Mori-Zwanzig projection formalism: from linear to nonlinear

TL;DR

This paper explores the Mori-Zwanzig projection formalism, demonstrating its relation to the linear and nonlinear regimes, and derives dynamic equations for collective coordinates in Hamiltonian systems.

Contribution

It shows that Mori's and Zwanzig's procedures are limiting cases of each other and derives dynamic equations for collective coordinates.

Findings

Mori's and Zwanzig's projections are mutual limits depending on Hilbert space size

Derived dynamic equations for collective coordinates in Hamiltonian systems

Clarified the relationship between linear and nonlinear projection formalisms

Abstract

The Mori-Zwanzig projection formalism is widely used in studying systems with many degrees of freedom. We used a system-bath Hamiltonian system to show that the Mori's and Zwanzig's projection procedures are mutual limiting cases of each other depending on the size of the projected Hilbert space. We also derived the dynamic equations of collective coordinates of a Hamiltonian system.