Symplectic Constraints in Classical Reaction Dynamics: From Gromov's Camel to Reaction Rates

Abstract

We investigate whether ideas from symplectic topology, in particular Gromov's non-squeezing theorem and symplectic capacity, can provide useful geometric insight into classical reaction dynamics near an index-1 saddle. Using Poincar\'e-Birkhoff normal form theory, we describe the phase-space structures that organize transport through the transition-state region, including dividing surfaces, normally hyperbolic invariant manifolds (NHIMs), and the associated bath-action geometry. For quadratic saddle-center and saddle-center-center models, the normal-form geometry identifies natural bath-action area scales associated with the reactive bottleneck. For anharmonic systems (Eckart-Morse and Eckart-Morse-Morse), we formulate corresponding candidate symplectic width scales -- based on transverse bath actions -- using high-order normal forms for bounded local neighborhoods associated with the reaction bottleneck near the saddle. We then present two numerical illustrations: the backward propagation of a locally coupled phase-space ball to examine linear non-squeezing behavior, and a bath-localized ensemble calculation in an anharmonic normal-form model. These computations are consistent with the idea that heavily biasing the initial phase-space distribution of an ensemble toward the high-action boundaries of the bath modes can induce a severe finite-time dynamical delay, influencing reactivity in ways not captured by total phase-space volume or flux alone. The results suggest a new geometric perspective on mode selectivity and reaction bottlenecks, while highlighting open mathematical questions concerning the precise relation between these candidate width scales and genuine symplectic capacities of suitably defined reactive neighborhoods.