Simulation study of the Green-Kubo relations for dilute granular gases

The Green-Kubo relations for dilute granular gases are employed to compute their transport coefficients by means of the direct simulation Monte Carlo method. This requires not only to follow the dynamics of the system, but also to identify some modified fluxes appearing in the time-correlation functions. The results are compared with those obtained from the Boltzmann equation by means of the Chapman-Enskog procedure in the first Sonine approximation. A good agreement is found for the shear viscosity over a wide range of inelasticities. Nevertheless, for the two transport coefficients associated with the heat flux, significant discrepancies appear for strong inelasticity. Their origin is discussed, showing that they are partially due to the presence of velocity correlations in the homogeneous cooling state of a dilute granular fluid.

Figure 1

Plot of ${\left(\ln X\right)}^{\prime }\equiv \partial \ln {\chi }_{\mathit{HCS}}∕\partial c$ as a function of $c$ for $\alpha =0.9$. The circles are the numerical derivative of the simulation results, the solid line the fitted function, while the dashed and dot-dashed lines are the Gaussian and first Sonine approximations for this quantity, respectively. Quantities are measured in the dimensionless units defined in the main text.

Figure 2

The same as Fig. 1 but for $\alpha =0.6$.

Figure 3

Time evolution of the temperature, measured in the units defined in the text, in the steady state. Time is measured in terms of the number of accumulated collisions per particle, $\tau$. The solid line corresponds to $\alpha =0.9$, the dotted line to $\alpha =0.6$, and the dashed line to $\alpha =0.3$.

Figure 4

Time evolution of the correlation function $J_{\eta }\left(t\right)$ for $\alpha =0.95$. The circles are the results of the simulation, while the solid line is the best fit to an exponential. Quantities are measured in the units defined in the main text.

Figure 5

The same as in Fig. 4 but for $\alpha =0.5$.

Figure 6

Dimensionless reduced shear viscosity coefficient, $\eta *$, as a function of $\alpha$. The solid line is the theoretical prediction derived in Ref. 7, while the symbols are from the simulations: the circles are obtained using the true (from the simulation) velocity distribution for the modified flux, the triangles with the Gaussian approximation, and the squares correspond to the first Sonine approximation.

Figure 7

Decay of the time correlation function $J_{\kappa }$, measured in the units defined in the main text, for $\alpha =0.95$. The circles are from the simulations, while the solid line is the best exponential fit.

Figure 8

The same as Fig. 7 but for $\alpha =0.5$.

Figure 9

The dimensionless reduced coefficient of thermal conductivity, $\kappa *$, as a function of $\alpha$. The symbols are from the simulations, while the solid line is the theoretical prediction derived in Ref. 7.

Figure 10

The dimensionless reduced coefficient of diffusive heat conductivity $\mu *$ as a function of $\alpha$. The circles are from the simulations using the Green-Kubo expression, and the solid line is the theoretical prediction derived in Ref. 7. The triangles are from an independent study of this transport coefficient in vibrated systems [11].

Figure 11

Simulation results for the correlation function $J_{\eta }^{\prime }\left(0\right)$ defined in Eq. (43) (filled circles), and its diagonal part, $J_{\eta }^{\prime \left(1\right)}$ (empty circles), as a function of the restitution coefficient $\alpha$. Quantities are measured in the units defined in the text.

Figure 12

Simulation results (filled circles) for the $N$-particle correlation function $J_{\kappa }^{\prime }\left(0\right)$. The empty circles are its diagonal part, $J^{\prime \left(1\right)}$. Quantities are measured in the units defined in the main text.