Some basic conventions

Billiards

A billiard is a dynamical system describing the motion of a point particle in a connected, compact domain Q ⊂ 𝕋^{d}. The boundary of the domain is assumed to be piecewise C^{3}-smooth. Inside Q the motion is uniform while reflection at the boundary ∂Q is elastic. As the absolute value of the velocity is a first integral of motion, the phase space of the billiard flow is fixed as M = Q × S^{d−1} – in other words, every phase point x is of the form x = (q, v) with q ∈ Q and v ∈ ℝ^{d}, |v| = 1. The Liouville probability measure μ on M is essentially the product of the Lebesgue measures, i.e. dμ = const. dqdv. The resulting dynamical system (M, S^{ℝ}, μ) is the (toric) billiard flow. The boundary ∂Q defines a natural cross-section, the Poincaré section for the billiard flow. Namely consider

M = ∂M = {(q, v) | q ∈ ∂Q, ⟨v, n(q)⟩ ≥ 0},

the Poincaré phase space (This set actually has a natural bundle structure.) The Poincaré section map T, also called the billiard map of Poincaré map (denoted by (M, T, μ_{1})) is defined as the first return map on M. The invariant measure for the map is denoted by μ_{1}, and we have dμ_{1} = const. |⟨v, n(q)⟩| dqdv. Recall the usual notation: for (q, v) ∈ M one denotes π(q, v) = q, the natural projection.

Semi-dispersing and Dispersing billiards Let n(q) denote the unit normal vector of a smooth component of the boundary ∂Q at the point q, directed inwards Q. A billiard is called semi-dispersing (dispersing) if for every q ∈ ∂Q the second fundamental form K(q) of the boundary component is non-negative (positive, resp.).

The Lorentz process

The Lorentz process is the natural ℤ^{d} cover of a toric billiard. More precisely: consider Π : ℝ^{d} → 𝕋^{d} the factorisation by ℤ^{d}. Its fundamental domain D is a d-dimensional cube (semi-open, semi-closed) in ℝ^{d}, so ℝ^{d} = ∪_{z∈ℤ^{d}}(D + z), where D + z is the translated fundamental domain. By denoting Q̃ = Π^{−1}Q, M̃ = Q̃ × S^{d−1}, etc., the Lorentz dynamics is (M̃, S̃^{ℝ}, μ̃) and its Poincaré section map is (∂M̃, T̃, μ̃_{1}). (The Lorentz dynamics is actually a billiard whose invariant measure is infinite.) By dropping the natural periodicity property of the scatterers of the ‘periodic’ Lorentz process one obtains the model of the Lorentz process, in general.

The Lorentz gas

The Lorentz gas is the motion of a countable number of non-interacting Lorentz particles. The natural invariant measure is the Gibbs-measure, which in this case is a Poissonian measure with density

ρ exp(−v^{2}/2) dx dv.