Relativistic Langevin dynamics in expanding media

We study the consequences of different realizations of diffusion processes in relativistic Langevin simulations. We elaborate on the Ito-Stratonovich dilemma by showing how microscopically calculated transport coefficients as obtained from a Boltzmann–Fokker-Planck equation can be implemented to lead to an unambiguous realization of the Langevin process. Pertinent examples within the prepoint (Ito) and postpoint (Hänggi-Klimontovich) Langevin prescriptions are worked out explicitly. Deviations from this implementation are shown to generate variants of the Boltzmann distribution as the stationary (equilibrium) solutions. Finally, we explicitly verify how the Lorentz invariance of the Langevin process is maintained in the presence of an expanding medium, including the case of an “elliptic flow” transmitted to a Brownian test particle. This is particularly relevant for using heavy-flavor diffusion as a quantitative tool to diagnose transport properties of QCD matter as created in ultrarelativistic heavy-ion collisions.

Figure 1

Distribution $dN/d{p}_z$ from Langevin simulations for heavy quarks with mass $m=1.5\mathrm{GeV}$, diffusing in a static medium at temperature $T=0.18\mathrm{GeV}$, compared to calculations with the corresponding analytical phase-space distributions: (a) prepoint Langevin scheme; (b) postpoint Langevin scheme. See text for more details.

Figure 2

Langevin simulation results for heavy quarks ($m=1.5\mathrm{GeV}$) diffusing in a flowing medium ($T=0.18\mathrm{GeV}$, $v_z=0.9$) compared to calculations with analytical phase-space distributions: (a) prepoint Langevin scheme; (b) postpoint Langevin scheme. The distribution obtained with a variant of the postpoint scheme and the corresponding blast-wave distribution are also shown. See text for more details.

Figure 3

Langevin simulation of the $p_T$ spectrum of a heavy quark ($m=1.5\mathrm{GeV}$) diffusing in a fireball ($T_i=0.33\mathrm{GeV}$, $T_f=0.18\mathrm{GeV}$), compared to the direct blast-wave calculations. (a) Prepoint Langevin scheme; (b) postpoint Langevin scheme. See text for more details.

Figure 4

Langevin simulation results for $v_2\left(p_T\right)$ for a heavy quark ($m=1.5\mathrm{GeV}$) diffusing in a QGP ($T_f=0.18\mathrm{GeV}$) compared to direct blast-wave calculations with the flow field and temperature of the background medium (fireball). (a) Prepoint Langevin scheme; (b) postpoint Langevin scheme.