From Deterministic to Stochastic
Papers XII–XIII established deterministic gradient flow on the closure functional: states evolve by steepest descent toward closure-stable configurations. This produces attractor basins, interaction potentials, and field-mediated forces.
But deterministic dynamics cannot account for probabilistic outcomes — identical initial conditions always produce identical trajectories. It cannot explain fluctuations around stable states, spontaneous transitions between basins, or the discrete excitation spectra observed in nature.
The solution is minimal: add noise.
The Langevin Equation
The noise amplitude σ controls the balance between deterministic relaxation and stochastic exploration. Three regimes:
- σ → 0 — recovers the deterministic gradient flow of Papers XII–XIII
- σ small — fluctuations near attractors, occasional transitions between basins
- σ large — free exploration of the state space, closure structure washed out
The Fokker–Planck Equation
The Langevin equation governs individual trajectories; the Fokker–Planck equation governs the probability density over the full state space. The two are mathematically equivalent descriptions of the same stochastic process.
The Stationary Distribution
The stationary distribution concentrates on the constraint manifold Mcl. Closure-stable states (F = 0) are probability maxima. States with large closure residual are exponentially suppressed. As σ → 0, probability collapses onto Mcl.
Emergent Excitation Spectra
Near a closure-stable state ψ0, the closure functional expands quadratically:
F ≈ ½(ψ − ψ0)T H(ψ0)(ψ − ψ0)
where H(ψ0) is the Hessian of F at the attractor. The local distribution is Gaussian with covariance proportional to H−1. The Hessian eigenvalues {hk} define the excitation spectrum:
- Large hk — tightly confined direction (stiff mode)
- Small hk — loosely confined direction (soft mode)
- Zero hk — symmetry direction or marginal mode
Discrete spectral structure emerges from the curvature of F at attractors — without postulating quantum energy levels.
Transition Rates
Kramers-type barrier crossing between attractor basins. The transition rate from basin A to basin B is:
High barriers produce exponentially suppressed transitions — stable particles. Low barriers produce enhanced transitions — short-lived states. This is a structural analogue of decay rates, without quantum decay theory.
Path-Integral Formulation
Trajectories following gradient flow are most probable. Deviations are exponentially penalised. The most probable transition path between two basins — the path minimising the Onsager–Machlup action — is the closure-framework analogue of an instanton.
Measurement as Basin Selection
Under stochastic dynamics, the state fluctuates among attractor basins. The probability of finding the system in basin B(ψ*) is:
P(basin ψ*) = ∫B(ψ*) Pst dψ
Measurement is reinterpreted as observing which basin the stochastic trajectory currently occupies. Outcomes are probabilistic — different observations may find different basins — without postulating wavefunction collapse or the Born rule. The probabilities are determined by the stationary distribution, which is itself determined by the closure functional.
How It Compares to Quantum Mechanics
| Quantum Mechanics | Stochastic Closure Dynamics |
|---|---|
| Hilbert space | Admissible state space Sadm |
| Hamiltonian Ĥ | Closure functional F |
| Schrödinger equation | Langevin / Fokker–Planck |
| Energy eigenvalues | Hessian eigenvalues at attractors |
| Born rule | Stationary distribution Pst |
| Quantum tunnelling | Kramers barrier crossing |
| Path integral (eiS/ℏ) | Onsager–Machlup (e−action/σ²) |
| Measurement / collapse | Stochastic basin selection |
| ℏ | σ (noise amplitude) |
| Unitary evolution | Not present (dissipative + stochastic) |
| Superposition | Not postulated |
The Analogy and Its Limits
What works
- Probabilistic distribution over states ↔ Born rule
- Discrete excitation spectra from Hessian curvature ↔ energy levels
- Exponentially suppressed barrier crossing ↔ quantum tunnelling
- Path-integral formulation (Onsager–Machlup) ↔ Feynman path integral
What doesn’t
- Dynamics is dissipative, not unitary — no conservation of probability current in the quantum sense
- Path integral is real-valued, not oscillatory — no interference
- Superposition is not a feature of the framework
- σ ≠ ℏ — the noise amplitude is a framework parameter, not Planck’s constant
Whether these gaps can be bridged — whether a modified closure dynamics could recover unitarity or interference — is an open question.
Stated Limitations
- No equivalence with quantum mechanics is claimed
- σ ≠ ℏ — noise amplitude is not identified with Planck’s constant
- Dynamics is dissipative, not unitary
- No interference effects — path integral is real-valued
- No Born rule derivation — the stationary distribution is an analogue, not a proof
- Finite-dimensional setting — extension to field-theoretic (infinite-dimensional) case not established
- Basin selection is not yet predictive — requires specification of measurement coupling
- The closure functional F is not uniquely derived from first principles
Probability without quantum axioms. Spectra without Hamiltonians. Transitions without tunnelling operators. All from the geometry of the closure landscape.