At the Top of the Mountain, the World can Look Boltzmann-Like: Sampling Dynamics of Noisy Double-Well Systems

The success of the transistor as the cornerstone of digital computation motivates analogous efforts to identify an equivalent hardware primitive, the probabilistic bit or p-bit, for the emerging paradigm of probabilistic computing. Here, we uncover a fundamental ubiquity in the stochastic dynamics of double well energy systems when initialized near the barrier top. Using a topological framework grounded in Morse theory and singularity theory, we make use of the result that all smooth, even double well potentials reduce near the saddle point to a canonical quartic normal form. Within this regime, the interplay of noise, synaptic bias, and potential curvature produces a topologically robust short time evolution characterized by a tanh like response. This enables Boltzmann like sampling that is largely independent of the detailed shape of the potential, apart from its effective temperature scaling. Analytical derivations and numerical simulations across multiple representative systems corroborate this behavior. Our work provides a unifying foundation for assessing and engineering a broad class of physical platforms, including oscillators, bistable latches, and magnetic devices, as p-bits operating within a synchronous framework for stochastic sampling and probabilistic computation.

💡 Research Summary

The paper presents a unifying theoretical framework for realizing probabilistic bits (p‑bits) using the stochastic dynamics of double‑well energy systems when they are initialized at the top of the barrier. By invoking Morse theory and catastrophe (singularity) theory, the authors show that any smooth, even double‑well potential can be locally transformed near its non‑degenerate saddle point into the canonical quartic normal form (the A₃ “pitch‑fork” singularity): U(ϑ)=α ϑ⁴/4 − r ϑ²/2. Adding a linear bias h (which represents the synaptic input) yields U(ϑ)=α ϑ⁴/4 − r ϑ²/2 − h ϑ, and the corresponding gradient‑flow dynamics become ẋ=−U′(ϑ)=h+α r ϑ−α ϑ³.

Close to the barrier (ϑ≈0) the cubic term is negligible, so the dynamics reduce to ẋ≈h+α r ϑ. By defining an observable s through a mapping f(ϑ)=tanh