We propose a route for parameterizing isotropic (generalized) Langevin [(G)LE] thermostats with the aim to correct the dynamics of coarse-grained (CG) models with pairwise conservative interactions. The approach is based on the Mori–Zwanzig formalism and derives the memory kernels from Q-projected time correlation functions. Bottom-up informed (GLE and LE) thermostats for a CG star-polymer melt are investigated, and it is demonstrated that the inclusion of memory in the CG simulation leads to predictions of polymer diffusion in quantitative agreement with fine-grained simulations. Interestingly, memory effects are observed in the diffusive regime. We demonstrate that previously neglected cross-correlations between the “irrelevant” and the CG degree of freedom are important and lie at the origin of shortcomings in previous CG simulations.

To reduce the numerical complexity in computer simulations of soft matter, coarse-grained (CG) models are commonly used. The systematic derivation of these models frequently targets at structural and thermodynamical properties based on procedures that map a high-dimensional space of atomistic degrees of freedom (DoFs) on a lower dimensional space of CG DoFs. The interaction sites (“beads”) of the resulting CG models usually correspond to the centers of mass of several atoms grouped together to represent, e.g., a small molecule, functional group, monomer, thermal blob, or entire macromolecule.1–8 Contrary to static properties, the dynamic properties of these CG models are usually poorly represented in standard molecular dynamics simulations.9–12 Despite progress made in recent years, challenges remain, in particular, in representing the dynamic properties of material-specific CG models on all relevant time scales.

The Hamiltonian dynamics of a fine-grained (FG) system can be represented with a CG model when non-Markovian interactions, represented by friction and noise terms, are introduced in the equation of motion (EoM). The Mori–Zwanzig theory13–15 provides an exact theoretical framework to link the FG and CG dynamics and has been explored in recent years with varying success.16–23 The difficulties encountered are usually caused by incomplete time scale separation of the relevant (CG) and the removed DoFs. Therefore, configuration-dependent frictions, e.g., pairwise, distance-dependent, and time-independent friction coefficients in dissipative particle dynamics (DPD) models, cannot readily be computed. A conceptually simpler approach involves the use of configuration-independent friction coefficients or memory kernels in a Langevin (LE) or a generalized Langevin (GLE) thermostat coupled to the CG DoFs.24,25

Methods for constructing (G)LE thermostats, which do not rely on an a posteriori optimization scheme,25–27 have in common that FG trajectories are analyzed by separating forces into contributions from an assumed conservative force field, governing structure and thermodynamical properties, and fluctuating contributions that are assigned to frictional and random interactions.16–23 In early studies, Lei et al.24 using this approach attempted to correct the self-diffusivity of a liquid composed of CG fluid particles representing clusters of Lennard-Jones (LJ) particles. A Markovian Langevin thermostat was applied where the friction coefficient was parameterized based on the integral of the force auto-correlation function (FACF) of the fluctuating forces. This approach was derived based on the EoM derived by Kinjo and Hyodo28 by neglecting cross-correlations between different CG beads and subsequently assuming the Markovian behavior. The authors found that the self-diffusion coefficient was underestimated by a factor of 4 in the CG Langevin models, while DPD-type models performed significantly better for the same system,24 indicating that the mismatch was due to the missing configuration dependency in the frictional and random forces. In the following years, the same research group focused their efforts on improving the methodology to derive more accurate DPD-type models by, e.g., the introduction of memory effects.19–21 Yet, the magnitude of the mismatch in self-diffusion coefficients in the Langevin models is intriguing as one could reasonably assume that configuration dependencies would average out in isotropic systems, similar to the case of single-particle implicit solvent GLE models, which can capture the underlying FG dynamics exactly in the absence of conservative interactions.29 In principle, the parametrization of (generalized) Langevin models can be optimized iteratively until the fine-grained dynamics is matched.25–27 While this is a valid approach and widely used in the derivation of conservative interactions for CG models,1,4,30 such methods obscure the origin of the poor performance of bottom-up derived friction coefficients and memory kernels for LE and GLE models.

Herein, we will demonstrate that through the splitting of interactions into conservative and fluctuating forces on the CG level, frictional contributions arise, which were not accounted for in earlier studies. The calculation of these contributions from Q-projected time correlation functions provides a route to compute memory kernels with which dynamic consistency is achieved in GLE simulations. It furthermore provides improved insight into interactions that lead to memory effects in soft matter systems manifested on diffusive time scales.

