A Physical Analogy between Molecular Ordering and SAT-to-Ising Annealing
ShivKishan Dubey, Rohit Sharma
TL;DR
The work establishes a direct thermodynamic analogy between cooling-induced molecular order and SAT solving by mapping CNF clauses to a pairwise Ising Hamiltonian via clause gadgetization, then applying simulated annealing. Physics-inspired observables, including energy $E_{logic}$, magnetization $|M|$, and backbone size $b$, reveal a strong energy–order correspondence: as energy decreases, global coherence rises and a fixed backbone emerges, indicating a unified minimization principle for computation and thermodynamics. Using the UF20-91 benchmark near the satisfiability transition, the study reports near-zero final energies and near-unit magnetization ($|M_f|\approx 0.95$) across 10 instances, supporting the notion of energetic crystallization of satisfiable formulas. The results motivate extending the framework to larger and structured SAT problems and to quantum/hybrid annealing, with potential analytical models linking entropy, constraint density, and phase behavior to computational hardness. $E_{logic}(x)=\sum_{j=1}^{m} E_j(x)$ with ground state $E_{min}=0$, $s_i\in\{-1,+1\}$, and $H(s)$ crafted so that satisfiability corresponds to $H_{min}$, illustrating a thermodynamic lens on logical coherence.
Abstract
As temperature drops, molecular systems may undergo spontaneous ordering, moving from random behavior to orderly structure. This research demonstrates a direct analogy between this type of thermodynamic ordering in molecular systems and the development of coherent logic in computationally complex problem sets. We have proposed a mapping of Boolean SAT problem instances to pairwise Ising Hamiltonian models. Using simulated annealing, we then applied phenomenal cooling to the system through thermal evolution from high entropy random assignment to lower entropy, ordered assignments (the energy minima) using molecular cooling analogs. This indicated that there was a rapid "first-order" or "logical crystallization" of satisfiable logical configurations. The degree of backbone rigidity did not strongly correlate with the level of physical ordering observed in the system; thus, it appears that there is primarily a local alignment of constraint satisfaction occurring in the system. Thus, we have provided empirical evidence that satisfiable logical configurations are analogous to the low energy crystalline states observed in molecular systems and provide evidence for a unified thermodynamic view of computational coherence and complexity.