Thermodynamics analysis of a Reaction-Diffusion matrix multiplication computing unit under the Linear Non-Equilibrium Regime

Implementations of matrix multiplication via diffusion and reactions, thus eliminating the need for electronics, have been proposed as a stepping stone to realize Molecular Nano-Neural Networks (M3N). This work examines whether such “matrix multiplication units” can function spontaneously, i.e., without continuous external en- ergy input. We employ the theory of Local Non-Equilibrium Thermodynamics in the linear regime, modeling the system through coupled reaction–diffusion equations and deriving the resulting entropy production. Numerical simulations on a 2D computa- tional mesh confirm that correct matrix multiplication and strictly increasing entropy can be attained under two key conditions: negligible cross-diffusion among distinct species and sufficiently sharp membranes to prevent back-diffusion. When these con- straints are met, the system concentrations naturally converge to the desired results, suggesting that autonomous chemical computing can be realized if the design parameters align with thermodynamic requirements.