Application of the projection operator formalism to non-Hamiltonian dynamics

TL;DR

This paper extends the projection operator formalism to non-Hamiltonian systems by mapping them to Hamiltonian systems, enabling the derivation of generalized Langevin equations with fluctuation-dissipation relations, useful for model reduction.

Contribution

It develops a generalized projection formalism for non-Hamiltonian systems by leveraging a recent mapping to Hamiltonian systems, establishing fluctuation-dissipation relations.

Findings

Validated the formalism with an analytically solvable example.

Numerical tests on a chemical network confirmed the approach.

Potential applications to biological networks and system robustness.

Abstract

Reconstruction of equations of motion from incomplete or noisy data and dimension reduction are two fundamental problems in the study of dynamical systems with many degrees of freedom. For the latter extensive efforts have been made but with limited success to generalize the Zwanzig-Mori projection formalism, originally developed for Hamiltonian systems close to thermodynamic equilibrium, to general non-Hamiltonian systems lacking detailed-balance. One difficulty introduced by such systems is the lack of an invariant measure, needed to define a statistical distribution. Based on a recent discovery that a non-Hamiltonian system defined by a set of stochastic differential equations can be mapped to a Hamiltonian system, we develop such general projection formalism. In the resulting generalized Langevin equations, a set of generalized fluctuation-dissipation relations connect the memory kernel and the random noise terms, analogous to Hamiltonian systems obeying detailed balance. Lacking of these relations restricts previous application of the generalized Langevin formalism. Result of this work may serve as the theoretical basis for further technical developments on model reconstruction with reduced degrees of freedom. We first use an analytically solvable example to illustrate the formalism and the fluctuation-dissipation relation. Our numerical test on a chemical network with end-product inhibition further demonstrates the validity of the formalism. We suggest that the formalism can find wide applications in scientific modeling. Specifically, we discuss potential applications to biological networks. In particular, the method provides a suitable framework for gaining insights into network properties such as robustness and parameter transferability.