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Entropy production in random billiards

  • * Corresponding author: Timothy Chumley

    * Corresponding author: Timothy Chumley 
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  • We consider a class of random mechanical systems called random billiards to study the problem of quantifying the irreversibility of nonequilibrium macroscopic systems. In a random billiard model, a point particle evolves by free motion through the interior of a spatial domain, and reflects according to a reflection operator, specified in the model by a Markov transition kernel, upon collision with the boundary of the domain. We derive a formula for entropy production rate that applies to a general class of random billiard systems. This formula establishes a relation between the purely mathematical concept of entropy production rate and textbook thermodynamic entropy, recovering in particular Clausius' formulation of the second law of thermodynamics. We also study an explicit class of examples whose reflection operator, referred to as the Maxwell-Smoluchowski thermostat, models systems with boundary thermostats kept at possibly different temperatures. We prove that, under certain mild regularity conditions, the class of models are uniformly ergodic Markov chains and derive formulas for the stationary distribution and entropy production rate in terms of geometric and thermodynamic parameters.

    Mathematics Subject Classification: Primary: 37A25, 80A99; Secondary: 60J25.


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  • 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

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