Dependence of the event fractal dimensions on the colliding nuclei masses

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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 ?

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0 r

rND

rKM

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4log

3log

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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


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”


2 dimension random distribution 2 dimension random distribution

-1 < x < 1, 0 < y < 2

Npoints

Dcorr


multiplicity dependence multiplicity dependence

Au+Au s = 200 GeV

minbias


Au+Au Au+Au s = 200 GeVs = 200 GeV

Fractal dimensions in and pt spaces



Fractal dimensions in pT space


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


STAR Au+Au STAR Au+Au s = 200 GeV s = 200 GeV

Ptmax of events (pt > .15 GeV/c)



Nucl. Phys. A757 (2005) 102



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


HIJING Cu+Cu HIJING Cu+Cu s = 62&200 GeV s = 62&200 GeV


HIJING Au+Au HIJING Au+Au s = 62&200 GeV s = 62&200 GeV








HIJING d+Au HIJING d+Au s = 62&200 GeV s = 62&200 GeV


HIJING HIJING s = 200 GeV SiSi&InIns = 200 GeV SiSi&InIn


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.




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



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


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:



Fractal dimensions in and pt

spaces



Fractal dimensions in pT space

Documents

Dependence of the event fractal dimensions on the colliding nuclei masses