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.

Lorentz Gas (Billiard)

Point particle specularly reflected from fixed disks in a box: chaotic paths and MSD growth toward diffusion.

About this model

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.

Continue with similar topics in this category — or all 46 in Thermodynamics.

View category →

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Lorentz Gas (Billiard)

About this model

How it works

Frequently asked questions

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About this model

How it works

Frequently asked questions