Observations from high-multiplicity proton-lead ($p$-Pb) collisions indicate that small systems may exhibit collective behavior in both heavy and light hadrons. This work investigates the roles of initial- and final-state interactions in shaping the nuclear modification factor and elliptic flow of $D$ mesons measured in $p$-Pb collisions. Initial-state effects, including the Cronin and shadowing effects, are considered in the heavy-quark initial conditions, while final-state interactions are simulated through Langevin evolution combined with the coalescence model of hadronization. Different initial geometries attributed to fluctuations in the medium's energy density are parametrized and translated into the momentum anisotropies of both light and heavy quarks. The corresponding $R_{p\mathrm{Pb}}$ and $v_2$ of $D$ mesons in 8.16 TeV $p$-Pb collisions are calculated under different assumptions for the final-state interactions. Assuming that the initial-state effects only modify the transverse-momentum spectra without altering the azimuthal distribution of heavy quarks, the measured $R_{p\mathrm{Pb}}$ of $D$ mesons can be qualitatively reproduced by the combined influence of initial- and final-state effects. However, the observed $v_2$ cannot be accounted for by final-state interactions alone. These results suggest that additional contributions to azimuthal anisotropies of heavy quarks originating from initial-state effects are required to explain the experimentally observed $v_2$.

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• Collective flow
• Jets & heavy flavor physics
• Quark-gluon plasma
• Relativistic heavy-ion collisions
• Langevin algorithm