Coherence Explorer
What is coherence?
Coherence ≈ 0 is like a crowd of strangers in a plaza — everyone moving independently, no shared rhythm. Coherence ≈ 1 is like a marching band — everyone in lockstep. This curve shows how a system transitions between the two as density increases.
This tool plots the coherence function — the single equation at the heart of Synchronism. It takes a density (ρ) and returns a coherence value between 0 (independent/quantum-like) and 1 (collective/classical-like).
What this shows / what it gets wrong: the tool shows how the curve's shape responds to its three parameters — that part is faithful. What the formula gets wrong, and you should know before sliding: the γ = 2/√Ncorr sharpness rule runs backwards against real condensed-matter physics (the most correlated systems get the flattest curves; real BCS transitions are among nature's sharpest), and the “ρcrit” slider marks a saturation knee, not the curve's midpoint — at ρcrit and γ = 2 the curve already reads C ≈ 0.88, and the true C = 0.5 midpoint sits near 0.32·ρcrit. Both numbers are γ-specific: the live readout by the ρcrit slider recomputes them as you drag γ. Details in the caveats below.
⚠ Terminology note for physicists: “Coherence” here means classical collective ordering (C ≈ 0 = independent/quantum-like; C ≈ 1 = classically ordered). This is anti-correlated with quantum phase coherence as used in condensed-matter physics, where BEC/BCS condensates — the most quantum-coherent systems known — would sit at low C by this measure (due to their large Ncorr). The two axes are orthogonal: macroscopic quantum states are simultaneously quantum and collective. The site uses “coherence” in the ordering/classicality sense, not the off-diagonal-long-range-order sense.
γ = 2/√Ncorr controls the transition sharpness. High γ (> 1.4, small Ncorr) = single-particle / uncorrelated regime (ideal gases, free atoms); γ ≈ 1 = the boundary where chemistry and biology happen; low γ (< 0.6, large Ncorr) = collective / correlated regime (BEC, superconductors, superfluids). Note: these labels describe the number of correlated degrees of freedom, not the standard quantum/classical distinction — BEC and BCS superconductors appear in the “collective” basin, which is correct (they have large Ncorr).
⚠ Sign-inversion caveat: γ = 2/√Ncorr is structurally inverted relative to real condensed-matter physics — more correlation (larger Ncorr) gives asmaller γ, i.e. a flatter transition, assigning the flattest curve to BCS superconductors (which have a sharp real Tc) and the sharpest curve to an ideal gas (which has no real phase transition). See the γ Calculator for the full sign-inversion accounting.
What to notice: Move γ from 2.0 downward and watch the curve flatten. γ ≈ 1 is the regime boundary where chemistry and biology happen — not a steepness extremum. (The slope is actually largest at ρ = 0 on a linear ρ axis; on a log axis, the peak slope grows as γ increases — e.g. peak dC/d(log₁₀ρ) ≈ 0.375 at γ=2 vs ≈ 0.25 at γ=0.5 — consistent with γ=2 being the sharpest curve.) The slider goes down to γ = 0.01: on the canonical ladder BEC sits near γ ≈ 2×10−3 (Ncorr = 106) and BCS superconductors near γ ≈ 6×10−4(Ncorr = 107) — both below the slider's floor (use the γ Calculator to reach those regimes). At very low γ the curve is nearly flat — high coherence at almost all densities, which is the strongly collective regime.
Why γ is decoupled from Ncorr here: This tool sets γ directly to explore how curve shape depends on the sharpness parameter, independently of any physical system. The relationship γ = 2/√Ncorr links γ to real systems — use the γ Calculator to enter a physical Ncorr and see where a real system lands.
- ln(x) — logarithm: it compresses huge ranges into small ones. A density 1,000× bigger becomes only ~7 units bigger. This is why the curve works across quantum to cosmic scales.
- tanh(u) — the “S-curve” shape. For very negative u it returns ≈ 0; for large positive u it returns ≈ 1; near zero it rises steeply. Think of a dimmer switch that snaps rather than fading gradually.
- γ — controls how quickly the snap happens. Large γ (γ = 2, free atoms) = a sharp cliff. Small γ (γ ≈ 6×10⁻⁴, superconductors) = a long gentle ramp.
- ρcrit — the density where the middle of the S-curve sits; a reference point set by fitting, not a physical critical point.
Adjust γ and ρcrit to see how the coherence function C(ρ) = tanh(γ · ln(ρ/ρcrit + 1)) responds.
Collective (γ < 0.6)
Higher γ = sharper, more abrupt snap to coherent. Lower γ = gentler slope. Depends on Ncorr (correlated particle count): γ = 2/√Ncorr.
Opens at γ ≈ 0.49, the value SPARC galaxy data actually prefers when fit freely. γ = 2 was the framework's original galaxy guess — refuted (drag the slider up to see it).
What the default view actually shows (added 2026-08-08): MOND. Set γ = 1/2 exactly and the compander collapses algebraically to C(x) = x/(x+2) = μsimple(x/2) — MOND's simple interpolating function, the factor of 2 absorbed by the fitted ρcrit. SPARC's 0.489 is 2.2% from that point, so the curve you are looking at on load is MOND's μ, not an alternative to it. Drag γ away from 0.5 and you are leaving MOND; the data pull it back.
Lower ρcrit = transition starts at lower density (shifts curve left). Higher = transition occurs at higher density.
This slider carries no shape information — and that is the point (added 2026-08-08). C depends on ρ and ρcrit only through the single ratio x = ρ/ρcrit, exactly, not just asymptotically. On the log axis this slider therefore performs a rigid horizontal translation by −log10ρcrit and changes nothing about the curve's shape: the tool is a one-parameter family plus a pan control, presented as two parameters. That is not a UI nit — it is the γ↔ρcrit degeneracy that makes SPARC's γ = 0.489 non-identifiable against a rescaled critical density, which is the crux of the whole galaxy sector, made visible. Read the abscissa as log10(ρ/ρcrit) — dimensionless — and the degeneracy is the only thing the slider can express. Raised by a visitor researcher pass.
Note: ρcrit is a saturation knee, not a critical density in the phase-transition sense. At ρ = ρcrit and the current γ = 0.49, C = tanh(0.49·ln2) ≈ 0.327 — the “+1” regulator inside the ln breaks sigmoid symmetry, so ρcrit is not the curve's midpoint. The C = 0.50 midpoint sits at ρ ≈ 2.07·ρcrit. This offset is γ-dependent and moves as you drag the slider (corrected 2026-07-27 — this note used to quote the γ=2 value 0.32 as if it were a general property, which goes wrong the moment the slider it accompanies is touched: at the SPARC-preferred γ = 0.49 the midpoint is at 2.07·ρcrit, i.e. above the knee).
Key Values
C(ρcrit)
0.3272
C(10ρcrit)
0.8259
C(100ρcrit)
0.9785