![Page 1: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/1.jpg)
REGGAE – Generator for Uniform Filling of LIPS
Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý
Zimanyi Winter School 6. 12. 2012
![Page 2: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/2.jpg)
e+e- event s at 4 5 +4 5 GeV / c
D E L PH I
A u+A u E vent s at 10 0 +10 0 GeV / c
S T A R
e+e- , L E P H eavy ion, RH I C
2/20
LEP:LHC (pp):LHC (Pb Pb):
n ≥ 4 n ≥ 5,6,8 n ≥ 100
Multiparticle phase space?
![Page 3: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/3.jpg)
Lorentz invariant phase space - LIPS
dynamics
kinematics & statistics
3/20
LIPS:
![Page 4: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/4.jpg)
• to calculate σ by Monte Carlo integration
• to generate events
Uniform filling of LIPS?
4/20
![Page 5: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/5.jpg)
Monte Carlo Integration
x1x2x3 x4 x5x6 x7 x8
Sample mean method
xi have to be uniformly distributed
5/20
Φn
f = |M|2
![Page 6: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/6.jpg)
Event generation
• Take event uniformly distributed in
• Calculate weight w = |M|2 for this event
• Accept this event with probability w/wmax
wmax
w = |M|2
Φn
w
Φn
6/20
![Page 7: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/7.jpg)
GENBOD generator (F. James)
7/20
fills LIPS uniformly
![Page 8: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/8.jpg)
• 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
M2
wM3
wmax
Generate events with Genbod
8/20
M22 = (p1 + p2)
2
M32 = (p1 + p2 + p3)
2
![Page 9: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/9.jpg)
GENBODw/wmax very lowGood for n < 30
RAMBOw/wmax much better, = 1 for massless particlesGood for n < 100 relativistic particles
NUPHAZw/wmax best so far, = 1 for massless particlesBetter than RAMBO, relativistic particles
GENBOD vs other generators
9/20
![Page 10: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/10.jpg)
REGGAE (Tomášik, Mereš, Melo,Balek,Černý)
• 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
Aim to generate pure phase space events with high multiplicity and efficiency for both relativistic and nonrelativistic particles
(REscattering after Genbod GenerAtor of Events)
θ
10/20
Computer Physics Communications 182 (2011) 2561-2566.
![Page 11: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/11.jpg)
Small w Large w
Gas in a box
For large n → Maxwell-Boltzmann with temperature T
11/20
![Page 12: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/12.jpg)
E(GeV)12/20
![Page 13: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/13.jpg)
LIPS
Microcanonical ensemble Boltzmann
LIPS-Boltzmannlarge n
large n
E 13/20
Darwin-Fowler method
V n
![Page 14: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/14.jpg)
Information entropy
14/20
![Page 15: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/15.jpg)
REGGAE vs other generators numerical integration
15/20
![Page 16: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/16.jpg)
n = 30 particles with mass 1 GeV, pa+pb = (100 GeV, 0, 0, 0)
n = 60 particles with mass 1 GeV
16/20
![Page 17: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/17.jpg)
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
![Page 18: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/18.jpg)
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
Vn
19/20
σ
|M|2 = 2E1 2E
2 …. 2E
n Vn
![Page 19: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/19.jpg)
Statistical model of hadronization
fireball hadrons
finite V
20/20
Described by microcanonical phase space
Heavy ioncollision
![Page 20: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/20.jpg)
BACKUP
![Page 21: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/21.jpg)
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
![Page 22: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/22.jpg)
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
Mn Mn-1 M3 M2
n-1n 3 2
1. . .
GENBOD generator (F. James)
Each 2-body decay evaluated in the CM frame of 2 daughters
![Page 23: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/23.jpg)
Standard Numerical Methods of Integration
a b
Rectangular
Trapezoidal
Simpson
6/20
![Page 24: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/24.jpg)
where
12 variables → 2
M22 = (p1 + p2)
2 M32 = (p1 + p2 + p3)
2
GENBOD generator (F. James)
10/20
![Page 25: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/25.jpg)
Pure phase space(Lorentz Invariant Phase Space, LIPS)
numerically
… complicated
1
Pure multiparticle phase space = LIPS
kinematics & statistics4/20
![Page 26: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/26.jpg)
Standard methods vs MC
SimpsonTrapezoidalRectangular
Standard methods
Monte Carlo Number of dimensions
1
2
d
1/n 1/n2 1/n4 1/√n
1/√n 1/n 1/n2 1/√n
1/√n1 1 1
n1/d n2/d n4/d
(Error scaling with n)
8/20
![Page 27: REGGAE – Generator for Uniform Filling of LIPS Ivan Melo, Boris Tomášik, Michal Mereš, Vlado Balek, Vlado Černý Zimanyi Winter School 6. 12. 2012](https://reader030.vdocuments.mx/reader030/viewer/2022032802/56649e175503460f94b029e2/html5/thumbnails/27.jpg)