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arxiv: 2503.20457 · v6 · submitted 2025-03-26 · 🧮 math-ph · cond-mat.stat-mech· math.MP

On the generalized Langevin equation and the Mori projection operator technique

Christoph Widder , Johannes Zimmer , Tanja Schilling This is my paper

Pith reviewed 2026-05-22 22:51 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechmath.MP
keywords Mori projectiongeneralized Langevin equationorthogonal dynamicsVolterra equationsstrongly continuous semigroupsprojection operator formalismfluctuation dissipation theorem
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The pith

The orthogonal dynamics for Mori projections is defined directly via linear Volterra equations and forms a strongly continuous semigroup generated by QLQ.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper resolves the well-posedness of the orthogonal dynamics in the Nakajima-Mori-Zwanzig formalism for rank-one Mori projections by defining its orbit maps as solutions to linear Volterra equations. From this definition the authors prove that the orthogonal dynamics is a strongly continuous semigroup whose generator is the operator QLQ, and they recover the generalized Langevin equation together with the second fluctuation-dissipation theorem. One route uses the bounded-perturbation theorem; an alternative route replaces the variation-of-constants formula by a limiting process that avoids any differentiability assumption on the fluctuating forces. The entire development applies to any autonomous dynamical system whose time evolution is given by a strongly continuous semigroup on a Hilbert space.

Core claim

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 QL Q = QLQ, where L is the generator of the time evolution operator, and P=1-Q is the Mori projection operator. The results apply to general autonomous dynamical systems whose time evolution is given by a strongly continuous semigroup.

What carries the argument

The definition of orthogonal-dynamics orbit maps as solutions of linear Volterra equations, from which the semigroup property and the generator QLQ are derived without invoking differentiability of fluctuating forces.

Load-bearing premise

The time evolution of the system must be given by a strongly continuous semigroup on a Hilbert space.

What would settle it

A concrete dynamical system whose evolution is a strongly continuous semigroup on a Hilbert space, yet whose memory kernel obtained from the Volterra-defined orthogonal dynamics differs from the kernel obtained via the bounded-perturbation route, would falsify the claimed equivalence.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript resolves the well-posedness of the orthogonal dynamics for rank-one Mori projections in the Nakajima-Mori-Zwanzig formalism. For systems whose time evolution is a strongly continuous semigroup on a Hilbert space, the orthogonal dynamics is constructed either via the bounded perturbation theorem (yielding the GLE and 2FDT by variation of constants) or, alternatively, by directly defining its orbit maps as solutions of linear Volterra equations; all semigroup properties, strong continuity, and the generator identification with the closure of QLQ are then proved from this definition. The results require no extra regularity on the fluctuating forces and apply to general autonomous C0-semigroup dynamics.

Significance. If the derivations hold, the paper supplies a self-contained, rigorous foundation for the Mori projection technique that closes an acknowledged gap in the literature. The Volterra-equation route is especially useful because it derives the semigroup property and generator directly from the GLE/2FDT uniqueness without invoking the bounded perturbation theorem or differentiability assumptions, thereby extending the applicability of the formalism to a broad class of autonomous dynamical systems.

minor comments (3)
  1. [Introduction / §2] The abstract states that the results apply to 'general autonomous dynamical systems whose time evolution is given by a strongly continuous semigroup'; the precise statement of this standing hypothesis (including the Hilbert-space setting and the definition of the Mori projection P) should appear explicitly in the introduction or §2.
  2. [Main text (generator identification)] Notation for the closure of QLQ is introduced as 'overline{QL}Q = QLQ'; a short remark clarifying whether the domain of the generator is taken to be the domain of L or a suitable core would improve readability.
  3. [Discussion / concluding section] The paper mentions an alternative limiting-process proof that avoids differentiability; a brief comparison table or paragraph contrasting the two routes (bounded perturbation vs. Volterra) would help readers assess the scope of each method.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of our results on the well-posedness of the orthogonal dynamics and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained within C0-semigroup theory

full rationale

The paper defines the orthogonal dynamics orbit maps directly as solutions to linear Volterra equations (or via bounded perturbation + variation of constants) and derives semigroup properties, generator identification, GLE, and 2FDT from that definition plus the standing hypothesis that the underlying dynamics is a strongly continuous semigroup on a Hilbert space. All steps remain inside standard functional-analysis results for C0-semigroups; no parameter fitting, self-referential definitions, or load-bearing self-citations that reduce the central claim to its own inputs are present. The construction is independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of semigroup theory rather than new postulates; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The time evolution is given by a strongly continuous semigroup on a Hilbert space
    Explicitly stated as the setting to which the results apply (final sentence of abstract)
  • domain assumption The projection operator is the rank-one Mori projection
    The paper restricts attention to rank-one projections (Mori's projection) throughout

pith-pipeline@v0.9.0 · 5871 in / 1529 out tokens · 122001 ms · 2026-05-22T22:51:06.315184+00:00 · methodology

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