We shall be interested in a GLE of the form

F(t)=FC(t)m0tdsK̃(ts)v(s)+F̃R(t),
(1)

in which F(t) is the total force on a CG particle, FC(t) is the conservative force corresponding to a CG force field, m is the particle mass, and v(t) is its velocity. One-dimensional descriptions are used throughout, which for isotropic systems trivially hold for all Cartesian dimensions. The random force F̃R(t) and the memory kernel K̃(t) are related through the fluctuation–dissipation theorem (FDT)15,31 and represent the GLE thermostat. The question of interest herein is how the microscopic interactions encode K̃(t).

To this end, it is instructive to first consider the GLE for a single particle in an isotropic medium. Expressed in the Mori–Zwanzig formulation, the EoM is

F(t)=m0tdsK(ts)v(s)+FQ(t),
(2)

in which FQ(t) denotes the Q-projected total force on the particle.15,32 Its relation to K(t) is provided by the FDT according to

K(t)=FQ(t)FQ(0)/mkBT,
(3)

where kB is the Boltzmann constant, T is temperature, and ⟨⋯⟩ denotes averaging over configurations of the microscopic (FG) system. To make the evaluation of Eq. (2) tractable in computer simulations, FQ(t) has to be interpreted to be a random force FR(t), in which case the EoM reads

F(t)=m0tdsK(ts)v(s)+FR(t).
(4)

The Q-projected total force in Eq. (3) can be written as FQ(t)=FQC(t)+δFQ(t). The kernel can therefore be expressed as

mkBTK(t)=FQ(t)FQ(0)=FQC(t)FQC(0)+δFQ(t)δFQ(0)+2FQC(t)δFQ(0)=mkBTKC(t)+mkBTKδ(t)+2mkBTKX(t).
(5)

A similar splitting of contributions to K(t) in terms of attractive and repulsive interactions in a LJ fluid was carried out by Carof et al.32 The above splitting of FQ(t) in Eq. (2) generally leads to correlated Q-projected conservative and fluctuating forces, i.e., FQC(t)δFQ(0)0. The random force, F̃R(t), in Eq. (1) is often interpreted to stem from the same fluctuations that determine δFQ(t).24 This however leads to the tacit assumption that FQC(t)δFQ(0)=0, and correspondingly, KX(t) = 0, which can only be assumed if FQC(t) and δFQ(t) evolve on separated time scales.

The above analysis of the single-particle model has consequences for interpreting the memory kernel in Eq. (1). The CG conservative forces are explicit in Eq. (1) and, therefore, do not contribute to the friction represented by the kernel K̃. We may therefore approximate K̃(t) as

K̃(t)Kδ(t)
(6)

if we assume that KX(t) = 0. In the Markovian limit, the Q-projected forces may be replaced with the real forces (“Q-approximation”) so that

K̃(t)δF(t)δF(0)/mkBT.
(7)

Based on these approximations, Lei et al.24 performed Langevin dynamics simulations of CG LJ clusters in which the applied friction coefficient γFACF was obtained from

γFACF=0τdtδF(t)δF(0)/mkBT.
(8)

In the general case, Eq. (5) however suggests that the kernel K̃ in Eq. (1) should be written as

K̃(t)=Kδ(t)+2KX(t).
(9)

It thus contains a contribution of Q-projected correlations between fluctuating forces, encoded by the term Kδ(t)=δFQ(t)δFQ(0)/mkBT, and a contribution corresponding to Q-projected cross-correlations of fluctuating and conservative forces, encoded by KX(t)FQC(t)δFQ(0)/mkBT. The remaining contribution KC(t)=FQC(t)FQC(0)/mkBT encodes the friction and memory due to the conservative interactions FC(t). The Q-projected time correlation functions are herein computed on the basis of the backward orthogonal dynamics method proposed by Carof et al.32 while using the second order integration scheme proposed by Jung et al.26 for numerical evaluation.

Alternatively, different memory kernels can be defined by multiplying Eq. (1) or Eq. (4) with v(0) and subsequent averaging, leading to Volterra equations. These equations can be numerically inverted to derive memory kernels from time correlation functions of mapped FG trajectories. In particular, we define the memory kernels KV(t), K̃V(t), and KCV(t) linked to the force–velocity correlation functions F(t)v(0), δF(t)v(0), and FC(t)v(0), respectively. The Volterra inversion approach is discussed in more detail in the supplementary material. In Table I, definitions of memory kernels both in terms of the backward orthogonal dynamics method and the Volterra inversion method are listed, as are their interrelations. Memory kernels derived from the inversion of Volterra equations are denoted by the superscript V. We note that a Volterra inversion applied to Eq. (4) provides KV(t) = K(t).29,33–36 Solving the Volterra equation corresponding to Eq. (1) however yields K̃V(t)=Kδ(t)+KX(t)K̃(t). This will be demonstrated numerically in Sec. II A and discussed analytically in the supplementary material.

