On the Limiting Markov Process of Energy Exchanges in a Rarely Interacting Ball-Piston Gas

Abstract

We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a ‘piston’, i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered as the elementary building block of a spatially extended high-dimensional billiard modeling heat transport in a class of hybrid materials exhibiting the kinetics of gases and spatial structure of solids. Using heuristic arguments and numerical analysis, we argue that, in a regime of rare interactions, the billiard process converges to a Markov jump process for the energy exchanges and obtain the expression of its generator.

Appendices

Appendix 1: Collision Volume

To determine the volume $|\Gamma |$ of configuration space, note that when $q_3 > (1 - \lambda )/2$, the volume of all possible positions $q_1$ and $q_2$ is

$$\begin{aligned} 4 \int _0^{(1 - \lambda )/2} \mathrm {d}\, q \left[ \tfrac{1}{2} - \sqrt{\rho ^2 - \left( q - \tfrac{1}{2} \right) ^2} \right] = 1 - \lambda - \rho ^2 \big ( \pi - 4 \arctan \lambda \big ) \,. \end{aligned}$$

(41)

When $q_3 < (1 - \lambda )/2$, we must subtract from the above area the quantity

$$\begin{aligned}&2 \int _{q_3}^{(1-\lambda )/2} \mathrm {d}\, q \left[ \tfrac{1}{2} - \sqrt{\rho ^2 - \left( q - \tfrac{1}{2} \right) ^2} \right] = \frac{1}{2} - \frac{\lambda }{4} - q_3 - \frac{1 - 2 q_3}{4} \sqrt{4 \rho ^2 - (1 - 2 q_3)^2} \nonumber \\&\quad + \rho ^2 \left[ \arctan \lambda - \arctan \frac{1 - 2 q_3}{\sqrt{4 \rho ^2 - (1 - 2 q_3)^2}} \right] . \end{aligned}$$

(42)

We therefore obtain the total volume of configuration space (10) by multiplying Eq. (41) by $\lambda + 2 \delta$ and subtracting the integral of Eq. (42) over $q_3$ from $(1 - \lambda )/2 - \delta$ to $(1 - \lambda )/2$.

To compute the collision surface integral, note that the position coordinates on $\partial \Gamma _{\textsc {bp}}$ are bounded according to

$$\begin{aligned}&\frac{1}{2}(1 - \lambda ) - \delta \le q_1 = q_3 \le \tfrac{1}{2} (1 - \lambda ) \,, \nonumber \\&\quad -\frac{1}{2} + \sqrt{\rho ^2 - (q_1-\tfrac{1}{2})^2} \le q_2 \le \tfrac{1}{2} - \sqrt{\rho ^2 - (q_1 - \tfrac{1}{2})^2}. \end{aligned}$$

(43)

Its projection on the $(q_1, q_2)$ plane is the area (42) evaluated at $q_3 = (1 - \lambda )/2 - \delta$. Since the surface itself makes an angle $\pi /4$ with respect to the $(q_1, q_2)$ plane, we obtain for $|\partial \Gamma _{\textsc {bp}}|$ the expression given by Eq. (11).

Appendix 2: Ball-Wall and Piston-Wall Return Times

Wall collision return times of the piston and ball can be computed in ways similar to Eq. (9),

$$\begin{aligned} \overline{\tau }_{\textsc {pw}}&= \frac{4 |\Gamma |}{|\partial \Gamma _{\textsc {pw}}|}\,, \nonumber \\ \overline{\tau }_{\textsc {bw}}&= \frac{4 |\Gamma |}{|\partial \Gamma _{\textsc {bw}}|}\,, \end{aligned}$$

(44)

where $|\partial \Gamma _{\textsc {pw}}|$ and $|\partial \Gamma _{\textsc {bw}}|$ are the areas of piston-wall and ball-wall collisions. The former corresponds to the area of all positions $q_1$ and $q_2$ such that $q_3 = ( 1 \pm \lambda )/2 \pm \delta$, which is parallel to the $(q_1, q_2)$ plane and twice the area (41) minus the projection of the collision surface $|\partial \Gamma _{\textsc {bp}}|$ (11) on this plane, and the latter to the positions $q_1, q_2$ such that $(q_1 \pm \tfrac{1}{2})^2 + (q_2 \pm \tfrac{1}{2})^2 = \rho ^2$ while $q_1 < q_3$, with $q_3$ integrated over the interval (1). That is,

$$\begin{aligned} |\partial \Gamma _{\textsc {pw}}|&= 2\Big [1 - \lambda - \rho ^2 \big ( \pi - 4 \arctan \lambda \big ) \Big ] - \frac{1}{\sqrt{2}}|\partial \Gamma _{\textsc {bp}}|\,, \end{aligned}$$

(45)

and

$$\begin{aligned} |\partial \Gamma _{\textsc {bw}}|&= \rho ( \lambda + 2 \delta ) \Big ( 8 \arcsin \frac{1 - \lambda }{2 \sqrt{2} \rho } - \arcsin \frac{\lambda + 2 \delta }{2 \rho } + \arcsin \frac{\lambda }{2 \rho } \Big ) \\&\quad + \rho \big [1 - \sqrt{1 - 4 \delta ( \lambda + \delta ) } \big ] \,. \end{aligned}$$

(46)

Appendix 3: Restriction of the Invariant Measure on ${M}_{\textsc {bp}}$ to Fixed Energy Configurations

