Why align before moving?

This is the usual discrete-time Vicsek update: neighbors are defined from positions at the current timestep, the new heading follows the local alignment rule, then each particle moves one step along that heading.

Is this a sharp phase transition?

In large systems, increasing density or lowering noise leads to a sharp rise in polarization; on a finite box the transition is blurred, but the qualitative crossover remains: more neighbors inside R and smaller η tend to raise V_a.

Why periodic boundaries?

A torus removes walls and keeps the density statistically uniform, matching the standard numerical setup used in Vicsek-model discussions.

Vicsek Model (Active Matter)

Self-propelled particles on a torus align headings with neighbors within radius R, add noise η, then drift at v₀; polarization V_a = |⟨e^{iθ}⟩| captures the flocking crossover with density ρ = N/L².

About this model

The Vicsek model is the canonical active-matter toy for flocking: self-propelled particles on a torus align their heading with the average direction of neighbors within interaction radius R, add uniform noise ξ ∈ [−η, η], and advect at speed v₀. The polarization V_a = |N⁻¹ Σ_i e^{iθ_i}| measures global orientational order; at fixed box area, ρ ∝ N. This page uses the standard discrete-time update, draws the square torus with velocity arrows, shows V_a as a gold pointer, and plots V_a(t); sweeping η, R, v₀, and N reproduces the familiar noise–density crossover (rounded on finite N).

Who it's for: Students in statistical physics of complex systems, soft matter, and collective behavior.

Key terms

Vicsek model
Active matter
Flocking
Order parameter
Alignment noise
Periodic boundaries

How it works

Self-propelled particles on a torus align their headings with neighbors within distance R, then move at speed v₀. Uniform noise η in the update competes with local coupling to produce a polarization order parameter V_a — the textbook Vicsek active-matter toy for flocking.

Key equations

θ_i(t+Δ) = Arg(Σ_{|r_ij|<R} e^{iθ_j}) + ξ_i with ξ_i ~ Uniform(−η, η), then r_i ← r_i + v₀ e^{iθ_i} Δ with periodic boundaries. Polarization V_a = |N⁻¹ Σ_i e^{iθ_i}|.

Frequently asked questions

Why align before moving?: This is the usual discrete-time Vicsek update: neighbors are defined from positions at the current timestep, the new heading follows the local alignment rule, then each particle moves one step along that heading.
Is this a sharp phase transition?: In large systems, increasing density or lowering noise leads to a sharp rise in polarization; on a finite box the transition is blurred, but the qualitative crossover remains: more neighbors inside R and smaller η tend to raise V_a.
Why periodic boundaries?: A torus removes walls and keeps the density statistically uniform, matching the standard numerical setup used in Vicsek-model discussions.

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About this model

How it works

Key equations

Frequently asked questions

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How it works

Key equations

Frequently asked questions