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.