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THEORY-692: Near equilibrium, steady states minimise entropy production and every fluctuation decays; far from it, the excess entropy production has no fixed sign, so a fluctuation can grow past an instability of the thermodynamic branch into a structure held up only by flows across the boundary, and thermodynamics supplies only a sufficient condition for such a state's stability, not a principle selecting which structure appears

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Properties3
StatusActive
Tagscomplex-systems, thermodynamics
SourceLIT-516 — Time, Structure and Fluctuations, LIT-527 — Pattern formation outside of equilibrium

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Source

Prigogine (1977), LIT-516, Nobel lecture “Time, Structure and Fluctuations”, read in full in NOTE-424. Cross & Hohenberg (1993), LIT-527, Reviews of Modern Physics 65:851–1112, skimmed in NOTE-418 (about 30 of 262 pages, chosen for this question). The lecture states the theorems and cites their proofs; it does not prove them.

What was actually shown

Near equilibrium (§§2–3 of the lecture). With local equilibrium and linear laws J = LX with Onsager symmetry, “for steady states sufficiently close to equilibrium entropy production reaches its minimum” (p. 266). That needs “even more restrictive conditions” than linearity: constant coefficients. The second-order excess entropy δ²S is negative and its time derivative is the entropy production, so δ²S is a Lyapounov function and “ensures the damping of all fluctuations”.

Far from equilibrium (p. 269). Around a non-equilibrium steady state ½ ∂δ²S/∂t = Σ δJ_ρ δX_ρ, the excess entropy production, which “has generally not a well-defined sign”. If it stays non-negative, the state is stable. That is sufficient; its violation is not shown to imply instability.

Dissipative structures. Beyond an instability of the branch continuous with equilibrium, a new state can appear: in Bénard convection, “a giant fluctuation stabilized by exchanges of energy with the outside world” (p. 267). The Brusselator, with its inputs held constant, has steady state X₀ = A, Y₀ = B/A, unstable for B > 1 + A² (4.4), beyond which there is a limit cycle; with diffusion, stationary patterns (the “Turing bifurcation”). Autocatalytic steps are necessary but not sufficient for the chemical case (p. 271). Which branch is taken at a bifurcation depends on which fluctuation happened.

The review’s verdict (LIT-527, §XIII). In general “no such free energy (or other so-called Lyapunov potential) can be defined for nonequilibrium systems, though there are notable exceptions” (p. 857). “There is no evidence for the existence of any global minimization principles controlling the structure, except as a perturbative statement near threshold” (p. 1080). A closed chemical system “ultimately must come to equilibrium”, so non-equilibrium structure is a transient or a response to “external chemical pumping” (p. 865).

What could have come out otherwise. A general extremum principle for non-equilibrium steady states could have been found. As of 1993 none had produced physical results on pattern formation, the maximum-entropy formalism included.

What this does not say

  • That minimum entropy production governs dissipative structures. The lecture restricts it to the strictly linear regime, where there are no such structures. Far from equilibrium behaviour can be “even opposite to that indicated by the theorem” (pp. 266–267).
  • That no selection principle can exist. Only that none is established. The maximum-entropy-production proposal (THEORY-671) is a later candidate, and its derivation is contested (THEORY-676).
  • That dissipative structure is a mark of life. Convection rolls and chemical clocks meet the criterion and are not alive. NOTE-424 and NOTE-418 both read the physics as a necessary condition at most.
  • That irreversibility comes from dynamics. The lecture’s §§6–7 programme assumes an entropy operator exists, says that “is certainly not always possible” (p. 279), and links it to macroscopic thermodynamics only “at least in the linear region”.
  • That Bénard’s cells were buoyancy-driven. The lecture’s “Bénard instability” is Rayleigh’s problem. Bénard’s open-dish hexagons were surface-tension driven (Pearson 1958; LIT-527 §II.B.3).

Connections

  • Turing’s instability (THEORY-543) is the stationary diffusive case the lecture names after him, with a purely kinetic criterion.
  • The BZ reaction (THEORY-508): a closed batch oscillates only as a transient, and a fed reactor sustains it, which is this account’s open system in the laboratory.
  • Maximum entropy production (THEORY-671) proposes the selection principle this account says thermodynamics lacks. It does not contradict Prigogine’s theorem, which answers another question in another regime. No rivals edge is declared: this account says no principle is established, not that none holds.
  • Gradient degradation (THEORY-532) and driven matter (THEORY-412) are further candidates for what selects among far-from-equilibrium states.
  • Organisms against flames (THEORY-072) cites the lecture for the structures themselves and draws the line by closure of constraints, which this physics does not supply.

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