Dependence of the event Dependence of the event fractal dimensions on the fractal dimensions on the colliding nuclei massescolliding nuclei masses
Oleg Rogachevsky
VHMP 2007
Dubna
18 september 2007 VHMP-2007 2
What is a fractal ?What is a fractal ?
)/1(log
)(loglim
0 r
rND
rKM
...261.13log
4log
3log
4loglim'
n
n
n
arcsKochKMD
Koch’s arcA fractal is a shape made of parts
similar to the whole in some way
E.Feder
•r - scale•N(r) number of elements in fractal at given scale
The fractal dimension of sets observed in the dynamics can be used as
quantitative measure of the chaoticity
18 september 2007 VHMP-2007 3
Fractals in Heavy Ion CollisionsFractals in Heavy Ion Collisions
Collections of small entities (particles or whatever) behave haphazardly, even though physical laws govern the particles individually
Deterministic chaos chaos results from deterministic process
Fractals deal with geometric patterns (represented chaos) and quantitative way of characterizing these patterns. Chaos, in contrast, deals with time evolution and its underlying or distinguishing characteristics
Garnett P. Williams “Chaos Theory Tamed”
18 september 2007 VHMP-2007 4
2 dimension random distribution 2 dimension random distribution
-1 < x < 1, 0 < y < 2
Npoints
Dcorr
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multiplicity dependence multiplicity dependence
Au+Au s = 200 GeV
minbias
18 september 2007 VHMP-2007 6
Au+Au Au+Au s = 200 GeVs = 200 GeV
Fractal dimensions in and pt spaces
18 september 2007 VHMP-2007 8
STAR Au+Au STAR Au+Au s = 200 & 62 GeV s = 200 & 62 GeV
Fractal dimensions in pt space (pt > .15 GeV/c)
entropy events
jet events
pT max
– maximum transverse momentum among the particles in the event
18 september 2007 VHMP-2007 9
STAR Au+Au STAR Au+Au s = 200 GeV s = 200 GeV
Ptmax of events (pt > .15 GeV/c)
18 september 2007 VHMP-2007 10
STAR Au+Au STAR Au+Au s = 200 GeV s = 200 GeV
Nucl. Phys. A757 (2005) 102
18 september 2007 VHMP-2007 11
STAR Au+Au STAR Au+Au s = 200 GeV s = 200 GeV
Binary-scaled yield ratio RCP (pT ) for central (0-5%) relative to peripheral(40-60%, 60-80%) collisions for charged hadrons from 200 GeV Au+Au collisions
Entropy events make main yield in pT spectra from 2.5 to 4 GeV/c but the bump above 2.5 GeV/c in entropy events decreases with growth of centrality
Nucl. Phys. A757 (2005) 102
18 september 2007 VHMP-2007 19
SummarySummary
Fractal dimension analysis are feasible to study event
features in the heavy ion collisions
Different dynamics of the particle interactions reveals throw
changes of the event fractal dimensions
Changes of fractal dimension caused by jet quenching
become more evident with increasing collision energy and
masses of colliding nuclei
Fractal dimension analysis can be applied to study event
structure at LHC, CBM and NICA.
18 september 2007 VHMP-2007 21
Scale dependent analysis Scale dependent analysis
Factorial moments (A. Bialas et al.)
Fractal moments (R. Hwa)
Scaled correlation analysis (T. Trainor)
Wavelet decomposition analysis
Renyi entropy analysis (I. Dremin)
The scale dependent analysis is a space-scale decomposition where the space corresponds to any physical variable
and the scale is the space size
18 september 2007 VHMP-2007 22
STAR Au+Au STAR Au+Au s = 200 & 62 GeV s = 200 & 62 GeV
Fractal dimensions in pt space (pt > .15 GeV/c)
entropy events
jet events
The yield of entropy events increases with growth of multiplicity and collisions energy
18 september 2007 VHMP-2007 23
STAR events analysisSTAR events analysis
all charged particles with: < 1.Azimuthal angle 2pT > 0.15 GeV/c
Nhits > 25
Nhits / N poss.hits > .52
Au+Au s = 200 GeVSTAR ~ 2 000 K events
HIJING ~ 73 K events
Au+Au s = 62 GeVSTAR ~ 1100 K events
HIJING ~ 21 K events
events:
particles: