Chapter 6 Calculations with Chemical Processes calculations have shown a lifespan up to 50 fm/c before the drop bursts. Figure6.17displays the distribution function of the quarks with their time dependence after a condensation drop has been formed. The Ągure shows a decline of the quark number with time and mainly particles with high energies leave the system. This is reasonable because high energetic particles can leave the potential well, leading to a collective cooling of the remaining medium.
The calculations in this section have shown the very interesting complexity of such a simple initial condition like an expanding matter droplet. The chemical reactions between particles have a very strong impact on the system behavior and dramatically change both Ćuctuations and medium propagation within the expansion. However, a characteristic signature which allows a event-by-event discrimination between the diferent kind of couplings and phase transitions in this scenario has not been found, at least not for calculations with the crossover couplingg= 3.3 and the second-order transition coupling g= 3.63. Such a discrimination could be possible in a statistical investigation of the angular distribution of the quarks which are emitted from the Ćuctuating chiral Ąeld.
chosen V = (36 fm)3 and the hot matter droplet has an initial diameter ofd≈14 fm, roughly corresponding to a gold nuclei. The temperature was again chosen to be T = 175 MeV and the couplings remain atg= 3.3,g= 3.63 andg= 5.5. To increase the numerical precision and to avoid numerical problems the grid-size was increased to Ngrid= 1923 and the time step was reduced to
∆t= 0.001fm/c. The interaction volume was slightly increased toVinteraction= (0.4 fm)3. The total particle number was around 5 million particles with a test-particle multiplicator of 200. All other parameters have been kept unaltered.
Figure 6.19 and6.20 show the result for a calculation with a large expanding matter droplet.
Both Ągures compare the same calculations with and without chemical interactions and diferent couplings, g= 3.3 with a cross-over transition in Figure6.19and g= 5.5 in Figure 6.20.
The calculations without chemical interactions do not change very much in comparison to the smaller systems. The overall symmetry of the system stays efectively intact and known structures like the shell-structures and cold quark droplets are still present.
The picture changes a bit for calculations with chemical interactions, especially for larger couplings like g= 5.5. After a short expansion phase the matter droplet starts to form strong and non-isotropic structures in the quark density due to the annihilation and creation process. In Figure 6.19 Ąrst structures are already present att≈1fm/c. These structures blur while the matter expands.
In Figure 6.20 these structures are much stronger and much Ąner. The reason is the strong coupling between the Ąeld and particles, leading to fast annihilation and strong damping of the Ąeld by σ-decay. While expanding, the Ćuctuations on the σ-Ąeld become stronger. Around t= 5fm/c the annihilation-rate of the quarks is negligible because of the low particle density due to expansion. The σ-decay becomes the dominating efect, leading to a strong damping of the Ąeld and to a quasi-freeze out of the Ąeld leading to some kind of local bubble-formation. The resulting Ąeld distribution stays stable for several 10fm/c local bubbles converge slowly to a big drop. This Ąnal drop contains the already known cold quarks which were not able to escape the kinetic potential.
In the calculations for larger system a qualitatively diference between the couplings can be seen in the time evolution. While calculations for g= 3.3 and g= 3.63 behave similar, calculations for g = 5.5 show a very diferent behavior: the quark matter forms bubbles and freezes out in a relative long-living structure. One of the remaining questions is if this formation can be observed in experiment. A possible approach will be an investigation of the angular distribution of the emitted particles, which could be mapped to an angular distribution of measured particle multiplicity. This is subject to an upcoming publication.
Chapter 6 Calculations with Chemical Processes
Figure 6.19: Evolution of the quark density for a large hot matter droplet scenario withg= 3.3. The initial size of the droplet has an diameter of aroundd≈14 fm which corresponds to the size of a gold nuclei, the total size of the system isV = (36 fm)3. Left: Simulation without chemical interactions Right: simulation with chemical interactions. Chemical interactions lead to formation of non-isotropic structures but the overall expansion of the system is not strongly altered.
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Figure 6.20: Quark density for a large hot matter droplet scenario withg= 5.5. The initial size of the droplet has an diameter of aroundd≈14 fm which corresponds to the size of a gold nuclei, the total size of the system isV = (36 fm)3. Left: Simulation without chemical interactionsRight: simulation with chemical interactions. Chemical interactions lead to a generation of strong and detailed structures. After 5fm/c the expansion of the system is stopped by the strong damping of the Ąeld, leading to the formation of local bubbles.
Chapter 7
Particle-Wave Interaction Method
Creativity is intelligence having fun.
Albert Einstein
This chapter discusses a new method which allows a numerical treatment of noncontinuous interactions between Ąelds and particles, called the particle-Ąeld method. The chapter starts with a historical introduction of the wave-particle duality, which gave the basic idea for the particle-Ąeld method. In Section 7.2 existing approaches for interaction between Ąelds and particles and their drawbacks are discussed. This discussion includes the Vlasov-equation, interactions in Fourier-space and the Langevin-equation. Section 7.3introduces the ideas and concepts of the particle-Ąeld method. In Section 7.4several examples and simple scenarios are presented which implement the particle-Ąeld method with increasing levels of complexity and demonstrate the possible applications of this method. The last Section 7.5considers several general properties of the method.