Qualitative dynamics of generalized Langevin equations and the theory of chemical reaction rates

<jats:p>In this paper, we present an analysis of condensed phase chemical reactions from the perspective of qualitative dynamical systems theory. Our approach is based on a phenomenological phase space representation of the generalized Langevin equation (GLE). In general, the GLE with memory requires an infinite-dimensional phase space for its description. The phenomenological phase space is constructed by augmenting the physical phase plane (q,p) with additional variables defined as the convolution of the system momentum with the memory kernel and its time derivatives. The qualitative dynamics in this representation are then characterized in terms of the eigenvalues and eigenvectors of the linear system near the barrier top. The phase space decomposes into a single unstable direction and a complementary stable subspace. The rate of exponential growth along the unstable eigenvector is directly related to the rate of chemical reaction, and our linear analysis reproduces the Grote–Hynes expression for the reaction rate [R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980)]. In the presence of noise, the stable subspace can be identified with the stochastic separatrix, a manifold of initial conditions with a reaction probability of 0.5. Other dynamical processes, such as solvent caging, can also be given a simple geometric interpretation in terms of the qualitative dynamical analysis.</jats:p>