Onsager Symmetry beyond Linear Response in Chemically Coupled Systems

Onsager reciprocity is commonly regarded as a consequence of near-equilibrium linear response. We show that, for chemically coupled systems satisfying microscopic reversibility and Curie symmetry, the underlying process rate matrix remains exactly symmetric arbitrarily far from equilibrium. Considering two coupled reactions connected by a pair of symmetry-related trajectory families, we derive an exact nonlinear flux relation in which thermodynamic driving enters only through nonlinear thermodynamic variables, while the process rate matrix retains a symmetric structure for arbitrarily large chemical potential differences. The trajectory rate matrix is diagonal in the natural trajectory basis, and a Hadamard transformation maps this exact trajectory description to the experimentally observable process basis, yielding a symmetric process rate matrix with equal diagonal elements and reciprocal off-diagonal elements. In the linear-response limit, the full symmetric constitutive matrix coincides with the differential response matrix, recovering the Onsager reciprocal relations. Away from equilibrium the two are generally distinct: the structural symmetry of the process representation persists, whereas the differential response matrix need not remain symmetric. We further derive exact ratcheting relations for the observable forward and reverse rate constants, showing that thermodynamic affinities and intrinsic kinetic asymmetry play complementary roles in determining directionality and chemical coupling. These results establish a general symmetry framework for chemically coupled systems that extends beyond conventional linear irreversible thermodynamics.