Generalised Langevin Dynamics: Significance and Limitations of the Projection Operator Formalism

TL;DR

This paper examines the mathematical foundations of the Mori-Zwanzig projection operator formalism, highlighting its limitations, especially for unbounded perturbations, and clarifies the properties and applications of the generalized Langevin equation.

Contribution

It derives the formalism using semigroup theory, discusses the limitations for Zwanzig's projection, and clarifies the conditions under which the generalized Langevin equation is well-posed.

Findings

Mori's projection leads to well-posed Volterra equations for the Langevin dynamics.

Zwanzig's projection involves unbounded operators with unresolved existence issues.

Memory terms can vanish for projections onto 'fast' and 'slow' variable subspaces.

Abstract

We discuss some mathematical aspects of the Mori-Zwanzig projection operator formalism. The core of the Mori-Zwanzig formalism is the generalised Langevin equation, which is typically derived from the Dyson-Duhamel identity. We derive the projection operator formalism for Mori's projection by means of semigroup theory, and we illustrate where rigorous methods fail for the case of Zwanzig's projection. For bounded perturbations of the time-evolution operator (e.g. for Mori's projection), the Dyson-Duhamel identity coincides with the variation of constants formula. For unbounded perturbations (e.g. for Zwanzigs's projection), the Dyson-Duhamel identity should be considered an equation for the orthogonal dynamics, for which the existence of unique solutions has yet to be established. Then we recall that all properties of Mori's generalised Langevin equation follow directly from the well-posedness of Volterra equations, irrespective of the projection operator formalism. Further, we discuss the use of Mori's generalised Langevin equation as a coarse-grained model. Finally, we illustrate that the memory term is a coupling term that is not necessarily related to memory. To this end, we introduce projections onto subspaces of 'fast' and 'slow' variables that are associated with the spectral decomposition of skew-adjoint operators. For these projections, the memory term vanishes.