1 / 56

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

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.

dante-donovan
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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 Senigallia, September 2005

  2. 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 Senigallia, September 2005

  3. 1. Long range yields collective degrees of freedom • Ex. mollified Coulomb (Fourier truncated) : H(q,p) = åipi2/(2m) - kånåi,jkn½-2 cos kn.(qi-qj) dt2qj = (1/m) ån En(qj)En(x) = - kåjkn-1 sin kn.(x-qj) år,nAr,n(t) sin (kn.x - wr,nt)with envelopes A varying slowlyAntoni, Elskens & Sandoz, Phys. Rev. E 57 (1998) 5347 Senigallia, September 2005

  4. 1 wave and 1 particle • Integrable system • Locality in velocity : p-wj/kj2 ~ 4 kjIj1/2 Senigallia, September 2005

  5. Beam-plasma paradigm Underlying plasma electrostatic modes (Langmuir, Bohm-Gross) Senigallia, September 2005

  6. M waves and N particles • Effective lagrangian • Effective hamiltonian H(p, q, I, f) = åipi2/2 + åjwj Ij - åi,jkjIj1/2 cos (kjqi-fj)coupling type mean field (global), 2 speciesconstants : H, P = åipi + åjkj Ij Senigallia, September 2005

  7. Effective hamiltonian • Dynamical reduction to an effective lagrangian and hamiltonian (“good chaos” vs quasi-constants of motion) N0 >>M+ N1Ex. : 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) Senigallia, September 2005

  8. 2. micro- < ... < macroscopic :Kinetic concepts • Phase space for the dynamics : R6NInstantaneous state : x = ((q1,p1), ..., (qN,pN))Probability distribution : f(N)(x,t) dNxRealization : f(N)(y,t) = Pj=1Nd(yj-xj(t))Evolution (Liouville) : df/dt = -[H,f] tf + Sj (pj/m).f /qj+ SjFj(x).f /pj= 0 Senigallia, September 2005

  9. Kinetic concepts • Observations : m space (Boltzmann) R6Instantaneous 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 Sjf(1)(qj,pj,t) Senigallia, September 2005

  10. Kinetic concepts • Realization : f(1s)(y,t) = N-1 Sj=1Nd(yj-xj(t))Evolution (BBGKY) : tf(1) + (p/m).qf(1)+ F(q,p).pf(1)= 0with F(q,p) = F[f (N)] = ... Senigallia, September 2005

  11. 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 Senigallia, September 2005

  12. Kinetic concepts • Weak coupling : molecular independence approximation f(N)(q,p,t) Pjf(1)(qj,pj,t) ... coherent with Liouville ? No ! ... supported by dynamical chaos ?... good approximation? Senigallia, September 2005

  13. 3. Vlasov • Coupling of mean field type : F1(q,p) = F1[f (N)] = N-1 Sj=2NF1j(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 : VlasovSpohn, Large scale dynamics of interacting particles (Springer, 1991) Senigallia, September 2005

  14. 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 R6Nidea : test particles norm .weak enough for Dirac Senigallia, September 2005

  15. Vlasov • Ex. : g(1s)(y,0) “smooth” f(1s)(y,0) = N-1 Sj=1Nd(yj-xj(0))f(1s)(y,0) - g(1s)(y,0) <cN-1/2 • limNlimt  limtlimNFirpo, Doveil, Elskens, Bertrand, Poleni & Guyomarc'h, Phys. Rev. E 64 (2001) 026407 Senigallia, September 2005

  16. 4. M waves and N particles • Effective hamiltonian mean field, 2 speciesH(p, q, I, f) = åipi2/2 + åjwj Ij - åi,jkjIj1/2 cos (kjqi-fj) • for M fixed, N : Vlasov • M=1 : free electron laser, CARL, ... Senigallia, September 2005

  17. 4.1. Cold beam instability Senigallia, September 2005

  18. Cold beam instability Senigallia, September 2005

  19. Cold beam instability Senigallia, September 2005

  20. Cold beam instability Senigallia, September 2005

  21. 4.2. Instability and damping Warm beam : gL = c df/dv Senigallia, September 2005

  22. Warm beam instability Senigallia, September 2005

  23. Warm beam instability Senigallia, September 2005

  24. Warm beam instability N2 :gLt = 200 Senigallia, September 2005

  25. Warm beam instability N2 :gLt = 200 particles initially in range 0.99 < v < 1.00 1.03 < v < 1.04 Senigallia, September 2005

