Latent Memory in PID-Controlled Langevin Dynamics: An Ao Decomposition Perspective

This paper establishes a structural isomorphism between three-term PID control and the potential-flux decomposition of non-equilibrium stochastic dynamics (Ao, 2004). The integral (I) term is interpreted as a latent historical memory accumulator whose antisymmetric coupling in n ≥ 2 dimensions generates the solenoidal probability current J_curl — the hallmark of a non-equilibrium steady state. The proportional (P) and derivative (D) terms contribute exclusively to the dissipative current J_diss. Numerical verification includes 1D (no curl) vs 2D+antisymmetric (curl > 0) comparisons. Declaration: During the preparation of this work, the author used an AI language model to assist with mathematical formalization, code implementation, and manuscript refinement. After using this tool, the author reviewed and edited the content as needed and takes full responsibility for the final content. Keywords : PID control, non-equilibrium steady state, Fokker-Planck equation, probability current, Ao decomposition, latent memory, Langevin dynamics