Driven inelastic Maxwell gases

We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates ${\tau }_c^{-1}$ and ${\tau }_w^{-1}$, respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution $r$. The velocity change of a particle with velocity $v$, due to driving, is taken to be $\Delta v=-(1+r_w)v+\eta$, where $r_w\in [-1,1]$ and $\eta$ is Gaussian white noise. For $r_w\in (0,1]$, this driving mechanism mimics the collision with a randomly moving wall, where $r_w$ is the coefficient of restitution. Another special limit of this driving is the so-called Ornstein-Uhlenbeck process given by $\frac{dv}{dt}=-\Gamma v+\eta$. We show that while the equations for the $n$-particle velocity distribution functions ($n=1,2,...$) do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for $r_w\ne -1$, the system goes to a steady state. Also we obtain the exact tail of the velocity distribution in the steady state. On the other hand, for $r_w=-1$, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with $\Gamma \ne 0$, whereas for the purely diffusive driving ($\Gamma =0$), the system does not have a steady state.

Figure 1

The variance ${\Sigma }_1\left(t\right)$ and the two-particle correlation ${\Sigma }_2\left(t\right)$ of the velocities for a cooling inelastic gas with 1000 particles with $r=1/2$ in the absence of the driving (static walls) for the two cases: (a) The rate of collision is independent of the variance $g=1$ and (b) the rate of collision is proportional to the variance $g={\Sigma }_1^{1/2}\left(t\right)$. For $g=1$, the lines plot the exact analytical expressions given by Eq. (B1). For $g={\Sigma }_1^{1/2}\left(t\right)$, the lines plot the approximate expressions given by Eqs. (B5) and (B6), while the points are obtained by exact numerical evaluation of the equation Eq. (B3).

Figure 2

The variance ${\Sigma }_1\left(t\right)$ and the two-particle correlation ${\Sigma }_2\left(t\right)$ of the velocities, for an inelastic gas with 1000 particles with $r=1/2$ driven by wall collisions with $\sigma =1$ and $r_w=-1$ for the two cases: (a) The rate of collision is independent of the variance $g=1$ and (b) the rate of collision is proportional to the variance $g={\Sigma }_1^{1/2}\left(t\right)$. For $g=1$, the lines plot the exact analytical expressions given by Eq. (9). For $g={\Sigma }_1^{1/2}\left(t\right)$, the points are obtained by exact numerical evaluation of the equation Eq. (6).

Figure 3

The variance ${\Sigma }_1\left(t\right)$ of the velocities for an inelastic gas with 1000 particles with $r=1/2$ driven by wall collisions with $\sigma =1$ and $r_w=1$ for the two cases: (a) The rate of collision is independent of the variance $g=1$ and (b) the rate of collision is proportional to the variance $g={\Sigma }_1^{1/2}\left(t\right)$. For $g=1$, the line plots the exact analytical expressions given by Eq. (8). For $g={\Sigma }_1^{1/2}\left(t\right)$, the points are obtained by exact numerical evaluation of the equation Eq. (6). The dotted line highlights the steady-state value calculated from Eq. (10).

Figure 4

The summary of the results for the PDF of the velocity distribution for different cases of $r_w\in [-1,1]$. For $r_w=-1$, the system does not reach a steady state, and the average energy and the two-particle correlation eventually increases linearly with the time. For $r_w=1$, the steady-state PDF has an exponential tail, whereas for $-1<r_w<1$, the tail of the PDF for very large velocities is Gaussian.