Cumulant Mechanics: An Explicit Treatment for Fluctuation on Dynamics

An extended dynamics method for classical and cumulant variables is formulated to take fluctuation effects into account directly. In particular, we have derived the coupled equations of motion (EOM) for the position, momentum, and second-order cumulants of the product of the momentum and position fluctuation operators for both quantum and classical regimes. The second-order quantal and classical cumulant dynamics have almost the same structure with proper initial conditions and more terms to describe friction for latter equations. We demonstrated that the present methods give the exact answer for the harmonic oscillator and are applied to analyze the dynamical quantum isotope effects on a model proton transfer reaction in a Guannine-Cytosine base pair and to obtain a thermal equilibrium state of 7-par-ticles classical Morse cluster.