Lorentz Gas (Billiard)
The Lorentz gas is a minimal kinetic model: a single point particle moves at constant speed in a billiard table with fixed convex scatterers. Specular reflections make the map strongly chaotic (positive Lyapunov exponent) for typical Sinai-type geometries with finite horizon. On long times, the mean squared displacement from a fixed origin grows approximately linearly with time, ⟨r²⟩ ∝ t, signalling normal diffusion—one of the few rigorous results in chaotic billiards. This page uses a square enclosure with a periodic array of hard disks; sliders change flight speed and disk radius (geometry). The trail and MSD readout are pedagogical, not a quantitative diffusion coefficient measurement.
Who it's for: Undergraduates studying chaos, kinetic theory, or statistical mechanics who want a visual bridge between deterministic scattering and transport.
Key terms
- Lorentz gas
- Sinai billiard
- Specular reflection
- Chaos
- Mean squared displacement
- Finite horizon
How it works
Lorentz gas: a single particle in a billiard with convex scatterers. Deterministic chaos and, on large scales, normal diffusion — a paradigm for kinetic theory.
Frequently asked questions
- Why does the trajectory look random if the physics is deterministic?
- Specular billiards are deterministic but exhibit sensitive dependence on initial conditions: tiny changes in position or angle grow exponentially. A single trajectory can look “random” even though it is uniquely fixed by the initial data—this is the hallmark of deterministic chaos.
- Is the MSD shown here the same as Einstein’s diffusion coefficient?
- Not directly. In two dimensions with finite horizon one expects ⟨r²⟩ ≈ 4D t at long times for some diffusion constant D that depends on geometry. The simulator displays MSD in box units versus simulation steps as an intuition aid; extracting D would require careful ensemble or time averaging and calibration of the time step.
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