,e]
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Boltzmann
• Irreversibility... byproduct of symmetry
(microreversibility) of collisions
• H theorem : tool for existence and regularity
of solutions
Friedlander & Serre, eds, Handbook of mathematical fluid dynamics
(Elsevier, 2001,... )
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Landau,
Balescu-Lenard-Guernsey
• Forces with long range and collisions
tf(1s) + (p/m).qf(1s) + Fext(q,p).pf(1s)
= - p. kU.(p* - p) (f(1s) f*(1s)) dp*
U =  (...)dk (Coulomb, Fourier)
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Landau,
Balescu-Lenard-Guernsey
• H theorem, maxwellian equilibria
• Diagrammatic derivation... “challenge for
the future”
Balescu, Statistical dynamics (Imperial college press, 1997)
Spohn, Large scale dynamics of interacting particles (Springer, 1991)
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6. M waves and N particles
(weak Langmuir turbulence)
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M waves and N particles
• Effective hamiltonian
H(p, q, I, f)
= i pi2/2 + j wj Ij - i,j kj Ij1/2 cos (kjqi-fj)
mean field type coupling, 2 species
constants : H, P = i pi + j kj Ij
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1 wave and 1 particle
• Integrable system
• Locality in velocity : p-wj/kj 2 ~ 4 kj Ij1/2
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1 particle in 2 waves
• Resonance overlap
s = [2(k1I11/2)1/2+2(k2I21/2)1/2] /
/ w1/k1 - w2/k2 
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1 particle in M waves
Bénisti & Escande, Phys. Plasmas 4 (1997) 1576
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Quasilinear limit
• 0 < tcorr ~ M-1 < t < tQL (gas : cf. tfree)
dt q = v
dt v = j kj kj Ij1/2 sin (kjq - fj)
~ white noise
tQL > J-1/3 ln s4/3
(or larger)
• t > tbox : dynamical independence
tbox ~ J-1/3
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Stochasticity in parameters
 dynamical chaos
(1 particle in M waves)
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Stochasticity in parameters
 dynamical chaos
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Quasilinear limit
resonance box (Bénisti & Escande)
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Quasilinear limit :
M (s), fj random
• Dense wave spectrum vj+1-vj = Dvj ~ M-1 :
“particle diffusion” (Smoluchowski-FokkerPlanck)
t f = v (2 a J v f )
• Coupling coefficients
a(v) = a(wj/kj) = p N kj2/4
• Waves : J(v) = J(wj/kj) = kj Ij /(N Dvj)
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Quasilinear limit :
M (s), N
• Dense wave spectrum vj+1-vj = Dvj ~ M-1 :
t f = v Q
• Many particles, poorly coherent :
induced and spontaneous emission
t J = Q
• Reciprocity of wave-particle interactions
Q = 2 a J v f – Fspont f
Fspont(v) = - 2 a /(N Dvj)
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Quasilinear limit
• H theorem
S = -  [f ln (f /f0) + (2a)-1 Fspont ln J] dv
• No Casimir invariants for f(v,t)
• Phenomenological equations of markovian
type : regeneration of instantaneous
stochasticity by “good dynamical chaos”
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Conclusion
• Long-range  mean field, collective
degrees of freedom + fewer particles
• Smooth  Vlasov (+ macroparticle)
• Mean field (e.g. charged particles) simpler
than short range (gas) for H-theorem and
kinetic eqn
• limN limt  limt limN
• N<  finite grid
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Landau damping
(non dissipative)
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Landau damping
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Landau damping
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Landau damping
Dynamics : non-linear regimes (trapping)
Canonical ensemble : phase transition
Firpo & Elskens, Phys. Rev. Lett. 84 (2000) 3318
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