Effect of finite particle number sampling on baryon number fluctuations

The effects of finite particle number sampling on the net baryon number cumulants, extracted from fluid dynamical simulations, are studied. The commonly used finite particle number sampling procedure introduces an additional Poissonian (or multinomial if global baryon number conservation is enforced) contribution which increases the extracted moments of the baryon number distribution. If this procedure is applied to a fluctuating fluid dynamics framework, one severely overestimates the actual cumulants. We show that the sampling of so-called test particles suppresses the additional contribution to the moments by at least one power of the number of test particles. We demonstrate this method in a numerical fluid dynamics simulation that includes the effects of spinodal decomposition due to a first-order phase transition. Furthermore, in the limit where antibaryons can be ignored, we derive analytic formulas which capture exactly the effect of particle sampling on the baryon number cumulants. These formulas may be used to test the various numerical particle sampling algorithms.

Figure 1

Time dependence of the enhancement of the scaled variance of the net baryon number, with respect to a simulation without an unstable phase in the EoS. The spatial volume in which the net-baryon number is calculated in each event is fixed by the condition $-0.3<z<0.3$ fm, while we integrate over the transverse directions. The enhancement originates from mechanical instabilities due to a first-order phase transition. The results shown are for central collisions of Pb nuclei at a fixed target beam energy of $E_{\mathrm{lab}}=3.5A$ GeV.

Figure 2

Variance over the mean of the net-baryon number in a given spatial volume restricted by a maximum $z$-coordinate value $z$. The hydro result is obtained at a time $t=3$ fm/$c$ for collisions of Pb+Pb at a beam energy of $E_{\mathrm{lab}}=3.5A$ GeV and fixing the number of participants to $N_{\mathrm{part}}=397$. The red line with square symbols depicts results for the pure hydro simulation before particle production. The black lines with small symbols are the results after particle production via the Cooper-Frye sampling (C-F) with a given number of test particles per real particle $Q_B=1/N_T$. The green lines are the expected results for a multinomial distribution [Eq. (11)] based on the pure hydro cumulants.

Figure 3

Same as in Fig. 2 but for the skewness.