Substituting the parametrisation (17) of the velocity vector $\mathbf {v}\in \mathbb {S}^{2}$ and evaluating its scalar product with the normal vector (6), the velocity integral in Eq. (19) splits into two contributions, integrated over an arc-length proportional to the angle along the arcs:

$$\begin{aligned} \int _{2\times \mathbb {S}^{1}: \, \mathbf {v}\cdot \mathbf {n} \ge 0} \mathrm {d}\mathbf {v}\,&(\mathbf {v}\cdot \mathbf {n}) = \frac{1}{\sqrt{2}} \int _{\mathbb {S}^{1}: \, \sqrt{2\epsilon _{\textsc {p}} } \ge \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha } \mathrm {d}\alpha \, (\sqrt{2\epsilon _{\textsc {p}} } - \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha ) \\&\quad + \frac{1}{\sqrt{2}} \int _{\mathbb {S}^{1}: \, \sqrt{2\epsilon _{\textsc {p}} } \le -\sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha } \mathrm {d}\alpha \, (-\sqrt{2\epsilon _{\textsc {p}} } - \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha ) \, ; \end{aligned}$$

(47)

see Fig. 6 for a graphical representation. Two separate regimes arise.

Fig. 6

Equation (47) has either one ($1/4 \le \epsilon _{\textsc {p}} \le \tfrac{1}{2}$) or two ($0 \le \epsilon _{\textsc {p}} < 1/4$) contributions, given by the integrals along the arc-circles on the hemisphere of velocity coordinates whose projection along the normal to the collision surface (6) is positive. Here $\epsilon _{\textsc {p}} = 1/8$ and the two arc-circles at $v_3 = \pm \tfrac{1}{2}$ contributing to Eq. (47) are the portions (in green) of the corresponding full circles above the plane tangent to the collision surface (the excluded parts of those circles are shown in white) (Color figure online)

On the one hand, when the piston’s energy is less than the ball’s, the condition $\sqrt{2\epsilon _{\textsc {p}} } \ge \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha$ in the first of the two integrals on the right-hand side of Eq. (47) is equivalent to

$$\begin{aligned} \arccos \sqrt{\frac{\epsilon _{\textsc {p}} }{\tfrac{1}{2} - \epsilon _{\textsc {p}} }} \le \alpha \le 2 \pi - \arccos \sqrt{\frac{\epsilon _{\textsc {p}} }{\tfrac{1}{2} - \epsilon _{\textsc {p}} }} \,. \end{aligned}$$

(48)

Performing the integral, we obtain the contribution,

$$\begin{aligned} \frac{1}{\sqrt{2}} \int _{\mathbb {S}^{1}: \, \sqrt{2\epsilon _{\textsc {p}} } \ge \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha }&\mathrm {d}\alpha \, (\sqrt{2\epsilon _{\textsc {p}} } - \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha ) \\&= 2\sqrt{\epsilon _{\textsc {p}} } \left( \pi - \arccos \sqrt{\frac{\epsilon _{\textsc {p}} }{\tfrac{1}{2} - \epsilon _{\textsc {p}} }} \right) + 2 \sqrt{ \tfrac{1}{2} - 2 \epsilon _{\textsc {p}} } \,. \end{aligned}$$

(49)

Likewise, the condition $\sqrt{2\epsilon _{\textsc {p}} } \le - \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha$ in the second of the two integrals on the right-hand side of Eq. (47) is equivalent to

$$\begin{aligned} \pi - \arccos \sqrt{\frac{\epsilon _{\textsc {p}} }{\tfrac{1}{2} - \epsilon _{\textsc {p}} }} \le \alpha \le \pi + \arccos \sqrt{\frac{\epsilon _{\textsc {p}} }{\tfrac{1}{2} - \epsilon _{\textsc {p}} }} \,. \end{aligned}$$

(50)

Thus the second integral yields the contribution

$$\begin{aligned} \frac{1}{\sqrt{2}} \int _{\mathbb {S}^{1}: \, \sqrt{2\epsilon _{\textsc {p}} } \le - \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha }&\mathrm {d}\alpha \, \left( -\sqrt{2\epsilon _{\textsc {p}} } - \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha \right) \\&= - 2\sqrt{\epsilon _{\textsc {p}} } \arccos \sqrt{\frac{\epsilon _{\textsc {p}} }{\tfrac{1}{2} - \epsilon _{\textsc {p}} }} + 2 \sqrt{ \tfrac{1}{2} - 2 \epsilon _{\textsc {p}} } \,. \end{aligned}$$

(51)

The contribution to Eq. (19) from the velocity integral in the corresponding energy interval is thus given by the sum of Eqs. (49) and (51).

On the other hand, when the piston’s energy is larger than the ball’s, the condition $\sqrt{2\epsilon _{\textsc {p}} } \ge \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha$ in the first of the two integrals on the right-hand side of Eq. (47) holds true for all angles $\alpha$. The result of the integration,

$$\begin{aligned} \frac{1}{\sqrt{2}} \int _{\mathbb {S}^{1}: \, \sqrt{2\epsilon _{\textsc {p}} } \ge \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha } \mathrm {d}\alpha \, \left( \sqrt{2\epsilon _{\textsc {p}} } - \sqrt{1 - 2\epsilon _{\textsc {p}} } \cos \alpha \right) = 2 \pi \sqrt{\epsilon _{\textsc {p}} } \,, \end{aligned}$$

(52)

yields the only contribution to the velocity integral in Eq. (19).