On the generalized Langevin equation and the Mori projection operator technique
Christoph Widder, Johannes Zimmer, Tanja Schilling
TL;DR
The paper provides a rigorous foundation for the Mori–Zwanzig projection technique in Hilbert spaces, focusing on the generalized Langevin equation (GLE) and the second fluctuation-dissipation theorem (2FDT) for rank-one projections. It offers two independent approaches: a semigroup-based method using the bounded perturbation theorem and a Volterra-equation method that derives the GLE/2FDT without differentiability assumptions on the fluctuating forces; both approaches identify the orthogonal dynamics with a strongly continuous semigroup generated by $\overline{\mathcal{QL}}\mathcal{Q}$ (or $\mathcal{QLQ}$) and recover the memory kernel $K(t)$ and fluctuating forces $\eta_t$ from the dynamics. The results extend to classical statistical mechanics via the Koopman/Liouville framework and to non-autonomous systems with time-dependent projections, clarifying when the Dyson identity holds and highlighting limitations for non-autonomous and infinite-rank projections. Overall, the work strengthens the mathematical underpinnings of projection-based reductions and clarifies the conditions under which GLEs and 2FDT remain valid.
Abstract
In statistical physics, the Nakajima-Mori-Zwanzig projection operator formalism is used to derive an integro-differential equation for observables in a Hilbert space, the generalized Langevin equation (GLE). This technique relies on the splitting of the dynamics into a projected and an orthogonal part. However, the well-posedness of the abstract Cauchy problem for the orthogonal dynamics remains an open problem. Moreover, it is rarely discussed under which assumptions the Dyson identity, which is used to derive the GLE, holds. In this article, we address this issue for rank-one projections (Mori's projection). For the Mori projection, the orthogonal dynamics is obtained from the bounded perturbation theorem. The variation of constants formula for strongly continuous semigroups then yields the GLE and the second fluctuation dissipation theorem (2FDT). We show that the variation of constants can be replaced by a limiting process in order to give a general proof of the GLE and 2FDT that does not require the differentiability of the fluctuating forces. In addition, we offer an alternative approach that does not require the bounded perturbation theorem. Our starting point is the observation that the GLE and 2FDT uniquely determine the fluctuating forces as well as the memory kernel. Furthermore, the orbit maps for the orthogonal dynamics can be directly defined via solutions of linear Volterra equations. All desired properties of the orthogonal dynamics are then proven directly from this definition. In particular, the orthogonal dynamics is a strongly continuous semigroup generated by $\overline{\mathcal{QL}}\mathcal{Q}=\mathcal{QLQ}$, where $\mathcal{L}$ is the generator of the time evolution operator, and $\mathcal{P}=1-\mathcal{Q}$ is the Mori projection operator. Our results apply to general autonomous dynamical systems whose time evolution is given by a strongly continuous semigroup.