Semigroup properties of orthogonal dynamics for the non-linear GLE with potential of mean force

Ascertain whether the semigroup properties of the orthogonal dynamics—such as strong continuity and unitary behavior established for the linear GLE under the Mori projection—also hold for the widely used non-linear generalized Langevin equation that includes a potential of mean force, typically formulated via the Zwanzig projection operator.

Background

Within the paper, the orthogonal dynamics is shown to be a strongly continuous semigroup (or unitary group when L is skew-adjoint) for the linear GLE with the finite-rank Mori projection. These properties underpin the well-posedness and interpretation of fluctuating forces.

The non-linear GLE with a potential of mean force is commonly expressed using the infinite-rank Zwanzig projection operator, where rigorous semigroup results are delicate. The authors explicitly state uncertainty about whether their semigroup results extend to this non-linear setting.