TABLE I.

Summary of important definitions and relations of the memory kernels considered.

Backward orthogonal dynamicsVolterra equationsAuxiliary relations
K(t)=FQ(t)FQ(0)/mkBT F(t)v(0)=m0tdsKV(ts)v(s)v(0) Kδ,X(t) = Kδ(t) + KX(t
KC(t)=FQC(t)FQC(0)/mkBT δF(t)v(0)=m0tdsK̃V(ts)v(s)v(0) K̃(t)=Kδ(t)+2KX(t) 
Kδ(t)=δFQ(t)δFQ(0)/mkBT FC(t)v(0)=m0tdsKCV(ts)v(s)v(0) KC,X(t) = KC(t) + KX(t
KX(t)=FQC(t)δFQ(0)/mkBT  KV(t) = K(t
  K̃V(t)=Kδ,X(t) 
  KCV(t)=KC,X(t) 
Backward orthogonal dynamicsVolterra equationsAuxiliary relations
K(t)=FQ(t)FQ(0)/mkBT F(t)v(0)=m0tdsKV(ts)v(s)v(0) Kδ,X(t) = Kδ(t) + KX(t
KC(t)=FQC(t)FQC(0)/mkBT δF(t)v(0)=m0tdsK̃V(ts)v(s)v(0) K̃(t)=Kδ(t)+2KX(t) 
Kδ(t)=δFQ(t)δFQ(0)/mkBT FC(t)v(0)=m0tdsKCV(ts)v(s)v(0) KC,X(t) = KC(t) + KX(t
KX(t)=FQC(t)δFQ(0)/mkBT  KV(t) = K(t
  K̃V(t)=Kδ,X(t) 
  KCV(t)=KC,X(t) 

As a test case for the proposed methods, we consider a generic star-polymer melt system, where every star polymer in the microscopic (FG) description consists of one central bead with ten arms and three beads per arm, following Wang et al.25 We use the iterative Boltzmann inversion (IBI) method for the derivation of the conservative CG model in which every star polymer is represented as a single interaction site. A visualization of a single star-polymer in its FG and CG representation is shown in Fig. 1. Further details can be found in the supplementary material.

FIG. 1.

The FG and CG representation of a single star-polymer molecule. In the CG process, all 31 beads are mapped on the center of mass as RI=imimIRI,i, and accordingly, the CG velocity and the total force are calculated as vI=imimIvI,i and FI = ∑iFI,i, respectively, where the sum runs over the FG beads and mI = ∑imi. Here, the subscript I refers to the Ith star-polymer molecule and I, i to the ith monomer bead within molecule I.

FIG. 1.

The FG and CG representation of a single star-polymer molecule. In the CG process, all 31 beads are mapped on the center of mass as RI=imimIRI,i, and accordingly, the CG velocity and the total force are calculated as vI=imimIvI,i and FI = ∑iFI,i, respectively, where the sum runs over the FG beads and mI = ∑imi. Here, the subscript I refers to the Ith star-polymer molecule and I, i to the ith monomer bead within molecule I.

Close modal

We first demonstrate the equivalence of the derived memory kernels based on Volterra equations and the backward orthogonal dynamics method. In particular, we want to validate that the cross contribution KX(t) contributes equivalently to KCV(t) and K̃V(t) obtained by a Volterra equation approach (see the supplementary material). The memory kernels obtained by these methods are compared in Fig. 2. Except for small numerical deviations, we find that KV(t) = K(t), KCV(t)=KC,X(t), and K̃V(t)=Kδ,X(t) holds.

FIG. 2.

Comparison of the memory kernels obtained from Volterra equations (KV, KCV, and K̃V) and the corresponding kernels obtained from the backward orthogonal dynamics method of Carof et al.32 (K, KC,X, and Kδ,X).

FIG. 2.

Comparison of the memory kernels obtained from Volterra equations (KV, KCV, and K̃V) and the corresponding kernels obtained from the backward orthogonal dynamics method of Carof et al.32 (K, KC,X, and Kδ,X).

