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 energy 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 computational 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 constraints 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.