Global stability and $H$ theorem in lattice models with nonconservative interactions

In kinetic theory, a system is usually described by its one-particle distribution function $f(\mathit{r},\mathit{v},t)$, such that $f(\mathit{r},\mathit{v},t)d\mathit{r}d\mathit{v}$ is the fraction of particles with positions and velocities in the intervals $(\mathit{r},\mathit{r}+d\mathit{r})$ and $(\mathit{v},\mathit{v}+d\mathit{v})$, respectively. Therein, global stability and the possible existence of an associated Lyapunov function or $H$ theorem are open problems when nonconservative interactions are present, as in granular fluids. Here, we address this issue in the framework of a lattice model for granularlike velocity fields. For a quite general driving mechanism, including both boundary and bulk driving, we show that the steady state reached by the system in the long-time limit is globally stable. This is done by proving analytically that a certain $H$ functional is nonincreasing in the long-time limit. Moreover, for a quite general energy injection mechanism, we are able to demonstrate that the proposed $H$ functional is nonincreasing for all times. Also, we put forward a proof that clearly illustrates why the “classical” Boltzmann functional $H_B\left[f\right]=\int d\mathit{r}d\mathit{v}f(\mathit{r},\mathit{v},t)lnf(\mathit{r},\mathit{v},t)$ is inadequate for systems with nonconservative interactions. This is done not only for the simplified kinetic description that holds in the lattice models analyzed here but also for a general kinetic equation, like Boltzmann's or Enskog's.

Figure 1

Evolution of the functional $H_B$ for two initial conditions in a uniformly heated system. Both simulations start from the stationary Gaussian shape but with a homogeneous temperature slightly shifted from the stationary one: (i) $T(t=0)=1.1T_{\text{st}}$ (circles) and (ii) $T(t=0)=0.9T_{\text{st}}$ (triangles). As predicted by the linear approximation, both functionals are symmetric with respect to the stationary value, which is schematically illustrated by plotting their mean value (dashed line). A system with $N=330$ sites has been considered, with $\nu =20$ and $\xi =50$. Plots correspond to averages over 3000 trajectories.

Figure 2

Relaxation towards the USF state. The initial condition is Gaussian, centered in $u_{\text{st}}\left(x\right)$ and with variance $T=7T_{\text{st}}$. (a) Velocity distribution function at $x=1/4$ for four times. Inset: Evolution of the excess kurtosis. (b) Monotonic relaxation of the $H$ functional, in clear agreement with the proven $H$ theorem. Data are averaged over 6000 trajectories in a system with $N=660$ sites, $\nu =20$, and $a=5$. Solid lines correspond to the (theoretical) Gaussian distributions for the plotted times, except the last time, for which it represents the theoretical steady distribution.

Figure 3

The same plots as in Fig. 2, but starting from a different initial condition. Now, the initial PDF is a Gaussian centered in $u(x,0)=u_{\text{st}}\left(x\right)+4.4sin\left(2\pi x\right)$ and with variance $T(t=0)=T_{\text{st}}$. (a) Solid lines correspond to the theoretical PDFs for the initial time and the steady state. (b) $H$ decreases again monotonically towards its steady value, consistent with our theoretical prediction.

Figure 4

The same plots as in Fig. 2, but starting from an initial PDF with a divergent Gram-Charlier series. Concretely, the plots correspond to an exponential initial distribution centered in $u(x,t=0)=u_{\text{st}}\left(x\right)+4.4sin\left(2\pi x\right)$ and with $T(t=0)=0.1T_{\text{st}}$.

Figure 5

Numerical results for the uniformly heated system. The plots are analogous to those in Fig. 2. (a) Time evolution of the PDF and (b) time evolution of the $H$ functional. The system is initially prepared with an exponential PDF centered in $u(x,t=0)=4.4sin\left(2\pi x\right)$ and with $T(t=0)=0.1T_{\text{st}}$. Data are averaged over 3000 trajectories in a system with $N=330$ sites.