From long-range interactions to collective behaviour and from hamiltonian chaos

Senigallia, September 2005 1

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


• 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


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


1 wave and 1 particle

• Integrable system

• Locality in velocity : p-j/kj 2 ~ 4 j Ij1/2


Beam-plasma paradigm

Underlying plasma electrostatic modes (Langmuir, Bohm-

Gross)


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


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)


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


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)


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)] = ...


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


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 ?


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)


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


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


4. M waves and N particles

• Effective hamiltonian mean field, 2 species H(p, q, I, ) = i pi


• for M fixed, N : Vlasov

• M=1 : free electron laser, CARL, ...


4.1. Cold beam instability


Cold beam instability






4.2. Instability and damping

Warm beam : L = c df/dv


Warm beam instability





N2 : Lt = 200



N2 : Lt = 200particles initially in range0.99 < v < 1.00 1.03 < v < 1.04


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




4.3. Thermalization (M=1) Dynamics : non-linear regimes (trapping)

Canonical ensemble : phase transition

Firpo & Elskens, Phys. Rev. Lett. 84 (2000) 3318


Thermalization (M>>1)

Y. Elskens & N. Majeri (2005)


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)

...


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 !


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)


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*


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)


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]


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,... )


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)


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)


6. M waves and N particles(weak Langmuir turbulence)


M waves and N particles

• Effective hamiltonian H(p, q, I, ) = i pi


mean field type coupling, 2 species

constants : H, P = i pi + j kj Ij


1 wave and 1 particle

• Integrable system

• Locality in velocity : p-j/kj 2 ~ 4 j Ij1/2


1 particle in 2 waves

• Resonance overlaps = [2(1I1

1/2)1/2+2(2I21/2)1/2] / / 1/k1 - 2/k2


1 particle in M waves

Bénisti & Escande, Phys. Plasmas 4 (1997) 1576


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


Stochasticity in parameters dynamical chaos

(1 particle in M waves)


Stochasticity in parameters dynamical chaos


Quasilinear limitresonance box (Bénisti & Escande)


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)


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)


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”


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



Landau damping(non dissipative)


Landau damping


Landau damping


Landau damping Dynamics : non-linear regimes (trapping)

Canonical ensemble : phase transition

Firpo & Elskens, Phys. Rev. Lett. 84 (2000) 3318

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From long-range interactions to collective behaviour and from hamiltonian chaos