Fig. 4: Statics and dynamics of the tracer.
From: Emergent memory from tapping collisions in active granular matter
a Probability distribution P(V) of a Cartesian component of the velocity for several values of the packing fraction ϕ. b Kinetic energy \({{{{{{{{\mathcal{K}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}=M\langle {{{{{{{{\bf{V}}}}}}}}}^{2}\rangle /2\) as a function of ϕ. c Autocorrelation of the velocity CV(t) for several values of ϕ. d, e Typical times of CV(t) calculated from the profile \(\alpha {e}^{-t/{\tau }_{1}}+\beta {e}^{-t/{\tau }_{2}}\) as a function of ϕ. f Probability distribution of the angular velocity P(Ω) for several values of ϕ. g Rotational kinetic energy \({{{{{{{{\mathcal{K}}}}}}}}}_{{{{{{{{\rm{r}}}}}}}}}=J\langle {\Omega }^{2}\rangle /2\) as a function of ϕ. h Angular velocity autocorrelation CΩ(t) for several values of ϕ. i Typical time of CΩ(t), extracted from the profile \(\sim {e}^{-t{\tau }_{\Omega }}\) as a function of ϕ. j Ratio 〈Ω2〉τΩ as a function of ϕ. k, l Mean-square displacement \({{{{{{{\rm{MSD}}}}}}}}(t)=\langle {\left({{{{{{{\bf{X}}}}}}}}(t)-{{{{{{{\bf{X}}}}}}}}(0)\right)}^{2}\rangle\), and mean-square angular displacement \({{{{{{{\rm{MSAD}}}}}}}}(t)=\langle {\left(\Phi (t)-\Phi (0)\right)}^{2}\rangle\), for several values of ϕ. m Diffusion coefficients, \({D}_{{{{{{{{\rm{t}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\) (left y-scale) and \({D}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\) (right y-scale), as a function of ϕ. The units of \({D}_{{{{{{{{\rm{t}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\) and \({D}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\) are 1.18 mm2 s−1 and 1.18 rad2 s−1, respectively. n Einstein relations for the rotational degree of freedom, \(\langle {\Omega }^{2}\rangle {\tau }_{\Omega }/{D}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\). o Conventional Einstein relation for the translational degree of freedom, \(\langle {{{{{{{{\bf{V}}}}}}}}}^{2}\rangle M/({\Gamma }_{{{{{{{{\rm{r}}}}}}}}}{D}_{{{{{{{{\rm{t}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}})\). p Active Einstein relation for \({D}_{{{{{{{{\rm{t}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\), given by Eq. (7). In all the panels of this figure, points reflect experimental data, solid colored lines are exponential fitting functions justified by our coarse-grained models, black dashed lines denote linear or constant scaling, black dashed-dotted lines evidence ballistic behavior, and solid black lines are guides for the eyes. In (b), (d), (e), (g), (i), (j), (m–p), error bars represent standard deviation.
