Model

This is work in progress.

We model planar gas of hard spheres in a strip between two walls. The particles move straight and collide with each other as usual, but collisions with the walls are controlled by Maxwell demons which follow specific rules. This induces a non-equilibrium steady state, e.g. a shear flow.

We look into how the collision rules relate to the thermodynamic entropy production. On the graph, the table is divided into a number of horizontal strips, and vertical axis is the y-coordinate of the strip. In each strip, we compute various averages. Say, VX is the average of the x component of velocity, while TX and TY are covariances of x and y components of the velocity respectively, measuring temperatures in the strips.

Wall collision rules

As in Chernov-Lebowitz (1997), we measure the collision angles with respect to \(x\) axis, positive at the top wall and negative at the bottom wall. We denote the incoming angle by \(\varphi\) and the outgoing angle by \(\psi\). Both are in \((0, \pi)\), and in case of specular reflection, \(\psi = \varphi\).

• a-rule: with a parameter \(\alpha \in (0, \pi)\), \[ \psi = \begin{cases} \varphi, & \varphi \in (0, \alpha], \\\\ \frac{\pi - \varphi}{\pi - \alpha} \alpha, & \varphi \in (\alpha, \pi). \end{cases} \]
• b-rule: with a parameter \(b > 0\), \[ \psi = (\pi + b) - [(\pi + b)^2 - \varphi(\varphi + 2b)]^{1/2} . \]
• c-rule: with a parameter \(c \in (0,1]\), \[ \psi = c \varphi . \]
• d-rule: As suggested by David Huse, the outgoing angle is always \(\alpha\), where \(\alpha \in (0, \pi)\) is a parameter
• m-rule: Maxwellian rule: incoming velocity \((v_x^-, v_y^-)\) is ignored, and outgoing velocity has distribution with density \[ \rho(v_x^+, v_y^+) = (2 \pi T_w^3)^{-1/2} v_y^+ \exp \Bigl( - \frac{(v_x^+ - \alpha)^2 + (v_y^+)^2}{2 T_w} \Bigr) . \] Here \(\alpha\) and \(T_w\) are the drift and temperature parameters.
• n-rule: At each wall, if the incoming velocity is \((v_x^-, v_y^-)\), then the outgoing velocity \((v_x^+, v_y^+)\) has the same magnitude and \[ \frac{v_x^+}{v_y^+} = - \frac{v_x^-}{v_y^-} + \mathcal{N}(\text{bias}, 1) , \] where \(\text{bias}\) is adjustable above, and \(\mathcal{N}\) is the normal random variable. When \(\text{bias}\) is positive, the bottom wall naturally pushes particles to the right, and the top wall to the left.
• s-rule: specular reflection, \(\psi = \varphi\)