Coupled Kinetic Processes and the Maintenance of Biological Function

This second paper in the Kinetic Organization and Biological Function series develops a mathematical-physics account of how interacting kinetic processes sustain selected functions while continuously changing their shared material and energetic conditions. Extending the momentum-bearing transporter investigated in Paper I, the work constructs a coupled stochastic system comprising a four-phase ion transporter, a passive ionic return pathway, finite ionic and chemical reservoirs, reversible chemical regeneration, and an inventory-derived membrane capacitor. The complete diffusion–jump generator incorporates exact conservation laws, finite electrochemical work, reactant-dependent transition feasibility, and consistent free-energy, caloric, and entropy accounting. For the registered system at nonzero mechanical coupling, kinetic Lyapunov estimates, positive transition kernels, guarded graph accessibility, and Harris ergodicity establish unique invariant probability laws for the full coupled dynamics. The closed chemical regime admits an explicit physically reversible Gibbs equilibrium. Under external chemical regeneration, the maintained source supports strictly positive stationary pump–leak circulation, with equal and opposing mechanism-resolved ionic currents despite zero mean net charge accumulation. Physical path-space reversal establishes an entropy-production rate of (6J) and an external caloric-input rate of (15J/2), where (J>0) is the stationary circulation. Deterministic source-corrector calculations bound this current within ([0.033765,,0.099287]) ions per source-time unit. The analysis further establishes quantitative properties of functional persistence, recovery, and fluctuation. For the selected charge band (1≤ Q≤3), chemical maintenance produces higher stationary occupancy, lower stationary exit flux, and greater continuous-survival probabilities at two specified horizons than closed equilibrium. An exact finite Gibbs–Bessel calculation determines the closed-equilibrium exit flux as approximately (0.167171094040872), with corresponding Palm mean inside- and outside-band excursion lengths. Uniform passive-assisted recovery and almost-sure eventual exit establish complementary properties of the same stochastic organization. Graph-cycle observability results also demonstrate how distinct physical histories can share an identical electrical trace while carrying different chemical energetic costs. Motivated by the historical relationship between 氣機 (qìjī, organized movement) and 氣化 (qìhuà, material and physiological transformation), the investigation gives a precise physical realization of reciprocal conditioning between movement and transformation. Transport changes shared resources and electrical conditions, which in turn govern subsequent kinetic activity. The resulting mathematics establishes how continuing material circulation, energetic transformation, and statistically persistent organization can coexist within a single coupled physical system. The functional observables and parameters are defined for this synthetic model, providing a foundation for subsequent investigation of collective kinetic organization and experimentally identified biological transport systems. The accompanying materials include the mathematical derivations, formal source correspondences, numerical methods, and a reproducible finite-sum calculation of the closed-equilibrium stationary exit flux.