Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description

Abstract

A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics is described through the corresponding Master Equation for the time evolution of the probability distribution. In the continuum limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to hydrodynamic equations of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible.

Appendix: Gaussian Character of the Noises

In the large system size limit $L\gg 1$, the current noise introduced in the Sect. (5.1) is white. We can introduce a new noise field $\tilde{\xi }(x,t)$ by

$$\begin{aligned} \xi ^{(j)}(x,t)=L^{-1/2}\tilde{\xi }(x,t) \end{aligned}$$

(89)

and $\tilde{\xi }(x,t)$ remains finite in the large system size limit $L \rightarrow \infty$,

$$\begin{aligned} \langle \tilde{\xi }(x,t)\rangle =0, \, \quad \langle \tilde{\xi }(x,t)\tilde{\xi }(x',t')\rangle \sim 2\,T(x,t) \delta (x-x')\delta (t-t'). \end{aligned}$$

(90)

Here we show that all the higher-order cumulants of $\tilde{\xi }(x,t)$ vanish in the thermodynamic limit as $L \rightarrow \infty$. Let us consider a cumulant of order n of the microscopic noise $\xi _{l,p}$ that is equal to the nth order moment of the $\xi$ plus a sum of nonlinear products of lower moments of $\xi$. A calculation analogous to the one carried out for the correlation $\langle \xi ^{(j)}_{l,p}\xi ^{(j)}_{l',p'}\rangle$ shows that the leading behaviour of any moment is of the order of $L^{-1}$, which is obtained when all the times are the same. Therefore, the moment $\langle j_{l,p}j_{l',p'}\ldots j_{l^{(n)},p^{(n)}}\rangle$ gives the leading behaviour of the considered cumulant, which is thus of the order of $L^{-1}$ for $p=p'= \cdots =p^{(n)}$; any other contribution to the cumulant is at least of the order of $L^{-2}$. We have that

$$\begin{aligned} \langle j_{l,p}j_{l',p'}\ldots j_{l^{(n)},p^{(n)}}\rangle \sim L^{-1}\langle C_{l,p}\rangle \delta _{l,l'}\delta _{l',l''}\delta _{l^{(n-1)},l^{(n)}}\cdots \delta _{p,p'}\delta _{p',p''}\delta _{p^{(n-1)},p^{(n)}}, \end{aligned}$$

(91)

where $\langle C_{l,p}\rangle$ is certain average that remains finite in the large system size limit as $L \rightarrow \infty$. In the continuous limit, each current introduces a factor $L^{2}$ due to the scaling introduced in Sect. 5. Moreover, we take into account the relationship between Kronecker and Dirac $\delta$’s in the continuum limit to write the cumulants $\langle \langle \cdots \rangle \rangle$ of the rescaled noise introduced in (89) as

$$\begin{aligned}&\langle \langle \tilde{\xi }(x,t)\tilde{\xi }(x',t')\cdots \tilde{\xi }(x^{(n)},t^{(n)})\rangle \rangle \sim {L^{3\left( 1-\frac{n}{2} \right) }}{\langle C(x,t)\rangle }\times \nonumber \\&\qquad \delta (x-x')\delta (x'-x'')\delta (x^{(n-1)}-x^{(n)})\cdots \delta (t-t')\delta (t'-t'')\delta (t^{(n-1)}-t^{(n)}). \end{aligned}$$

(92)

Thus, in the limit as $L \rightarrow \infty$,

$$\begin{aligned} \langle \tilde{\xi }(x,t)\tilde{\xi }(x',t')\cdots \tilde{\xi }(x^{(n)},t^{(n)})\rangle =0, \quad \text {for all}\; n>2, \end{aligned}$$

(93)

and the vanishing of all the cumulants for $n>2$ means that the momentum current noise is Gaussian in the infinite size limit.

The same procedure can be repeated for the energy current noise, by defining $\xi ^{(J)}(x,t)=L^{-1/2}\tilde{\eta }(x,t)$), with the result

$$\begin{aligned}&\langle \langle \tilde{\eta }(x,t)\tilde{\eta }(x',t')\cdots \tilde{\eta }(x^{(n)},t^{(n)})\rangle \rangle \sim {L^{3\left( 1-\frac{n}{2} \right) }}{\langle D(x,t)\rangle }\times \nonumber \\&\qquad \delta (x-x')\delta (x'-x'')\delta (x^{(n-1)}-x^{(n)})\cdots \delta (t-t')\delta (t'-t'')\delta (t^{(n-1)}-t^{(n)}). \end{aligned}$$

(94)

In the equation above, $\langle D(x,t)\rangle$ is a certain average, different from $\langle C(x,t)\rangle$, but also finite in the large system size limit. Thus, we have that

$$\begin{aligned} \langle \tilde{\eta }(x,t)\tilde{\eta }(x',t')\cdots \tilde{\eta }(x^{n},t^{n})\rangle =0, \quad \text {for all}\; n>2, \end{aligned}$$

(95)

and the energy current noise also becomes Gaussian in the continuum limit.

Note that the Gaussianity of the noises is independent of the validity of the local equilibrium approximation, which is only needed to write $\langle C(x,t)\rangle$ and $\langle D(x,t)\rangle$ in terms of the hydrodynamic fields u(x, t) and T(x, t). Besides, a similar procedure for the dissipation noise gives that the corresponding scaled noise vanishes in the continuum limit, since the power of L in the dominant contribution to the nth order cumulant is $3-5n/2$ instead of $3-3n/2$. This means that the dissipation noise is subdominant as compared to the currents noises in the continuum limit, and can be neglected.