Close modal

Figure 3 shows all memory kernels evaluated based on the backward orthogonal dynamics method and their respective integrals Γ(t). It can be seen that on small time scales, the total memory kernel K(t) is mainly governed by the fast DoFs, i.e., Kδ(t). On longer time scales, the conservative CG interactions, i.e., KC(t), dominate. The contributions from cross-correlations, i.e., KX(t), overall are small in amplitude. Yet, over long time scales, the mainly negative contributions accumulate, giving rise to an effective decrease in the total friction. A similar behavior was found for the cross-correlations of long-ranged and short-ranged interactions in LJ fluids.32 

FIG. 3.

Left: the memory kernel K(t) and all its contributions derived from the backward orthogonal dynamics method. Contributions from fluctuating forces (Kδ) decay quite fast, giving rise to quasi-instantaneous friction, while contributions from CG interactions (KC) show a slow decay. Contributions from cross-correlations (KX) are small in amplitude on all time scales, being positive on short time scales with a negative slowly decaying tail on long time scales. Right: integrals Γ(t)=0tdsK(s) of the memory kernel and its contributions. For t → ∞, these integrals represent the friction coefficients in the Markovian approximation. The total integral of KX(t) is negative, effectively reducing the total friction. The inclusion of KX(t) in the parameterization of CG models thus should increase the self-diffusion coefficient.

FIG. 3.

Left: the memory kernel K(t) and all its contributions derived from the backward orthogonal dynamics method. Contributions from fluctuating forces (Kδ) decay quite fast, giving rise to quasi-instantaneous friction, while contributions from CG interactions (KC) show a slow decay. Contributions from cross-correlations (KX) are small in amplitude on all time scales, being positive on short time scales with a negative slowly decaying tail on long time scales. Right: integrals Γ(t)=0tdsK(s) of the memory kernel and its contributions. For t → ∞, these integrals represent the friction coefficients in the Markovian approximation. The total integral of KX(t) is negative, effectively reducing the total friction. The inclusion of KX(t) in the parameterization of CG models thus should increase the self-diffusion coefficient.

Close modal

For the parameterization of the GLE-thermostat models, we fit the integrals of the memory kernels K(t), Kδ(t), Kδ,X(t), and K̃(t) with six damped oscillators, respectively (see the supplementary material for details). We chose to fit the integrals rather than the memory kernels to account for the accumulative nature of memory effects, which effectively govern the qualitative behavior. Furthermore, we want to emphasize the long time dynamics. In Fig. 4, the fitted functions are overlayed as dotted lines. For K(t), Kδ(t), and Kδ,X(t), the fitted functions well reproduce the integral of the memory kernels up to a time of t = 50. For K̃(t), we chose to fit the function only up to t ≈ 20. The reason is that the cross contributions KX(t) show a long-lived negative tail, leading to divergence in the plateau of the integral. All memory kernels that can be derived from Volterra equations seem not to suffer from this problem.

FIG. 4.

Left: memory kernels applied in CG simulations. The black dotted lines represent the fit functions used for the parameterization of the GLE thermostat. Right: respective integrals of the memory kernels and their fit functions (black dotted lines).

FIG. 4.

Left: memory kernels applied in CG simulations. The black dotted lines represent the fit functions used for the parameterization of the GLE thermostat. Right: respective integrals of the memory kernels and their fit functions (black dotted lines).

Close modal

For the parameterization of Markovian LE models, we chose to derive the friction coefficient from the plateau value of the fit functions. By this, the effective friction coefficient employed in the Markovian LE models is consistent with the memory kernels of the non-Markovian GLE models discussed in Sec. II C, which allows for an analysis of the relevance of memory effects. We will below refer to the different Markovian LE models as “γ-models” and adjust the notation for “γ” according to the corresponding kernels [i.e., γ-model ↔ K, γδ-model ↔ Kδ, γδ,X-model ↔ Kδ,X, γ̃-model K̃, and γFACF-model δF(t)δF(0)/mkBT].

