Model system for classical fluids out of equilibrium

A model system for classical fluids out of equilibrium, referred to as a dissipative particles dynamics (DPD) solid, is studied by analytical and simulation methods. The time evolution of a DPD particle is described by a fluctuating heat equation. This DPD solid with transport based on collisional transfer (high-density mechanism) is complementary to the Lorentz gas with only kinetic transport (low-density mechanism). Combination of both models covers the qualitative behavior of transport properties of classical fluids over the full-density range. The heat diffusivity is calculated using a mean-field theory, leading to a linear-density dependence of this transport coefficient, which is exact at high densities. Subleading density corrections are obtained as well. At lower densities the model has a conductivity threshold below which heat conduction is absent. The observed threshold is explained in terms of percolation diffusion on a random proximity network. The geometrical structure of this network is the same as in continuum percolation of completely overlapping spheres, but the dynamics on this network differs from continuum percolation diffusion. Furthermore, the kinetic theory for DPD is extended to the generalized hydrodynamic regime, where the wave-number-dependent decay rates of the Fourier modes of the energy and temperature fields are calculated.

Figure 1

Simulated values of the thermal diffusivity $R_3\left(\rho \right)=D_T∕D_{\infty }$ versus $\rho$ for the 3D heat conduction model with $N={10}^3$ DPD particles. Results from off-lattice simulations (●) and on-lattice simulations (▵) are compared with the mean-field value (3.12) (dashed line). At $\rho =3$ and 27 the system sizes are, respectively, $L∕r_c={(N∕\rho )}^{1∕3}\simeq 6.9$ and 3.3, which is rather small, and strong finite-size effects are to be expected.

Figure 2

2D simulations of the dimensionless heat diffusivity $D_T\left(\rho \right)∕{\lambda }_0{r}_c^2$ plotted versus $\rho$ to test the mean-field prediction $D_{\infty }\left(\rho \right)$ in Eq. (3.8) (dashed line), valid at high density. Labels ▵ and ● refer to the random solid, respectively, with and without Langevin force in systems with ${10}^4$ and ${10}^5$ DPD particles. $H$ refers to the Heaviside weight function.

Figure 3

2D simulations of the heat diffusivity $R_2\left(\rho \right)$ plotted versus $\rho$. (a), (b), and (c) show, respectively, the behavior near threshold, the crossover, and the approach to the saturation value at large $\rho$. For definitions of symbols and parameter values we refer to the previous figure.

Figure 4

Temperature profiles at times of measurement. For $\rho \gtrsim 3$ a steady linear profile is reached, but this is not the case for $\rho \lesssim 1.6$.

Figure 5

The plot shows that the dominant correction to $1-R_d\left(\rho \right)$ behaves in 2D-like $A_2∕\rho$ with $A_2\simeq 1.2$. The data refer to the deterministic case (vanishing Langevin force) with the Heaviside weight function.

Figure 6

(a) Decay rate of the 3D heat mode $\zeta \left(k\right)=D_T\left(k\right)k^2$ in units of ${\omega }_0$ and (b) heat diffusivity $D_T\left(k\right)=D_T-k^2{B}_T+\cdots$ in units of ${\omega }_0{r}_c^2$, plotted versus $k{r}_c$. Solid lines in (a) and (b) are calculated with the Lucy function in Eq. (3.11) and $\alpha =100$.