Dynamics of a colloidal particle coupled to a Gaussian field: from a confinement-dependent to a non-linear memory

The authors investigate the equilibrium dynamics of a colloidal particle embedded in a fluctuating medium described by a Gaussian field to which the particle is linearly coupled (the coupling amplitude is a nonlinear function of the colloid's position). Integrating out the Gaussian field at the dynamical level leads to an effective dynamics for the particle that features a position dependent memory kernal (and a corresponding multiplicative Gaussian white noise). The authors argue that these ingredients account for the observations of existing experiments.

Overall this work addresses tracer diffusion in a complex medium, which is a notoriously difficult problem. The model is carefully defined and the calculations are presented in a very well-organized fashion. Numerical simulations are also presented with a view to delineating the domain of validity of the perturbation expansion. Actually, my only critical comments are mostly concerned with discussion and presentation issues (in the introduction). If the authors addressed these comments, my opinion would be that the submission could be accepted.

1) In the model, the coupling of the particle to the medium appears to be linear in the field (which of course allows for integrating out of the \phi modes). A linear coupling affects the deterministic field profile, but leaves the field fluctuations unaffected (and there is no corresponding free energy contribution). The peculiar choice of a linear coupling (in \phi) deserves a deeper discussion (this is also discussed in the conclusion of [13], and likely the author common to [13] and to the present submission has an opinion on whether the linear nature of the coupling to the field matters or not). I am sure that the authors can do a little more than the one sentence that addresses this issue in the conclusion section.

2) In their introduction, the authors comment on the X dependence of \Gamma: I understand that in an introduction one has to stage ones' results, but perhaps some of the "surprise" could be toned down ("annoying feature", "undesired", really ?). I don't think that there is any prior expectation that \Gamma should be independent of X. In the standard Mori-Zwanzig projection formalism (repeated in [9]), such a dependence is actually a natural feature. It is only because additional hypotheses (in terms of separation of energy, length and time scales) are fulfilled that eventually Gamma can become independent of X (in short, the mobility in a thermostat can depend on the thermostated degrees of freedom: a textbook example is that of a freeely diffusing particle in the vicinity of a wall, where hydrodynamics makes the mobililty position dependent). I have the superficial impression that these arguments/comments are already clearly stated in [8,9] and it wouldn't hurt (in my opinion) for the clarity of the manuscript to plainly repeat them and then to insist on the core of the manuscript (how to describe the motion when the medium is a Gaussian field). In fact, and in my opinion, the manuscript is interesting in that it defines a model that is simple enough to be attacked by analytical means, but complex enough to the point of displaying such nontrivial features such as a position dependent mobility.

3) Just out of curiosity, isn't an UV cut-off on the q modes required? This would also exclude potential Ito/Strato issues at high q. (the "q infinity" mode is delta correlated, but perhaps with a negligible weight?).

4) Again, just a comment that perhaps invites some rephrasing here or there: the way the existence of an FDT is described almost makes the reader think this is a nontrivial feature (3.3 "Remarkably"...I don't think so). In general, integrating out degrees of freedom only reduces the entropy production. When it is zero to begin with (as considered by the authors) there is no reason that integrating the field degreees of freedom out could lead to a violation of the FDT. It is true, however, that by choosing not to work within a path-integral formalism (Onsager Machlup or else), time-reversal for the X dynamics is somewhat obscured.

5) Sections 4 & 5 are the core of the manuscript, and in themselves they are interesting and self-contained. The medium/particle coupling is used as a perturbation parameter. Surely the linear part could be trimmed down and the "critical" limit could be at least motivated by experiments (if any, would the colloids in lutidine studied by G. Volpe, without light activation part, be a potential candidate). We know that the power laws given by the Gaussian approximation are mean-field values. This could be stated.

6) In the conclusion the authors write "are confirmed by numerical simulations". I like better the wording adopted in the introduction: "the validity of the perturbation expansion is probed by numerical simulations". Here it would be nice if a disucssion of physical orders of magnitude (fom [8,9]) were inserted. This is a conclusion, so it doesn't need to be up to three digits, but a rough idea of where experiments stand wrt to the perturbation expansion of the authors would be welcome.

The authors provide a detailed theoretical study of the dynamics of a colloidal particle in a trap driven through a complex, phase-separating fluid. This is done based on a combination of an overdamped Langevin equation for the colloid and model-B dynamics for the (density or concentration) field of medium. By integrating out the medium they find a state-dependent memory kernel which depends on the trap features, consistent with recent experiments and simulations. Interestingly, the resulting kernel is still related to the (coloured) noise via a fluctuation-dissipation relation. Further, the authors propose a perturbation expansion in terms of the coupling parameter lambda between particle and medium to explicitly calculate the two-time correlation function of the particle position in dependence of the correlation length of the medium, finding (in the long-time limit) a universal algebraic decay which is only depends on the dimension (rather than on details of the trap). This is also reflected by the associated power spectral density which, importantly, can be measured experimentally.
Altogether, the authors present an interesting and careful study of a problem of current interest, that is, the non-Markovian response of a particle in a viscoelastic medium which cannot be described via simple (linear) kernels often used in the literature. They clarify the implicit dependencies on trap stiffness etc. found in earlier experiments and simulations, which is certainly a very relevant result. The here envisioned phase-separating medium can be studied experimentally, and I really like to dependency of the results on the underlying correlation length.

However, I am not convinced that the study contains sufficiently innovative pieces which would justify a publication in SciPost. My main concern is that the model equations (2) and (3) have already been used in the literature (see e.g. [21,22]), so the very idea of coupling the colloid to a density field to describe its dynamics is not new. Also the non-Markovian equation (16) leading to the non-trivial memory kernel has been considered previously (see Eq. (33) in [21]), and a (technically slightly different) perturbation analysis in terms of lambda has been proposed earlier in [21,22], as noted by the authors themselves. So in my impression, the main new approach here is the analysis of the role of the underlying correlation length... (?) In any case, the authors need to better clarify these issues in the manuscript. In the present form, the manuscript seems more suitable for a specialized journal.

Some other points: - Equation (1): I understand that an effective Hamiltonian quadratic in the field is convenient for the later analysis, since the equations of motion are linear in the field. However, why is this ansatz still justified for a fluid close to the critical point - shouldn't one go (at least) to fourth order to ensure stability? Or do these higher-order terms cancel out anyway in the later analysis? - it seems to be important to have a rotationally invariant potential V(x), see, e.g., Eq. (23) below. The authors should comment on the physical implications of this restriction. - Beginning of Sec. 4.1: I do not really understand what the authors mean by "equations .... are made non-linear by their coupling lambda". After all, the equations ARE linear in lambda ....