In Fig. 5, the mean square displacement (MSD), scaled by 16t is plotted in the linear regime and compared to the FG results. The scaled MSD is constant for the shown time scale for all models and represents the self-diffusion coefficient. As expected, the γ-model, for which all interactions are encoded in γ obtained by integrating K(t), accurately reproduces the self-diffusion coefficient of the FG system and can be seen as a validation of the method for calculating the memory kernels. For the remaining models, the LE thermostat is used in conjunction with CG conservative interactions based on the IBI method. For the γFACF-model [Eq. (8)], we find that the self-diffusion coefficient is underestimated by ≈25%. This is in line with the findings of Lei et al.,24 who found an even larger deviation of approximately a factor of 4 for LJ clusters. The finding that LE models based on the FACF of the fluctuating forces δF(t) show significantly slower dynamics appears to be quite general. Comparing the result for the γFACF-model with the γδ-model shows that the main error source for the deviation does not lay in the Q-approximation as the γδ-model predicts even lower self-diffusion coefficients. The γδ,X-model, in which the friction coefficient is identical to the one obtained by integrating the kernel K̃V(t) obtained from the Volterra equation (cf. Table I), improves the result significantly but still leads to an underestimation of the self-diffusion coefficient. The γ̃-model further enhances the dynamics compared to the γδ,X-model but is found to overshoot, effectively leading to an overestimation of the self-diffusion coefficient by a similar margin as the γδ,X-model shows an underestimation of the self-diffusion coefficient.

FIG. 5.

Scaled mean square displacement (MSD) for the evaluation of self-diffusion coefficients in CG LE models. FG denotes the evaluation based on the mapped all-atom trajectory.

FIG. 5.

Scaled mean square displacement (MSD) for the evaluation of self-diffusion coefficients in CG LE models. FG denotes the evaluation based on the mapped all-atom trajectory.

Close modal

The very fast self-diffusion of the γ̃-model can be understood in terms of memory effects, which are neglected in Markovian models. Generally, when applying a non-Markovian GLE thermostat, the effects on the dynamics depend on the density of states governed by the conservative interactions.37,38 In a simple model system, this feature of GLE thermostats was nicely demonstrated by Kappler et al. by applying a GLE thermostat to a particle in a one-dimensional double well potential.39,40 They have shown that the application of an exponential memory kernel, which decays on much longer time scales than the typical time scale of fluctuations within one minimum of the double well, leads to a slow down of the barrier crossing dynamics. In particular, the introduction of a second exponential term in the memory kernel has shown that the effects of the shorter lived memory kernel dominate.40 This underlines the feature that dynamic modes in the system more strongly couple to a memory kernel that relaxes on comparable time scales.

Differences in the self-diffusion coefficients obtained from Markovian and non-Markovian models parameterized based on the same memory kernel provide indications for memory effects in the long time dynamics. The preceding arguments applied to the CG model used in this work can be used to explain the role of memory. The IBI-model is effectively purely repulsive. As we consider a dense star-polymer melt, the overall dynamics is dominated by short-lived collisions of the beads with their nearest neighbors. The positive amplitudes of the memory kernel on short time scales couple strongly to the dominant dynamic modes. The introduction of the cross contributions KX(t) in the memory kernel, as demonstrated in Figs. 3 and 4, introduces negative amplitudes on long time scales, which should mostly decouple from the short-lived fluctuations due to collisions. Including contributions on all time scales indiscriminately in the derivation of the friction coefficient of the LE-thermostat leads to an increase in the diffusion coefficient compared to non-Markovian models, which correctly account for the frequency or time scale dependence of dissipative interactions. We will validate this interpretation of the data in Sec. II C.

The dynamics of the non-Markovian models is evaluated based on the velocity auto-correlation function (VACF) and compared to the FG model in Fig. 6. We do not show the non-Markovian counterpart of the γFACF-model for which the functional form of the memory kernel is assumed not to be physically meaningful due to a lack of time scale separation. The model without conservative interaction (K, blue line) reproduces the FG VACF and its integral D(t)=0tdsv(s)v(0) on all time scales and is shown here to validate the method for the calculation and fitting of the memory kernel K(t). For the non-Markovian models, we again find that the models not including the cross-correlation contributions [Kδ(t)] or including it only once [Kδ,X(t)] lead to an underestimation of the self-diffusion coefficient. The self-diffusion coefficient is well matched for K̃(t). With this kernel, the VACF is also better matched in terms of its amplitudes on small time scales compared to Kδ(t) and Kδ,X(t). The origin of the remaining deviations from the FG VACF can be attributed to errors in the conservative interactions and is discussed in the supplementary material.

FIG. 6.

VACFs and their integrals D(t)=0tdsv(s)v(0) for the FG simulation compared to the GLE CG models.

FIG. 6.

