Fig. 6: Statics and dynamics of the tracer in a passive bath.
From: Emergent memory from tapping collisions in active granular matter
a, b Photo and diagram of a passive bath particle (appropriately labeled with a capital “P'', for “Passive''). c Photo of the experiment for a tracer in a passive bath at packing fraction ϕ = 0.22. The white scale bar reads 30 mm. d Trajectories of the tracer for ϕ = 0 (blue) and ϕ = 0.22 (red). The black scale bar reads 50 mm. e Kinetic energy \({{{{{{{{\mathcal{K}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}=M\langle {{{{{{{{\bf{V}}}}}}}}}^{2}\rangle /2\) as a function of ϕ. We also report as a black solid line the linear scaling of \({{{{{{{{\mathcal{K}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}\) for a tracer in an active bath, see Fig. 4b. f Typical time τ1 of the autocorrelation of the velocity CV(t) calculated from the profile \(\alpha {e}^{-t/{\tau }_{1}}\) as a function of ϕ. g CV(t) for several values of ϕ. Here, the solid black line marks the double exponential decay observed in Fig. 4c for ϕ = 0.15. h Rotational kinetic energy \({{{{{{{{\mathcal{K}}}}}}}}}_{{{{{{{{\rm{r}}}}}}}}}=J\langle {\Omega }^{2}\rangle /2\) as a function of ϕ. i Typical time of angular velocity autocorrelation CΩ(t), extracted from the profile \(\sim {e}^{-t/{\tau }_{\Omega }}\) as a function of ϕ. The solid black line marks the value of τΩ as a function of ϕ in the presence of the active bath, see Fig. 4I. j Einstein relation for the rotational degree of freedom, \(\langle {\Omega }^{2}\rangle {\tau }_{\Omega }/{D}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\). k Conventional Einstein relation for the translational degree of freedom, \(\langle {{{{{{{{\bf{V}}}}}}}}}^{2}\rangle M/({\Gamma }_{{{{{{{{\rm{r}}}}}}}}}{D}_{{{{{{{{\rm{t}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}})\). In all the panels, points reflect experimental data and black dashed lines denote linear or constant scaling. In (e), (f), (h–k) error bars represent one standard deviation.
