Spontaneous emergence of macroscopic order from microscopic interactions in driven, dissipative systems — no external template imposed, order set by intrinsic dynamics. Canonical models: Bak-Tang-Wiesenfeld 1987 abelian sandpile and…
self-organization
Bak-Tang-Wiesenfeld 1987 abelian sandpile: self-organised criticality
Bak-Tang-Wiesenfeld 1987 (Phys. Rev. Lett. 59:381) introduced the *self-organised criticality* (SOC) concept: a slowly driven,…
Prigogine 1947 theorem of minimum entropy production
Prigogine 1947-1977 (Thermodynamics of Irreversible Processes; Nobel Prize 1977) theorem — a *linear* non-equilibrium system with fluxes…
Lotka-Volterra predator-prey: conservative oscillation as self-organisation
Lotka 1920 (PNAS 6:410) and Volterra 1926 (Nature 118:558) independently derived the two-species predator-prey equations: ẋ = ax - bxy…
BTW 1D sandpile: critical slope z_c = 2, deterministic linear avalanches
Exact result for the 1D abelian sandpile. The toppling rule 'if z_i ≥ z_c, move z_c grains: one to each neighbor and z_c-2 to ground (z_c…
Prigogine min-σ theorem: X_2^ss = -L_{12}/L_{22}·X_1 (2-flow residual ≡ 0)
Exact algebraic instance of the Prigogine theorem for two coupled linear fluxes. Entropy production σ = L_{11}X_1² + 2 L_{12} X_1 X_2 +…
Lotka-Volterra conserved quantity dH/dt ≡ 0 on every trajectory
Exact symbolic verification of conservation. With ẋ = ax - bxy, ẏ = cxy - dy and H(x,y) = cx + by - d ln(x) - a ln(y), direct…