Hydrodynamic correlation functions of a driven granular fluid in steady state

We study a homogeneously driven granular fluid of hard spheres at intermediate volume fractions and focus on time-delayed correlation functions in the stationary state. Inelastic collisions are modeled by incomplete normal restitution, allowing for efficient simulations with an event-driven algorithm. The incoherent scattering function $F_{\mathrm{incoh}}(q,t)$ is seen to follow time-density superposition with a relaxation time that increases significantly as the volume fraction increases. The statistics of particle displacements is approximately Gaussian. For the coherent scattering function $S(q,\omega )$, we compare our results to the predictions of generalized fluctuating hydrodynamics, which takes into account that temperature fluctuations decay either diffusively or with a finite relaxation rate, depending on wave number and inelasticity. For sufficiently small wave number $q$ we observe sound waves in the coherent scattering function $S(q,\omega )$ and the longitudinal current correlation function $C_{\mathrm{l}}(q,\omega )$. We determine the speed of sound and the transport coefficients and compare them to the results of kinetic theory.

Figure 1

Incoherent intermediate scattering function for several values of volume fraction $\eta$ and coefficient of restitution $\varepsilon$. All lines for $N=200,000$, where $\varepsilon =0.8$, 0.9, and 1.0, are indicated by dashed, solid, and dot-dashed lines, respectively. $\eta =0.05$ corresponds to the left, and $\eta =0.4$ to the right, lines. All error bars are of the order ${10}^{-3}$. Open diamonds and circles are for $N=10,000$, $\varepsilon =0.8$, and $\eta =0.05$ and 0.1, respectively.

Figure 2

Relaxation time of the incoherent scattering function as a function of volume fraction $\eta$ for several values of $\varepsilon$ for simulation runs with $N=200,000$ particles.

Figure 3

Mean square displacement for $\varepsilon =0.8$ and various volume fractions, ranging from $\eta =0.05$ [top (black) line] to $\eta =0.4$ [bottom (violet) line].

Figure 4

Intermediate incoherent scattering function $F_{\mathrm{incoh}}(\mathbf{q},t)$ using Eq. (7) (solid lines) and, for comparison, the Gaussian approximation using Eqs. (8) and (9) (dashed lines).

Figure 5

Self-diffusion constant $D$ as a function of volume fraction $\eta$. Filled symbols, with lines to guide the eye, were obtained via linear fits to the mean square displacement for large times. Garzó results correspond to Eq. (2.10) of Ref. [28], and the Enskog result corresponds to Eq. (d5) of Ref. [21].

Figure 6

Time-density superposition for the incoherent scattering function.

Figure 7

Intermediate coherent scattering function $F(q,t)$.

Figure 8

Longitudinal current correlation as a function of time for several densities. Inset: Variation with $q$.

Figure 9

Longitudinal current correlation as a function of angular frequency.

Figure 10

Position of the maximum ${\omega }_{\max }$ longitudinal current correlation $C_l(\mathbf{q},\omega )$. Solid lines and filled symbols are for $\epsilon =0.9$, and dashed lines indicate linear fits with slopes as listed in Table . Inset: Comparison ${\omega }_{\max }$ for $\epsilon =0.9$ (solid lines) and for $\epsilon =0.8$ (dash-dotted lines).

Figure 11

Dynamic structure factor $S(q,\omega )$ for $\eta =0.05,\varepsilon =0.8$, and $q=0.2$ – $0.5$. Symbols indicate simulation results obtained via Fourier transform of $F(q,t)$, and lines indicate fits with Eq. (33).

Figure 12

$S(q,\omega )$ for $\eta =0.05,\varepsilon =0.8$, and $q=1.0$ – $3.0$.

Figure 13

$S(q,\omega )$ for $\eta =0.2,\varepsilon =0.9$, and $q=0.5$–$0.7$.

Figure 14

$S(q,\omega )$ for $\eta =0.2,\varepsilon =0.9$, and $q=0.8$–$1.5$.

Figure 15

Thermal diffusivity $D_T(q)$ obtained via fits to $S(q,\omega )$ with Eq. (33). The arrows indicate the $q_c$ values from Table .

Figure 16

Longitudinal viscosity ${\nu }_{\mathrm{l}}(q)$ obtained via fits to $S(q,\omega )$ with Eq. (33). Arrows indicate the $q_c$ values from Table .