Why and how do systems react in thermally fluctuating environments?

Many chemical reactions, including those of biological importance, take place in thermally fluctuating environments. Compared to isolated systems, there arise markedly different features due to the effects of energy dissipation through friction and stochastic driving by random forces reflecting the fluctuation of the environment. Investigation of how robustly the system reacts under the influence of thermal fluctuation, and elucidating the role of thermal fluctuation in the reaction are significant subjects in the study of chemical reactions. In this article, we start with overviewing the generalized Langevin equation (GLE), which has long been used and continues to be a powerful tool to describe a system surrounded by a thermal environment. It has been also generalized further to treat a nonstationary environment, in which the conventional fluctuation-dissipation theorem no longer holds. Then, within the framework of the Langevin equation we present a method recently developed to extract a new reaction coordinate that is decoupled from all the other coordinates in the region of a rank-one saddle linking the reactant and the product. The reaction coordinate is buried in nonlinear couplings among the original coordinates under the influence of stochastic random force. It was ensured that the sign of this new reaction coordinate (= a nonlinear functional of the original coordinates, velocities, friction, and random force) at any instant is sufficient to determine in which region, the reactant or the product, the system finally arrives. We also discuss how one can extend the method to extract such a coordinate from the GLE framework in stationary and nonstationary environments, where memory effects exist in dynamics of the reaction.