Multi-Energy Analysis of Baryon Stopping: Extracting Non-Equilibrium Dynamics via Stochastic Rapidity Diffusion

We present a comprehensive extraction of baryon stopping transport properties derived from net-proton transverse momentum ($p_T$) spectra across a broad energy range, from $E_{\mathrm{l}\mathrm{a}\mathrm{b}}=2.0A\text{ GeV}$ ($\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.70\text{ GeV}$) to $\sqrt{s_{\mathrm{N}\mathrm{N}}}=62.4\text{ GeV}$ in central heavy-ion collisions ($\le 10\mathrm{\%}$ centrality). By performing a systematic, multi-rapidity blast-wave analysis of experimental invariant yields, we extract the continuous rapidity density profile ${\rho }_{\mathrm{n}\mathrm{e}\mathrm{t}}(y)$ and the transverse momentum broadening profile $⟨p_T^2{⟩}_{\mathrm{n}\mathrm{e}\mathrm{t}}(y)$. Utilizing a stochastic diffusion framework governed by relativistic Fokker-Planck dynamics, we analytically derive the longitudinal friction coefficient ${\alpha }_{\mathrm{b}\mathrm{w}}$, the rapidity diffusion coefficient $D_y$, the longitudinal relaxation time ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{x}}$, and the associated longitudinal Reynolds number ${\mathrm{R}\mathrm{e}}_L$. Our extracted central stopping fractions demonstrate qualitative consistency with the previous published two source thermal calculations. Crucially, the excitation function of ${\mathrm{R}\mathrm{e}}_L$ reveals a drop by a factor of $\sim 163$, from ${\mathrm{R}\mathrm{e}}_L\approx 4.88$ at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=2.70\text{ GeV}$ to ${\mathrm{R}\mathrm{e}}_L\approx 0.03$ at $62.4\text{ GeV}$. This provides a clear macroscopic baseline for the crossover from inertial, advection-dominated baryon stopping to thermal, diffusion-dominated nuclear transparency, establishing a solid data-driven baseline for the topological baryon junction dynamics and high-${\mu }_B$ matter anticipated at upcoming facilities, including FAIR, NICA, and the CEE.