On the geometry of transient relaxation

Abstract

Coupled chemical reactions are often described by (stiff) systems of ordinary differential equations (ODEs) with widely separated relaxation times. In the phase space Γ of species concentration variables, relaxation can be represented as a cascade through a nested hierarchy of smooth hypersurfaces (inertial manifolds) {Σ}: If d is the number of independent concentration variables, then Γ≡Γ_d⊇Σ_d−1⊇Σ_d−2⋅⋅⋅. The last three sets in this hierarchy have special chemical importance: Σ₀ is the stagnation point of the ODEs, i.e., chemical equilibrium; M(≡Σ₁) is the linelike slow manifold describing the dynamical steady state in closed systems; Σ(≡Σ₂) is the two‐dimensional surface containing the slowest transient flow that reaches M. Thus M and Σ are the structures underlying most steady‐state and transient kinetics experiments. The ODEs describe the velocity field in Γ, which may be used to define functional equations for M, Σ, and other members in the hierarchy {Σ}. These functional equations can be solved to give explicit formulas for M, Σ, etc. In a model three‐step mechanism, M is described by a vector functional equation involving ordinary derivatives, whereas Σ is described by a scalar functional equation involving partial derivatives. We show how Σ may be found by iterative solution of this functional equation if decay is monotonic, and comment on the complications introduced by oscillatory transients. The functional equation approach may be generally applied in higher dimensional systems.