The impossibility of this perpetuum mobile is (I think) a consequence of the fact that the phase flow preserves the volume dxdydtheta, AT LEAST IF THE PROBLEM IS MATHEMATICALLY FORMULATED in terms of classical billiards (I can imagine different formulations). Here is one possible mathematical formulation of the physical statements, avoiding the need to talk of the temperature. I apologize to experts for some of what follows. I claim that the phase volume of the billiard is a resonable interpretation of the energy of the corresponding set of photons". More precisely, our photons can be thought of as particles moving with unit speed and undergoing elastic collisions with obstacles. The phase space is three dimensional with coordinates x,y,theta where theta is the angle of the velocity with a fixed direction in the plane. To justify the above claim, let us start with a huge number of photons" equidistributed in the phase space: teh density is nearly constant. By volume preservation the density will remain constant for all time. Thus the volume really counts the number of photons, and thus the energy, since each ``photon" carries one unit of energy.

Now it is easy to see that the combined energy of ``photons" hitting an obstacle c
(e.g., a disk) equals the combined energy of the ones that bounce off (during the same time interval). Indeed, c is the projection onto the xy-plane of the cylinder c\times S^1 in the phase space. Half of the surface of this cylinder is entered by the fluid; another half is exited. By volume preservation, the entering flux equals the exiting flux. That is, as many photons hit the obstacle, as bounce off. Thus the temperature of the obstacle does not change.

The impossibility of this perpetuum mobile is (I think) a consequence of the fact that the phase flow preserves the volume $dx\ dy\ d\theta$, AT LEAST IF THE PROBLEM IS MATHEMATICALLY FORMULATED in terms of classical billiards (I can imagine different formulations). Here is one possible mathematical formulation of the physical statements, avoiding the need to talk of the temperature. I apologize to experts for some of what follows. I claim that the phase volume of the billiard is a reasonable interpretation of the energy of the corresponding set of "photons". More precisely, our photons can be thought of as particles moving with unit speed and undergoing elastic collisions with obstacles. The phase space is three dimensional with coordinates $x,y,\theta$ where $\theta$ is the angle of the velocity with a fixed direction in the plane. To justify the above claim, let us start with a huge number of "photons" equidistributed in the phase space: the density is nearly constant. By volume preservation the density will remain constant for all time. Thus the volume really counts the number of photons, and thus the energy, since each "photon" carries one unit of energy.

Now it is easy to see that the combined energy of "photons" hitting an obstacle $c$ (e.g., a disk) equals the combined energy of the ones that bounce off (during the same time interval). Indeed, $c$ is the projection onto the $xy$-plane of the cylinder $c\times S^1$ in the phase space. Half of the surface of this cylinder is entered by the fluid; another half is exited. By volume preservation, the entering flux equals the exiting flux. That is, as many photons hit the obstacle, as bounce off. Thus the temperature of the obstacle does not change.