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

Abir Hasan, Nikhil Shukla

TL;DR

The paper demonstrates that barrier-top initialization in smooth, even double-well energy landscapes universally yields Boltzmann-like sampling through short-time gradient-flow dynamics, irrespective of microscopic device details. By framing the dynamics with Morse/theory-based reductions to an A3 quartic form, the authors show a tanh-like relaxation that underpins Boltzmann statistics and enables synchronous, clocked p-bits across platforms. They validate the approach with two physical realizations—SHI-driven oscillators and MTJs—and multiple canonical potentials, revealing device-specific effective temperatures but a shared topological mechanism. This work provides a unifying theoretical foundation for designing scalable, noise-driven probabilistic computing architectures that operate synchronously. The results have practical implications for building robust probabilistic accelerators that leverage barrier-top dynamics in diverse hardware substrates.

Abstract

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.

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

TL;DR

The paper demonstrates that barrier-top initialization in smooth, even double-well energy landscapes universally yields Boltzmann-like sampling through short-time gradient-flow dynamics, irrespective of microscopic device details. By framing the dynamics with Morse/theory-based reductions to an A3 quartic form, the authors show a tanh-like relaxation that underpins Boltzmann statistics and enables synchronous, clocked p-bits across platforms. They validate the approach with two physical realizations—SHI-driven oscillators and MTJs—and multiple canonical potentials, revealing device-specific effective temperatures but a shared topological mechanism. This work provides a unifying theoretical foundation for designing scalable, noise-driven probabilistic computing architectures that operate synchronously. The results have practical implications for building robust probabilistic accelerators that leverage barrier-top dynamics in diverse hardware substrates.

Abstract

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.
Paper Structure (13 sections, 48 equations, 2 figures, 1 table)

This paper contains 13 sections, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Realizing Boltzmann Sampling Through System Dynamics
  3. Illustrative Examples
  4. Harmonic Oscillator under First & Second Harmonic Injection
  5. Magnetic Tunnel Junction (MTJ)
  6. Generalization to Smooth Even Double-Well Potentials
  7. Canonical Double Well Energy System
  8. Boltzmann Sampling using the $A_3$ Canonical Form
  9. Simulation Results
  10. Conclusion
  11. Formulating the Map $s=f(\vartheta)$
  12. Mean First-Passage Time for the Logistic Drift Model
  13. Even polynomials that yield a Double Well Energy Landscape

Figures (2)

  • Figure 1: Illustrative Double Well Energy Systems: Energy as a function of the internal state variable for (a) oscillator under SHI (b) $A_3$ normal form. Measured probability distribution ($p$) as a function of synaptic input for: (c) oscillator under SHI; (d) A3 normal form. The simulations have been performed under the following noise conditions:$K_n=0.1$ (blue); $K_n=0.2$ (cyan); $K_n=0.3$ (orange); $K_n=0.4$ (red). Inset shows the extracted inverse temperature $\beta$.
  • Figure 2: Other Double Well Energy Systems: Measured probability distribution ($p$) as a function of synaptic input for various potential functions: (a) Even order polynomial (b) Double Gaussian Potential (c) Cosh Potential. The exact potentials have been defined in the main text. The simulations have been performed under the following noise conditions:$K_n=0.1$ (blue); $K_n=0.2$ (cyan); $K_n=0.3$ (orange); $K_n=0.4$ (red). Inset shows the extracted inverse temperature $\beta$.