Application of the projection operator formalism to dissipative dynamics

Dimension reduction is a fundamental problem in the study of dynamical systems with many degrees of freedom. Efforts have been made to generalize the Mori-Zwanzig projection formalism, originally developed for Hamiltonian systems, to general non-Hamiltonian and dissipative systems. The difficulty lies in defining an invariant measure. Based on a recent discovery that a system defined by stochastic differential equations can be mapped to a Hamiltonian system, we developed a projection formalism for general dissipative systems. Our numerical test on a chemical network with end-product inhibtion demonstrates the validity of the formalism. We suggest that the formalism can find usage in various branches of science. Specifically, we discuss potential applications in studying biological networks, and its implications in network properties such as robustness, parameter transferability. It is common to study dynamics of a system with many degrees of freedom in almost every scientific field. In general it is impractical, and often unnecessary, to track all the dynamical information of the whole system. A common practice is projecting the dynamics of the whole system into that of a smaller subsystem through information contraction. The procedure leads to the celebrated Langevin and generalized Langevin dynamics. The Mori-Zwanzig formalism is a formal procedure of projection, especially for Hamiltonian systems [1, 2, 3, 4]. Inspired by its great success in irreversible statistical mechanics, Chorin and coworkers, have developed a version of the Mori-Zwanzig formalism for higher-order optimal prediction methods for general dynamical systems [5, 6]. The difficulty lies in choosing an invariant measure in defining an inner product (see below for details). The choice is straightforward for a Hamiltonian system, but not clear for a general system. Recently one of us has proved that one can map a system described by a set of stochastic differential equations ẋi = dxi/dt = Gi(x) + M