Holographic Brownian dynamics of a heavy particle in a boosted thermal plasma background

Abstract

In this work, we have performed a detailed holographic analysis of the stochastic dynamics of a heavy particle propagating through a strongly coupled plasma moving with a constant velocity along a fixed spatial direction. To model this scenario within the framework of the AdS/CFT correspondence, we consider a boosted AdS black brane geometry in the bulk. The boost corresponds to the uniform motion of the plasma on the boundary field theory side. The presence of this boost introduces a preferred direction, leading to an anisotropic environment in which the behavior of the Brownian particle differs depending on its direction of motion. Consequently, we examine two distinct cases, namely,Brownian motion parallel to the direction of the boost and motion perpendicular to it. In this work we have computed the diffusion coefficient for both along the boost and perpendicular to the boost directions. We have obtained the diffusion coefficient by following the two different approaches in both the cases. These complementary approaches yield consistent results, thereby reinforcing the reliability of the computations carried out. Additionally, we verify the fluctuation-dissipation theorem within this anisotropic setup, confirming its validity in both longitudinal and transverse to the direction of boost. Our findings provide deeper insight into the non-equilibrium transport properties of strongly coupled plasma and further elucidate the holographic description of Brownian motion in anisotropic backgrounds. Finally, we proceed to holographically compute the Butterfly velocity by using the entanglement wedge subregion duality and express the diffusion coefficients in terms of the chaotic observables.

Figures (7)

• Figure 1: In the left pannel of the figure we have represented the variation of diffusion coefficient along the boost with the boost parameter. We have plotted the diffusion coeffcient for different values of the IR cutoff $\epsilon$. The red, blue, and black curves represent the variation of $D^{\text{para}}$ with the boost parameter for $\epsilon = 0.4$, $0.5$, and $0.9$, respectively. The right pannel of the above figure represents the variation of $D^{para}$ with the IR-cutoff for fixed value of boost parameter. The red, blue, and black curves represent the variation of $D^{\text{para}}$ with the IR cutoff for $v = 0.5$, $0.7$, and $0.9$, respectively.
• Figure 2: The Figure above shows the variation of $(s^{2}_{\mathrm{reg}}(t))^{\mathrm{para}}_{\mathrm{Boson}}$ with respect to the observer’s time $t$. For this plot, we have considered the parameters $d = 4$, $\alpha = 10$,$\epsilon=0.5$ and $r_h = 20$. The red, blue, and black curves represent the RMSD results (as given in Eq. \ref{['diffpara']}) for boost velocities $v = 0$, $0.5$, and $0.9$, respectively.
• Figure 3: The Figure above shows the variation of $(s^{2}{\mathrm{reg}}(t))^{\mathrm{para}}_{\mathrm{Fermion}}$ with respect to the observer’s time $t$. For this plot, we have considered the parameters $d = 4$, $\alpha = 10$, and $r_h = 20$. The red, blue, and black curves represent the RMSD results (as given in Eq. \ref{['fpar']}) for boost velocities $v = 0$, $0.5$, and $0.9$, respectively.
• Figure 4: The left panel of the above Figure represents the variation of $D^{per}$ with respect to the boost velocity $v$. To do this plot we have set $d=4,\alpha=10, r_{h}=20$. On the other hand, the right panel of the Figure presents a comparison between the two diffusion coefficients, $D^{\mathrm{per}}$ and $D^{\mathrm{para}}$. The black and red lines represent the diffusion coefficients parallel and perpendicular to the direction of the boost respectively. Again to do this plot we have set $d=4,\alpha=10, r_{h}=20, v=0.9$.
• Figure 5: The above Figures represent the variation of RMSD with respect to the observer's time. In the left panel of the above Figure we have represented the variation of RMSD with respect to the observer's time by keeping the fact that, the particle is moving perpendicular to the direction of boost. On the other hand in the right panel we have graphically represented the comparison of RMSD for parallel and perpendicular case.