Entropy production in random billiards

Figure 1.  The random billiard map is the composition of two maps: the geodesic translation $ \mathcal{T} $ and scattering determined by the reflection operator $ P $. The distribution of the velocity $ V $ after reflection is given by $ \mathcal{B}_x = P_{\mathcal{T}(x)} $

Figure 2.  The two-masses system

Figure 3.  Configuration manifold for the two-masses random billiard system

Figure 4.  Depiction of some of the vectors appearing in the proof of Proposition 5

Figure 5.  A particle of mass $ m $ bounces back and forth between two plates kept at temperatures $ T_1 $ and $ T_2 $. For the reflection operator we use the Maxwell-Smoluchowski model with probabilities of diffuse reflection $ \alpha_1 $ and $ \alpha_2 $

Figure 6.  Entropy production for a billiard system whose billiard domain is formed by the union of two discs of equal radius $ r $ whose centers are $ a $ units apart. The ratio parameter is $ a/2r $ and the number above each graph is the temperature difference $ T_2-T_1 $. The vertical bars indicate 95% confidence intervals

Figure 7.  A simple random billiard heat engine. The difference in temperatures allows the system to do work against an external force

Figure 8.  When $ F = 0 $ the conveyor belt steadily drifts counterclockwise when $ T_h-T_c>0 $, and clockwise when the temperatures are reversed. The temperatures for the top $ 4 $ graphs are $ T_c = 1 $ and $ T_h = 1, 10, 25, 50 $, and $ T_c, T_h $ are reversed for the $ 3 $ lower graphs. The inset is the same as the graph for $ T_c = T_h = 1 $ but in a finer scale so that its stochastic character is more clearly apparent

Figure 9.  Efficiency of the billiard heat engine as a function of the force acting on the sliding wall