Mathematical models for surface corrosion, oxide formation, Ornstein-Uhlenbeck mean reversion, and dispersive failure rates, derived from first principles in Mathematical Geoenergy (GEM), Chapter 19 (Dissipative Energy: Resilience, Durability and Reliability).
Chapter 19 treats surface degradation and component failure as diffusion- dominated processes subject to MaxEnt disorder. The same dispersive diffusion kernel that appears in battery transport (Chapter 18) and porous-media flow (Chapter 20) drives oxide and rust growth here. Two extensions are developed:
Corrosion power law — Fickian t^(1/2) growth combined with a fresh- surface exposure mechanism yields the empirically observed t^0.7 scaling for marine and industrial corrosion.
Ornstein-Uhlenbeck (O-U) correction — Pure diffusion grows without bound; the O-U mean-reverting stochastic process introduces a restoring force that saturates growth at an asymptotic thickness W_∞.
Bathtub curve — Applying a MaxEnt mix of infant-mortality failure rates gives an exponentially decaying early hazard; adding a linear wear-out term produces the asymmetric V-shaped bathtub hazard familiar from reliability engineering.
Silicon dioxide growth is the canonical diffusion-controlled surface reaction. The Deal-Grove model treats oxide growth as Fickian (parabolic regime: W ∝ √t). Applying MaxEnt to the diffusion coefficient D — only the mean D₀ is known — replaces the Gaussian Green’s function with the dispersive Laplace kernel (Eq. 19-2). The resulting oxide layer width (Eq. 19-3) retains the √t scaling but with heavier tails.
Rust growth mirrors oxide growth but with an additional mechanism: as the oxide layer thickens, peeling exposes fresh metal surfaces at a rate that itself follows a power law. Combining diffusion-limited growth (∝ t^0.5) with linear fresh-surface exposure (∝ t^β, β ≈ 0.2) gives the combined exponent α = 0.5 + β ≈ 0.7 (Eq. 19-5), matching marine and industrial corrosion measurements.
Pure Fickian diffusion predicts unbounded growth, but real oxide layers reach a limiting thickness because the growing layer itself impedes further oxidant transport. The O-U process captures this through a restoring force −α(X − X̄) that limits the diffusion excursion. The stationary variance σ²/(2α) and asymptotic growth profile W_∞(1 − exp(−t/τ_c)) (Eq. 19-8) fit both urban/rural corrosion data and oxide films in mild environments.
Component failure combines three regimes visible in the bathtub curve: early infant-mortality (defects from manufacture), constant random failure, and wear-out. The MaxEnt argument applied to a population with unknown individual failure rates gives an exponentially decaying early hazard; adding a linear cumulative-damage term yields the asymmetric bathtub hazard h(t) = h₀·exp(−δt) + μ_f·t (Eq. 19-10).
Eq. 19-1 — MaxEnt distribution for diffusion coefficient
\[p(D) = \frac{1}{D_0}\cdot e^{-D/D_0}\]Eq. 19-2 — Dispersive diffusion kernel (MaxEnt average over D)
\[C(t, x) = \frac{1}{2\sqrt{D_0 t}}\cdot e^{-x/\sqrt{D_0 t}}\]Obtained via the Laplace-Bessel identity (Gradshteyn & Ryzhik §3.471):
\[\int_0^\infty \frac{e^{-Ax - B/x}}{\sqrt{x}}\\,dx = \sqrt{\frac{\pi}{A}}\cdot e^{-2\sqrt{AB}}\]Eq. 19-3 — Dispersive oxide layer width
\[W(t) = \sqrt{D_0\cdot t}\]Mean penetration of the dispersive kernel; equals the scale parameter of the one-sided exponential distribution Exp(1/√(D₀t)).
Eq. 19-4 — Fickian (Deal-Grove) reference width
\[W_\mathrm{Fickian}(t) = \sqrt{\frac{4 D_0 t}{\pi}}\]Half-normal mean of the Gaussian Green’s function. Both Eqs. 19-3 and 19-4 scale as √t; the dispersive result has a √(π/4) ≈ 0.89 smaller prefactor, reflecting the heavier tails of the Laplace kernel.
