A finite-time exponent for random Ehrenfest gas

• We present a finite-time exponent for particles moving in a plane containing polygonal scatterers.
• The exponent found recovers the Lyapunov exponent in the limit of the polygon becoming a circle.
• Our findings unify pseudointegrable and chaotic scattering via a generalized collision rule.
• Stretch and fold:shuffle and cut :: Lyapunov:finite-time exponent :: fluid:granular mixing.

We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit.