1. Event-by-event fluctuations as a sign of 1st order Quark-Hadron phase transition
Ivan Melo, Mikuláš Gintner
Žilina
in collaboration with
Boris Tomášik (Banská Bystrica)
Samuel Koróny (Banská Bystrica)
FÚ ČAV Praha, Jun 14, 2007 k

4. Where is the 1st order phase transition?
SPS?
RHIC

5. Spinodal fragmentation at phase transition
Example: linear sigma model coupled to quarks
[O. Scavenius et al., Phys. Rev. D 63 (2001) ]
spinodal
phase transition
What is fast?
bubble nucleation rate < expansion rate

6. Supercooling down to spinodal
O. Scavenius et al., Phys. Rev. D 63 (2001) :
compare nucleation rate with Hubble constant (1D)
likely to reach spinodal

7. Droplets and rapidity distributions
rapidity distribution in a single event
dN/dy
dN/dy y y
without droplets
with droplets
If we have droplets, each event will look differently

8. The measure of difference between events
Kolmogorov–Smirnov test:
Are two empirical distributions generated from the same underlying probability distribution? 1
D measures the
difference of two
empirical distributions D
maximum
distance y

9. Kolmogorov-Smirnov: theorems
How are the D's distributed?
Smirnov (1944): If we have two sets of data generated from the same underlying distribution, then D's are distributed according to
This is independent from the underlying distribution!
For events generated
from the same distribution,
Q's (Q=1-H)will be
distributed
uniformly.

10. KS test on simulated events
* artificial 'events' coming from uniform or half-normal distribution
* simulated events from heavy ion collisions with no quark-gluon plasma (UrQMD)
* simulated events from droplet generator

11. KS-test on generated distributions
- random data generated
according to half-normal
distribution
- peak close to Q = 1
(“too similar” events)
WHY? Q
- we use asymptotic formula valid for very large n
results improve with larger n and/or correction terms
- we have limited number of significant figures
Does it matter in other e-by-e studies?

12. UrQMD for Pb+Pb at 158 AGeV, b=0, 523 events (Marcus Bleicher)
KS-test on UrQMD
UrQMD for Pb+Pb at 158 AGeV, b=0, 523 events
(Marcus Bleicher)
... similar events

13. Droplet generator (B. Tomášik)
MC generator of (momenta and positions of) particles
some particles are emitted from droplets (clusters)
droplets are generated from a blast-wave source (tunable parameters)
droplets decay exponentially in time (tunable time, T)
tunable size of droplets: Gamma-distributed or fixed
no overlap of droplets
also directly emitted particles (tunable amount)
chemical composition: equilibrium with tunable params.
rapidity distribution: uniform or Gaussian
possible OSCAR output

14. Droplet generator: rapidity spectra
T = 175 MeV, various droplet sizes, all particles from droplets
No difference in distributions summed over all events!

15. Droplet generator: all from droplets
average droplet
volume 50 fm3
20 fm3 Q Q
Events are clearly different!
10 fm3 Q Q

16. Droplet generator: part from droplets
V = 10 fm3

17. Conclusions
a method to point out any differences between individual events
apply the method to data from NA49
in most central data: droplets would appear when we reach the phase transition line
Droplets hee

18. Bubble nucleation rate
eg. [L. Csernai, J. Kapusta: Phys. Rev. Lett. 69 (1992) 737]
nucleation rate:
difference in free energy:
critical size bubble:
at phase transition
=> must supercool

19. The size of droplets Minimize thermodynamic potential per volume where
[I.N. Mishustin, Phys. Rev. Lett. 82 (1998), 4779; Nucl. Phys. A681 (2001), 56c]
Minimize thermodynamic potential per volume
where
this gives
also [D.E. Grady, J.Appl.Phys. 53 (1982) 322]