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Application of the projection operator formalism to non-hamiltonian dynamics.

Authors
  • Xing, Jianhua
  • Kim, K S
Type
Published Article
Journal
The Journal of Chemical Physics
Publisher
American Institute of Physics
Publication Date
Jan 28, 2011
Volume
134
Issue
4
Pages
44132–44132
Identifiers
DOI: 10.1063/1.3530071
PMID: 21280712
Source
Medline
License
Unknown

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.

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