Potential Energy Landscape $V(x)$
Phase Plane $(x, \dot{x})$ & Metrics
Barrier Height $\Delta V$
4.80 J
Escape Energy Threshold
Lyapunov Curvature $V''(x)$
8.42
Restoring Stiffness ($\lambda_L$)
Kramers MFPT $\tau$
4.2 × 10³ s
Mean Escape Time under $D$
Current State Basin
Well A (Left)
Entrenchment Index: 0.86
Enables Waddington habituation: longer residency carves a deeper local potential basin.
In epistemic logic (Gärdenfors & Makinson AGM theory), entrenchment $\alpha \le_E \beta$ orders propositions by how stubbornly they are retained during belief contraction. Retracting a highly entrenched proposition forces all dependent, less-entrenched beliefs to be surrendered first with minimal retraction cost.
Status: Consistent Belief Set (0 Retracted)
1. Dynamical System Potential Entrenchment
A state $x^*$ is an attractor when $V'(x^*) = 0$ and $V''(x^*) > 0$. Under Langevin stochastic dynamics:
A state $x^*$ is an attractor when $V'(x^*) = 0$ and $V''(x^*) > 0$. Under Langevin stochastic dynamics:
m \ddot{x} = -\nabla V(x) - \gamma \dot{x} + \sqrt{2 D \gamma} \cdot \xi(t) + F_{\text{kick}}(t)
The entrenchment depth is quantified by the saddle activation barrier $\Delta V = V(x_{\text{saddle}}) - V(x^*)$.
2. Kramers First-Passage Escape Rate
The escape probability per unit time across barrier $\Delta V$ under thermal noise intensity $D$ follows Kramers formula:
The escape probability per unit time across barrier $\Delta V$ under thermal noise intensity $D$ follows Kramers formula:
r_K = \frac{\sqrt{V''(x^*) \cdot |V''(x_{\text{saddle}})|}}{2 \pi \gamma} \exp\left(-\frac{\Delta V}{D}\right), \quad \tau = \frac{1}{r_K}
Deeply entrenched states have exponential relaxation half-lives $\tau \propto e^{\Delta V / D}$, making escapes exponentially improbable under sub-critical perturbations.
3. Waddington Canalization (Evolving Basin Plasticity)
In biological morphogenesis and cognitive habituation, residency time deepens the basin through feedback:
In biological morphogenesis and cognitive habituation, residency time deepens the basin through feedback:
V(x, t) = V_0(x) - \gamma_c \int_{0}^{t} \mathcal{N}(x - x(\tau), \sigma^2) \, d\tau
The more often a trajectory passes through state $x$, the more resistant $x$ becomes to future perturbations.