VACFs and their integrals D(t)=0tdsv(s)v(0) for the FG simulation compared to the GLE CG models.

Close modal

A comparison of the derived self-diffusion coefficients between the Markovian and non-Markovian models allows us to assess the importance of memory effects in the models at hand. Most evidently, we find that the K̃(t)-model shows a significantly slower self-diffusion than the Markovian γ̃-model. The same holds true to a lesser extent when comparing the Kδ,x(t)-model and the γδ,x-model, while the self-diffusion coefficients for the Kδ(t)-model and the γδ-model are comparable. This finding is exactly in line with the discussion presented in Sec. II B.

In this work, we have presented a new method for the parameterization of LE thermostats and GLE thermostats for dynamic consistency in CG models. Recently, it has been demonstrated that GLE-thermostat models can be parameterized by means of optimization schemes,25,26 which ensure that the dynamical properties such as the VACF of CG beads are reproduced. We instead aimed to illuminate the main origins of the deficiencies in early attempts24 of constructing Langevin thermostat based CG models based on the Mori–Zwanzig theory. We have found that cross-correlations of CG and FG degrees of freedom must be considered to consistently model friction and memory effects on a CG level. In non-Markovian models, inclusion of the cross-correlated contribution to the memory leads to improved representability of the dynamics, also on short time scales. Neglecting the cross-correlated contribution to the memory kernel leads to dynamic inconsistencies of the CG model and fully accounts for the very slow dynamics predicted with methods solely based on the correlations of fluctuating forces.

Additionally, the comparison of Markovian and non-Markovian models has shown that, in the CG system studied in this work, memory effects manifest on the diffusive time scale. This has shown to be only the case for memory kernels, which exhibit a significant negative contribution on large time scales. In the supplementary material, we demonstrate that this memory effect is inverted for positive long-lived memory kernels and is less pronounced in dilute (implicit solvent) CG systems.

For the proposed approach to perform optimally, in principle, the conservative interactions would have to represent the exact multi-body potential of mean force. Deviations can lead to a discrepancy of the friction caused by the CG conservative interactions in sampling governed by the CG force field compared to the sampling of the microscopic (FG) system (see the supplementary material). Generally, the loosest requirement to have a fully dynamically consistent model, independent of the choice of how to model friction and random interactions, is an exact representation of the multi-body potential of the mean force on the CG level.11,41 This is, of course, not generally feasible, and compromises have to be made. While an iterative optimization technique25,26 can be used to optimally reproduce a single measure such as the VACF, this procedure does not guarantee that all conceivable dynamical properties are reproduced. This is related to the representability problem of CG conservative interactions,42–44 where, e.g., the IBI method only guarantees the reproduction of the radial distribution function (RDF) but is known to, e.g., overestimate the pressure, underestimate isothermal compressibilities,45,46 and does not guarantee to represent higher-order correlations.

In cases in which the exact reproduction of the VACF is required, the memory kernel obtained with the method proposed in this work can be used as an optimal initial guess for the memory kernel in iterative reconstruction methods, which have shown to need up to hundreds of iterations to reach convergence, when the initial guess is far from the optimal solution.26 

While in the past much effort has been put into the development of new methods for parameterizing CG methods with conserved dynamical properties,11 predictive modeling of the dynamic properties of the material-specific soft matter system still poses challenges.12 For non-Markovian models, the form of Eq. (1) is computationally the least demanding in both derivation and implementation with a Mori–Zwanzig based CG modeling approach. These types of models are not as rigorous as formulations including configuration-dependent memory kernels,27 but the ease of their implementation, in principle, allows for application to more complex problems. For future endeavors within this field of research, it is desirable to extend the efforts to more complex systems such that predictive capabilities of the existing approaches can be tested.

See the supplementary material for more details on the setup of the simulations and the derivation of the coarse-grained models and also for complementary data and discussions on memory effects, the origin of remaining inaccuracies in the coarse-grained models, and the Markovian, high mass limit for coarse-grained generalized Langevin models.

Financial support for this work was granted by the Deutsche Forschungsgemeinschaft (DFG) via SFB TRR 146 (Grant No. 233630050, Project A2). The authors thank Swaminath Bharadwaj, Marvin P. Bernhardt, Madhusmita Tripathy, and Friederike Schmid for inspiring discussions.

The data that support the findings of this study are available within the article and its supplementary material. Input parameters for simulations, analysis scripts, and raw data of the shown figures are available from the corresponding author upon reasonable request.

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