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).
⚠ 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).
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 decreases.) The slider goes down to γ = 0.01: BCS superconductors sit near γ ≈ 0.02 and BEC near γ ≈ 0.002 (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 ≈ 0.02, 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.
Single-particle (γ > 1.4)
Higher γ = sharper, more abrupt snap to coherent. Lower γ = gentler slope. Depends on Ncorr (correlated particle count): γ = 2/√Ncorr.
Ncorr = 1.0 — mean-field approximation weakens as Ncorr approaches 1
Lower ρcrit = transition starts at lower density (shifts curve left). Higher = transition occurs at higher density.
Note: ρcrit is a saturation knee, not a critical density in the phase-transition sense. At ρ = ρcrit with γ = 2, C = tanh(2·ln2) ≈ 0.88 — the function is already 88% saturated. The C = 0.50 midpoint occurs at ρ ≈ 0.32·ρcrit, well below ρcrit. The “+1” regulator inside the ln breaks sigmoid symmetry.
Key Values
C(ρcrit)
0.8824
C(10ρcrit)
0.9999
C(100ρcrit)
1.0000