1. “ Critical Point and Onset of Deconfinement ” Florence, July 2nd-9th 2006

2. Outline Gradual Freeze Out in a layer – brief review What is Freeze Out? How to model Freeze Out? Combining Bjorken hydro with gradual FO Results and discussion

4. Chemical freeze out describes many hadron ratios with only three parameters T, m and g Becattini, Florkowski, Keranen, Manninen, Tawfik, Redlich… Chemical Freeze Out s

5. Freeze Out as a discontinuity Theory of discontinuities in relativistic flow (only space- like), Taub, 1948. Generalization for both time- like and space-like discontinuities, Csernai, 1987. Matching conditions: Problem of negative contributions on the space-like part of FO hypersurface?

6. [L Bravina et al. (1995), UrQMD simulations] Kinetic Freeze Out Is there a sharp FO hypersurface? FO Layer ?

8. Gradual Freeze out description: effective kinetic model The particle number is conserved, thus: Gradual Freeze Out E d N d 3 p = R ¢ 4 V d 4 x 0 h p ¹ @ ¹ f i ( x 0 ; p ) i = R d 3 p p 0 R ¢ 4 V d 4 x 0 h p ¹ @ ¹ f i ( x 0 ; p ) i Kinetic Equation N f ( x ) ´ R S 0 2 N ¹ i ( x 0 ) d 3 ¾ ¹ ( x 0 ) ¡ R S 0 1 N ¹ i ( x 0 ) d 3 ¾ ¹ ( x 0 ) N f ( x ) = R ¢ 4 V d s ¹ ( x 0 ) d 3 ¾ ¹ ( x 0 ) h @ ¹ N ¹ i ( x 0 ) i f ( x ; ~ p ) = f f ( x ; ~ p ) + f i ( x ; ~ p )

9. The Boltzmann Transport Equation Nonlinear equation based on the following assumptions: oOnly binary collisions oThe molecular chaos – no correlations: o - is a smoothly varying function compared to the m.f.p. f ( x ; p 1 ; p 2 ) = f ( x ; p 1 ) £ f ( x ; p 2 ) 1 2 4 p Gain 3 4 2 p Loss p ¹ @ ¹ f = 1 2 Z 12 D 4 f 1 f 2 W p 4 12 ¡ 1 2 Z 2 D 34 ff 2 W 34 p 2 f ( x ; p )

10. The Boltzmann Transport Equation for Freeze Out Introducing the FO probability - such that FO probability not included !!! Rethermalization term p ¹ @ ¹ f f = + 1 2 Z 12 D 4 f i 1 f i 2 P FO W p 4 12 p ¹ @ ¹ f i = ¡ 1 2 Z 12 D 4 f i 1 f i 2 P FO W p 4 12 + 1 2 Z 12 D 4 f i 1 f i 2 W p 4 12 ¡ 1 2 Z 2 D 34 f i f i 2 W 34 p 2 P FO f f = P FO f f i = ( 1 ¡ P FO ) f p ¹ @ ¹ f = 1 2 Z 12 D 4 f 1 f 2 W p 4 12 ¡ 1 2 Z 2 D 34 ff 2 W 34 p 2 f ( x ; ~ p ) = f f ( x ; ~ p ) + f i ( x ; ~ p )

11. The approximate kinetic FO equations - escape rate @ s f i ( s ; p ) = ¡ P esc ( s ; p ) f i ( s ; p ) + £ f i eq ( s ; p ) ¡ f i ( s ; p ) ¤ 1 ¸ t h @ s f f ( s ; p ) = + P esc ( s ; p ) f i ( s ; p ) P esc ( s ; p ) 1 2 Z 12 D 4 f i 1 f i 2 P FO W p 4 12 ' P esc f i ( s ; p ) How to find the escape rate? L-s Cos q Probability not to collide increases with the decrease of remaining distance We want to model Freeze Out in a finite layer

12. Simple model for the FO in a finite layer

14. Post FO distribution for different LSpace-Like Time-Like Energy of interacting component Non thermal Post FO distributions

15. Post FO distribution for narrow layers The post FO distribution for the layers with thickness above several is independent on L ! Might justify the use of FO hypersurface, but with proper non-thermal post FO distribution But our model is oversimplified What happens if we take into account expansion of the fireball?

16. Bjorken model

17. Bjorken model with gradual Freeze Out Pure Bjorken Freeze out in a layer for Bjorken geometry Combined model

18. Bjorken model with gradual Freeze Out expansion terms Freeze Out terms

19. Bjorken model with gradual Freeze Out Initial state: Phase 1) Pure Bjorken hydrodynamics Phase 2) Bjorken expansion and gradual FO

20. Bjorken model with gradual Freeze Out Free component

21. Bjorken model with gradual Freeze Out Free component

22. Bjorken model with gradual Freeze Out Calculations are done for massless baryonfree gas, P=e/3

23. Bjorken model with gradual Freeze Out Energy conservation

24. Bjorken model with gradual Freeze Out

25. Expansion is necessary! This was pointed out in EPJ C30 (03) 255 Long gradual Freeze Out produces entropy ! Forbidden thickness ?

26. Bjorken model with gradual Freeze Out Wide layers The post FO distribution for the layers with thickness above several is independent on L ! Narrow layers “No expansion”

27. Conclusions We built a simple, but nontrivial, model, which incorporates Bjorken-like expansion of the fireball and Freeze Out within a layer of given thickness The post FO distribution for the layers with thickness above several m.f.p. is independent on L! Justifies the usage of FO hypersurface? Long gradual Freeze Out might lead to entropy production With non-thermal Post FO distributions?