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REGGAE – Generator for Uniform Filling of LIPS

Ivan Melo , Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý. REGGAE – Generator for Uniform Filling of LIPS. Zimanyi Winter School 6 . 12 . 2012. Multiparticle phase space?. LEP : LHC (pp): LHC (Pb Pb):. n ≥ 4 n ≥ 5,6,8 n ≥ 100. 2/20.

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REGGAE – Generator for Uniform Filling of LIPS

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  1. Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý REGGAE– Generator for Uniform Filling of LIPS Zimanyi Winter School 6. 12. 2012

  2. Multiparticle phase space? LEP: LHC (pp): LHC (Pb Pb): n ≥ 4 n ≥ 5,6,8 n≥100 2/20

  3. Lorentz invariant phase space - LIPS dynamics kinematics & statistics LIPS: 3/20

  4. Uniform filling of LIPS? • to calculate σ by Monte Carlo integration • to generate events 4/20

  5. Monte Carlo Integration f = |M|2 Φn x1 x2 x3 x4 x5 x6 x7 x8 Sample mean method xi have to be uniformly distributed 5/20

  6. Event generation w= |M|2 wmax w Φn • Take event uniformly distributed in • Calculate weight w = |M|2 for this event • Accept this event with probability w/wmax Φn 6/20

  7. GENBOD generator (F. James) fills LIPS uniformly 7/20

  8. Generate events with Genbod M22 = (p1 + p2)2 M3 M32 = (p1 + p2 + p3)2 w wmax M2 • Generate M2, M3 uniformly within kinematic limits • Calculate weight w • Accept M2, M3 with probability w/wmax • Generate angles, calculate momenta, boost to Lab system 8/20

  9. GENBOD vs other generators GENBOD w/wmax very low Good for n < 30 RAMBO w/wmax much better, = 1 for massless particles Good for n < 100 relativistic particles NUPHAZ w/wmax best so far, = 1 for massless particles Better than RAMBO, relativistic particles 9/20

  10. REGGAE (Tomášik, Mereš, Melo,Balek,Černý) (REscattering afterGenbodGenerAtor ofEvents) Computer Physics Communications 182 (2011) 2561-2566. Aim to generate pure phase space events with high multiplicity and efficiency for both relativistic and nonrelativistic particles • Use Genbod in the 1st step to generate event with any w • Let the particles in the event collide virtually to reach the most probable configurations in the phase space with w → 1 θ 10/20

  11. Gas in a box Large w Small w For large n → Maxwell-Boltzmann with temperature T 11/20

  12. E(GeV) 12/20

  13. large n LIPS-Boltzmann LIPS Darwin-Fowler method large n Microcanonical ensemble Boltzmann V n 13/20 E

  14. Information entropy 14/20

  15. REGGAE vs other generators numerical integration 15/20

  16. n = 30 particles with mass 1 GeV, pa+pb = (100 GeV, 0, 0, 0) n = 60 particles with mass 1 GeV 16/20

  17. What next? REGGAE can fill LIPS uniformly for fixed n and chemical composition … but can we predict if total CM energy prefers to convert to, say, n=50 or n=60 particles? I.e. generate events with different n? 17/20

  18. From LIPS to microcanonical phase space? • LIPS counts states in the momentum space, these states are asymptotic (infinite volume) • Microcanonical counts states both in momentum and configuration space which has finite volume V σ Vn |M|2 = 2E1 2E2 …. 2En Vn 19/20

  19. Statistical model of hadronization Heavy ion collision hadrons fireball finite V Described by microcanonical phase space 20/20

  20. BACKUP

  21. Question: Molecules colliding in a box give canonical Boltzmann, how is this different from REGGAE collisions which give LIPS-Boltzmann? Answer: Molecules in a box collide in both momentum and configuration space Question: Can we adjust REGGAE collisions to get canonical Boltzmann? Question: If we get canonical Boltzmann, do we also get uniform filling of the microcanonical phase space? 18/20

  22. GENBOD generator (F. James) Mn→ 1 + 2 + 3 + … + n Mn2 = (pa + pb)2 M2 → 1 + 2 M22 = (p1 + p2)2 M3→ 1 + 2 + 3 → 3 + M2 M32 = (p1 + p2 + p3)2 Each 2-body decay evaluated in the CM frame of 2 daughters n n-1 3 2 . . . Mn Mn-1 M3 M2 1

  23. Standard Numerical Methods of Integration Rectangular Trapezoidal Simpson a b 6/20

  24. GENBOD generator (F. James) M22 = (p1 + p2)2 M32 = (p1 + p2 + p3)2 where 12 variables → 2 10/20

  25. Pure phase space(Lorentz Invariant Phase Space, LIPS) numerically … complicated 1 Pure multiparticle phase space = LIPS kinematics & statistics 4/20

  26. Standard methods vs MC (Error scaling with n) Standard methods Number of dimensions Monte Carlo Rectangular Trapezoidal Simpson 1/n2 1/n4 1/√n 1/n 1 2 1/√n 1/n 1/n2 1/√n 1 1 1 d 1/√n n1/d n2/d n4/d 8/20

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