The Galton Board: Limit Theorems and Recurrence

The Galton board [21, Chapter V], also known as the quincunx or bean machine, is one of the simplest mechanical devices exhibiting stochastic behavior. It consists of a vertical (or inclined) board with interleaved rows of pegs. A ball thrown into the Galton board moves under gravitation and bounces off the pegs on its way down. If many balls are thrown into the quincunx, then one can observe a normal distribution of balls coming to rest on the machine floor. In this paper we deal with an idealized infinite Galton board: we consider a ball moving in a bean machine of infinite length under a constant external field. We neglect friction and the spin of the ball. Our pegs are convex obstacles (scatterers) positioned periodically on the board and satisfying the ‘finite horizon’ condition (the latter means that the ball cannot move in any direction indefinitely without meeting a scatterer). This model is identical to a periodic Lorentz gas. Historically, Lorentz gas (in the 3D space and without necessarily periodic position of scatterers) was introduced in 1905 (see [26]) to illustrate the transport of electrons in metals in a spatially homogeneous electric field. Periodic Lorentz gases were later studied mathematically [2, 3, 33]. Without external fields, the periodic Lorentz gas reduces to a billiard system on its fundamental domain (a torus minus scatterers). This is a dispersing billiard (Sinai billiard); it preserves a Liouville (equilibrium) measure and has strong ergodic and statistical properties. In particular it exhibits diffusive behavior; see [2, 3, 5, 33, 35]. Under a constant external field, the moving particle is likely to accelerate indefinitely, thus the system does not even have a stationary measure (physicists say that there is no steady state). Such a non-stationary behavior makes mathematical studies very difficult and may explain the lack of rigorous results (until now), despite persistent interest in the physics community [4, 23, 24, 27, 28, 30, 31]. To make things more tractable, one can remove the excess energy in various ways (deterministically or stochastically). One way to do that is to modify the equations of motion by introducing the so-called Gaussian thermostat [9, 10, 28]; this will keep the particle’s speed constant. A rigorous investigation of the Gaussian thermostatted Lorentz particle under a small external field is done in [9]: it is proven that the dynamics has a stationary measure (steady state), the particle exhibits diffusive behavior and, in addition, it slowly drifts with an average velocity proportional to