Eq. 19-5 — Corrosion power-law growth
\[W_\mathrm{corr}(t) = K\cdot t^\alpha, \qquad \alpha \approx 0.7\]The exponent α combines Fickian oxide growth (β_growth = 0.5) with a fresh-surface exposure mechanism (β_exposure ≈ 0.2):
\[\alpha = \beta_\text{growth} + \beta_\text{exposure} \approx 0.5 + 0.2 = 0.7\]The corresponding rate dW/dt = Kαt^(α−1) decreases over time for α < 1 (self-limiting growth). Marine and industrial corrosion data typically give α ∈ [0.6, 0.8]; mild-environment data diverge at longer times (see O-U correction below).
Eq. 19-6 — O-U stochastic differential equation
\[dX = -\alpha_\mathrm{ou}\cdot(X - \bar{X})\\,dt + \sigma_\mathrm{ou}\\,dW_t\]Eq. 19-7 — Stationary (asymptotic) variance
\[\mathrm{Var}_\infty = \frac{\sigma_\mathrm{ou}^2}{2\cdot\alpha_\mathrm{ou}}\]The time-dependent variance relaxes to this limit:
\[\mathrm{Var}(t) = \frac{\sigma_\mathrm{ou}^2}{2\cdot\alpha_\mathrm{ou}} \Bigl(1 - e^{-2\alpha_\mathrm{ou}\cdot t}\Bigr)\]Eq. 19-8 — Bounded (plateau) corrosion growth
\[W_\mathrm{OU}(t) = W_\infty\cdot\bigl(1 - e^{-t/\tau_c}\bigr)\]where $W_\infty = \sigma_\mathrm{ou}/\sqrt{2\alpha_\mathrm{ou}}$ is the asymptotic oxide thickness and $\tau_c = 1/(2\alpha_\mathrm{ou})$ is the O-U time constant.
Eq. 19-9 — Exponential (MaxEnt) failure distribution
\[f(t) = \lambda\cdot e^{-\lambda t}, \qquad h(t) = \lambda \text{ (constant)}\]MaxEnt with only mean lifetime 1/λ known uniquely selects this distribution (constant hazard rate).
Eq. 19-10 — Asymmetric bathtub hazard function
\[h(t) = h_0\cdot e^{-\delta t} + \mu_f\cdot t\]The corresponding cumulative hazard and survival function are:
\[H(t) = \frac{h_0}{\delta}\bigl(1 - e^{-\delta t}\bigr) + \frac{\mu_f}{2}\cdot t^2\] \[S(t) = \exp\bigl(-H(t)\bigr)\]The bathtub minimum occurs at:
\[t^* = \frac{1}{\delta}\ln\left(\frac{h_0\cdot\delta}{\mu_f}\right)\]| File | Purpose |
|---|---|
corrosion_symbolic.py |
Symbolic derivation of all Chapter 19 equations using SymPy |
corrosion_numerical.py |
Numerical implementation, validation, and composite figure |
corrosion_model_output.png |
Output figure (6 panels) generated by corrosion_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python corrosion_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python corrosion_numerical.py
Same dispersive kernel, new domain: the MaxEnt averaging of the Gaussian Green’s function over an exponential prior for D produces the Laplace dispersive kernel (Eq. 19-2) regardless of physical context — batteries (Ch. 18), corrosion (Ch. 19), porous media (Ch. 20). The unifying principle is disorder in the transport coefficient.
Power law ≈ 0.7 is not empirical magic: the observed t^0.7 corrosion exponent arises naturally from the sum of two MaxEnt exponents: t^0.5 for diffusion-limited growth and t^0.2 for fresh-surface exposure. No fitting is required beyond the identification of these two mechanisms.
O-U reversion explains the long-time plateau: the Ornstein-Uhlenbeck restoring force models the fact that a thickening oxide layer impedes further oxidant transport. The asymptotic thickness W_∞ = σ/√(2α) is set entirely by the noise-to-drag ratio of the O-U process.
Bathtub is asymmetric by construction: the infant-mortality side is exponential (fast decay, proportional to h₀·δ) while the wear-out side grows only linearly with μ_f. This asymmetry — a deeper left well than right — matches empirical failure data for mechanical and electronic components.
MaxEnt unifies all three failure regimes: the exponential failure distribution (Eq. 19-9) emerges from MaxEnt with a mean-lifetime constraint; mixing these over a population with uncertain individual rates gives the decaying infant-mortality term; adding diffusion-driven wear gives the full bathtub.
Minimal parameterisation: Eq. 19-5 needs two parameters (K, α); Eq. 19-8 needs two (W_∞, τ_c); Eq. 19-10 needs three (h₀, δ, μ_f). MaxEnt suppresses the need for higher-order parameters by selecting the maximum-entropy distribution consistent with the known constraints.