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From long-range interactions
to collective behaviour
and from hamiltonian chaos
to stochastic modelsYves Elskens
umr6633 CNRS — univ. ProvenceMarseille
Coulomb’05 High intensity beam dynamicsSeptember 12 - 16, 2005 – Senigallia (AN), Italy
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)
- n i,j kn-2 cos kn.(qi-qj) dt
2 qj = (1/m) n En(qj)
En(x) = - j kn-1 sin kn.(x-qj)
r,n Ar,n(t) sin (kn.x - r,nt)with envelopes A varying slowlyAntoni, Elskens & Sandoz, Phys. Rev. E 57 (1998) 5347
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1 wave and 1 particle
• Integrable system
• Locality in velocity : p-j/kj 2 ~ 4 j 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, ) = i pi
2/2 + j j Ij - i,j j Ij1/2 cos (kjqi-j)
coupling type mean field (global), 2 speciesconstants : 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) = j=1
N (yj-xj(t))
Evolution (Liouville) : df/dt = -[H,f] tf + j (pj/m).f /qj + j Fj(x).f /pj = 0
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Kinetic concepts
• Observations : space (Boltzmann) R6
Instantaneous state : {(q1,p1), ..., (qN,pN)}Marginal distribution :
f(1)(q1,p1,t) dq1dp1 = .. f(N)(q1,p1,t) j=2
N dqjdp1 ... symmetrized :
f(1s)(q,p,t) = N-1 j f(1)(qj,pj,t)
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Kinetic concepts
• Realization : f(1s)(y,t) = N-1 j=1N (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) dpn 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) j 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 j=2
N 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) et
: 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 j=1
N (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, ) = i pi
2/2 + j j Ij - i,j j Ij1/2 cos (kjqi-j)
• 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 : L = 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 : Lt = 200
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Warm beam instability
N2 : Lt = 200particles initially in range0.99 < v < 1.00 1.03 < v < 1.04
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Vlasov
• Casimir invariants dt f(1s)(q,p,t) = 0
dt [f(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 > 01 = sup limt ln x(t) / x(0) 1+2 = sup limt ln a12(t) /
a12(0) a12(t) = x1(t) x2(t)
...
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Chaos & entropy production
• Hamilton Poincaré-Cartan : dt j=1
3N dpj dqj = 0 symmetric spectrum 6N-j = -j
Liouville : dt j=13N dpjdqj = 0
sum j=16N j = 0
no attractor !
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Chaos & entropy production
• Dynamical complexity : entropy production per time unit
dSmacro/dt < kB hKS ~ kB j j+
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) 1Dorfman, 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), dilutionBoltzmann 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(, p*-p) d
dp*
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Boltzmann
• Valid with probability 1 in Grad limit : N , Nr2 = cst
for 0 < t < free/5or 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 > 0and = 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 collisionstf(1s) + (p/m).qf(1s) + Fext(q,p).pf(1s)
= - p. U.(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, ) = i pi
2/2 + j j Ij - i,j j Ij1/2 cos (kjqi-j)
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-j/kj 2 ~ 4 j Ij1/2
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1 particle in 2 waves
• Resonance overlaps = [2(1I1
1/2)1/2+2(2I21/2)1/2] / / 1/k1 - 2/k2
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1 particle in M waves
Bénisti & Escande, Phys. Plasmas 4 (1997) 1576
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Quasilinear limit
• 0 < corr ~ M-1 < t < QL (gas : cf. free) dt q = v dt v = j j kj Ij
1/2 sin (kjq - j) ~ white noise
QL > J-1/3 ln s4/3 (or larger)
• t > box : dynamical independencebox ~ 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 limitresonance box (Bénisti & Escande)
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Quasilinear limit : M (s), j random
• Dense wave spectrum vj+1-vj = vj ~ M-1 : “particle diffusion” (Smoluchowski-Fokker-Planck)
t f = v (2 J v f )
• Coupling coefficients (v) = (j/kj) = N j
2/4
• Waves : J(v) = J(j/kj) = kj Ij /(N vj)
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Quasilinear limit : M (s), N
• Dense wave spectrum vj+1-vj = vj ~ M-1 : t f = v Q
• Many particles, poorly coherent : induced and spontaneous emission
t J = Q
• Reciprocity of wave-particle interactions Q = 2 J v f – Fspont f Fspont(v) = - 2 /(N vj)
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Quasilinear limit
• H theorem S = - [f ln (f /f0) + (2)-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