Source
Toner and Tu (1995), LIT-1670, pp. 1–8 of the arXiv preprint adap-org/9506001; read in NOTE-1228.
What was actually shown
The variables. A coarse-grained velocity field v(r, t) obeying eq. (1) and a number density ρ obeying ∂ρ/∂t + ∇ · (vρ) = 0, which “reflects conservation of birds” (eq. 2, p. 3). The abstract offers it for “a large universality class of microscopic rules, including those recently simulated by Viscek et. al.” (p. 1).
The broken symmetry. For α > 0 the α and β terms “simply make the local v have a non-zero magnitude (= √(α/β)) in the ordered phase”, and v = v₀x̂∥ + δv with v₀x̂∥ = ⟨v⟩ “the spontaneous average value of v” (p. 3). The direction is arbitrary: “the direction in which we choose to break the symmetry of v was arbitrary” (p. 9).
The eliminated degree of freedom. “We will ignore fluctuations in the magnitude |v| from its optimal value of √(α/β) since they decay in a finite time (of order 1/α)” (pp. 3–4). The speed is thereafter fixed: v = (v⊥, √(1 − |v⊥|²)).
The modes that remain. The theory is then in v⊥ and δρ (eqs. 3, 4). “All of the correlation and response functions for this problem” have “two sharp peaks at ω = ±cq⊥”, with c = √(σ₁ρ₀) “the speed of sound” (p. 8), and their widths D⊥ᴿq⊥² + D∥q∥² vanish as q → 0. So the long-wavelength dynamics are propagating and slowly damped, and in d = 2 they scale hyperdiffusively, with z = 6/5 (pp. 7, 9).
This could have been otherwise as a description: a flock could have been modelled with individual trajectories, or with the speed as a dynamical variable at long wavelengths. The theory does neither.
What this does not say
- That the exponents are right. That is THEORY-1095’s question.
- That the universality claim holds. No microscopic rule is coarse-grained into eqs. (1, 2) in the paper.
- That the ordered state is an agent, an individual, or one system in the record’s sense. The paper’s order is a phase of a field; it has no account of agency, and its “birds” are in scare quotes throughout. Whether a broken-symmetry mode is a unit is for CLAIM-157 and QUESTION-281, not this paper.
- “Goldstone mode”. The paper does not use the term. That the transverse velocity is slow because the symmetry is broken is the standard reading of a theory of this form, and is how CLAIM-478 would use it; the paper states only the modes and their dispersion.