  26. 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 Senigallia, September 2005

  27. Warm beam instability Senigallia, September 2005

  28. 4.3. Thermalization (M=1) Dynamics : non-linear regimes (trapping)Canonical ensemble : phase transition Firpo & Elskens, Phys. Rev. Lett. 84 (2000) 3318 Senigallia, September 2005

  29. Thermalization (M>>1) Y. Elskens & N. Majeri (2005) Senigallia, September 2005

  30. 4.4. Chaos & entropy production • Chaos : Liapunov exponents > 0l1 = sup limtln dx(t)/dx(0)l1+l2 = sup limtln da12(t)/da12(0)da12(t) = dx1(t) dx2(t) ... Senigallia, September 2005

  31. Chaos & entropy production • Hamilton  Poincaré-Cartan : dt Sj=13N dpj dqj = 0  symmetricspectrum l6N-j = -lj Liouville : dt Pj=13N dpjdqj = 0  sumSj=16Nlj = 0 no attractor ! Senigallia, September 2005

  32. 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) 55Elskens, Physica A 143 (1987) 1Dorfman, An introduction to chaos in nonequilibrium statistical mechanics (Cambridge, 1999) Senigallia, September 2005

  33. 5. Kinetic approach : Boltzmann and variations • Forces with short range (collisions), dilutionBoltzmannAnsatz : 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* Senigallia, September 2005

  34. Boltzmann • Valid with probability 1 in Grad limit : N , Nr2 = cstfor 0 < t < tfree/5or for expansion in vacuum...  longer time ? open problem ! Spohn, Large scale dynamics of interacting particles (Springer, 1991) Senigallia, September 2005

  35. Boltzmann • Entropy :nsBoltzmann(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] Senigallia, September 2005

  36. 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,... ) Senigallia, September 2005

  37. 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) Senigallia, September 2005

  38. 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) Senigallia, September 2005

  39. 6. M waves and N particles(weak Langmuir turbulence) Senigallia, September 2005

  40. M waves and N particles • Effective hamiltonian H(p, q, I, f) = åipi2/2 + åjwj Ij - åi,jkjIj1/2 cos (kjqi-fj)mean field type coupling, 2 speciesconstants : H, P = åipi + åjkj Ij Senigallia, September 2005

  41. 1 wave and 1 particle • Integrable system • Locality in velocity : p-wj/kj2 ~ 4 kjIj1/2 Senigallia, September 2005

  42. 1 particle in 2 waves • Resonance overlaps = [2(k1I11/2)1/2+2(k2I21/2)1/2] / / w1/k1-w2/k2 Senigallia, September 2005

  43. 1 particle in M waves Bénisti & Escande, Phys. Plasmas 4 (1997) 1576 Senigallia, September 2005

  44. Quasilinear limit • 0 < tcorr ~ M-1 < t < tQL (gas : cf. tfree) dtq = v dtv = åjkjkjIj1/2 sin (kjq- fj) ~ white noisetQL>J-1/3 ln s4/3 (or larger) • t > tbox : dynamical independencetbox ~ J-1/3 Senigallia, September 2005

  45. Stochasticity in parameters dynamical chaos(1 particle in M waves) Senigallia, September 2005

  46. Stochasticity in parameters dynamical chaos Senigallia, September 2005

  47. Quasilinear limitresonance box (Bénisti & Escande) Senigallia, September 2005

  48. Quasilinear limit : M (s), fj random • Dense wave spectrum vj+1-vj = Dvj ~ M-1 : “particle diffusion” (Smoluchowski-Fokker-Planck) tf = v(2aJvf) • Coupling coefficients a(v) = a(wj/kj) = pNkj2/4 • Waves : J(v) = J(wj/kj) = kjIj /(N Dvj) Senigallia, September 2005

  49. Quasilinear limit : M (s), N • Dense wave spectrum vj+1-vj = Dvj ~ M-1 : tf= vQ • Many particles, poorly coherent : induced and spontaneous emission tJ = Q • Reciprocity of wave-particle interactions Q= 2aJvf – FspontfFspont(v) = - 2 a /(N Dvj) Senigallia, September 2005

  50. Quasilinear limit • H theorem S = - [f ln (f/f0) + (2a)-1Fspont ln J] dv • No Casimir invariants for f(v,t) • Phenomenological equations of markovian type : regeneration of instantaneous stochasticity by “good dynamical chaos” Senigallia, September 2005

More Related