1 Introduction

For the modeling of fluid flow, different methodologies are employed when the interest is on macroscopic or on microscopic scales. Macroscopic scales typically are described by the continuum or Navier-Stokes (NS) equations, whereas for microscopic scales, the molecular dynamics (MD) method with realistic potentials can be used. For the intermediate mesoscopic scales, NS and MD cannot be applied directly. The NS equations may be extended to the mesoscopic range by adding a stochastic stress, which results in the Landau-Lifshitz Navier-Stokes (LLNS) equations^[1]. Since the LLNS equations are based on the continuum hypothesis, they are still inappropriate when the spatial scales approach the dimension of molecules. Given an accurate force field, MD is valid at any length scale. However, it is computationally unfeasible to simulate a reasonably complex fluid flow with MD. A gap exists between the length scales that can be efficiently simulated by MD and those where the NS or LLNS equations become valid. As an attempt to bridge the gap, dissipative particle dynamics (DPD) was invented^[2-4]. It is a Lagrangian particle method operating at the mesoscopic scales. DPD has been successfully applied to study a wide range of phenomena, such as solubility of polymers^[4], rheology of colloids^[5], and dynamics of membranes^[6].

As the mesoscopic scales described by the DPD method overlap with macroscopic and microscopic scales, it is necessary to establish relations between DPD and MD for the small-scale limit and between DPD and NS/LLNS for the large-scale limit. It is difficult to develop generally valid formal relations for these limits. Assumptions and restrictions have always been made for each individual scenario. For example, a formal connection between DPD and MD has been established for structured fluids^[7-8] using the Mori-Zwanzig projection formalism^[9-10]. However, for a simple fluid, a correspondence between DPD particles and MD particles is unclear, and a formal connection is still under development^[11]. Other attempts to develop a formal relation between DPD and NS equations have also been made in several early studies. In Refs. [12] and [13], a kinetic theory was used to connect the DPD input parameters (without conservative force) to output transport coefficients. In Ref. [14], analytical expressions of wavenumber-dependent macroscopic properties have been derived for a two-dimensional system. The extension of such analysis to three dimensions is difficult. Moreover, often the assumption of exponential decay of the current auto-correlation function (CACF) is invoked to proceed with analytical derivations, which may be invalid, however, for small scales of DPD. Hence, the relations derived so far cannot cover the entire range of DPD parameters.

An interpretation of DPD from a top-down perspective has enabled a successful alternative approach. In a seminal paper^[15], Español and Revenga started with a Lagrangian discretization of the NS equations using the smoothed particle hydrodynamics (SPH) method^[16-17] and introduced thermal fluctuations directly on the Lagrangian particles following the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) framework^[18]. To reflect that it bridges DPD and SPH, this method is called smoothed dissipative particle dynamics (SDPD). Soon after its invention, SDPD has been applied to many problems, such as flows of red blood cells^[19], colloidal particles^[20], DNA chains^[21], and viscoelastic liquids^[22]. However, to capture unconventional phenomena beyond the continuum limit, SDPD has to be modified or extended^[23-24].

In this paper, we choose to analyze DPD. To be more specific about the possible significance of analyzing the small scale dynamics of DPD, we outline three illustrative examples as motivation for our work.

Soddemann et al.^[25] realized that DPD equations (with only friction and stochastic forces) can be utilized as an effective thermostat as alternative to the Langevin equations (LEs) for MD simulations. Due to momentum conservation, DPD indeed provides correct and unscreened hydrodynamic correlations of the MD system. Moreover, the effect of the DPD friction parameter on the intrinsic viscosity of microcanonical MD was studied extensively^[25-27]. Nevertheless, at least two questions remain open: (ⅰ) it is unclear how the friction parameter may affect small-scale dynamics of microcanonical systems, (ⅱ) whether the DPD thermostat effects are comparable to the Langevin thermostat at these small scales.

In another study^[28], the equation of motion for DPD particles without conservative force was rewritten as (see Eq. (13) in Ref. [28])

(1)

where an environment velocity V_i and the random force are defined using properties of the neighboring particles (the other terms are defined in Section 2). Equation (1) suggests that DPD shares similarities with the LE. Furthermore, in the regime of large overlapping parameter, or large number of neighboring particles, the velocity auto-correlation of a tagged particle is predicted well by the analytical expressions derived from Eq. (1). In other regimes, however, the relation between DPD and the LE is unknown.

DPD is a dissipative system with conservation of the momentum. Another well-known model with such properties is the hybrid Lagrangian-Eulerian approach^[29]. The Eulerian part serves to conserve momentum of the particle-based system, which is usually described by the LEs. A similar approach can also be applied to model rarefied gases^[30]. It appears that the application range of DPD may be extended to such situations, provided that the small-scale behavior of DPD is fully understood.

In the current work, we apply numerical analysis to investigate the behavior of DPD on small scales in three dimensions. In particular, we measure the CACFs of DPD in Fourier space and compare them with analytical expressions that can be derived for LLNS and LEs. We demonstrate similarities and differences of DPD with other systems and come to conclusions about the correspondence between DPD and LEs. The remainder of the paper is structured as follows: in Section 2, we describe the DPD method with a short review of previous findings on DPD macroscopic properties; in Section 3, we discuss properties of the CACFs of the LLNS equations and the LEs; in Section 4, the CACFs of DPD obtained from simulations are compared with the analytical solutions of CACF for the LE; in Section 5, we summarize our results and come to conclusions.

2 DPD model

DPD is a multi-particle model. A pairwise force F_ij acts on each particle i and shifts its locations

(2)

The force F_ij consists of three different parts: conservative force F_ij^C, dissipative force F_ij^D, and random force F_ij^R, that is,

(3)

In this paper, we consider DPD without the conservative force, F_ij^C=0. This choice is motivated by the objective of comparing thermostat characteristics of LEs with that of DPD. In addition, analytical properties of the corresponding LEs (without conservative force) can be derived straightforwardly.

Dissipative and random forces in DPD are defined as

(4)

(5)

where the weighting functions w^D(r_ij) and w^R(r_ij) as well as the dissipation coefficient γ and the random coefficient σ are related so that they satisfy the fluctuation-dissipation balance^[3]

(6)

We define r_ij = r_i -r_j, r_ij=|r_ij|, and as relative position, distance, and unit vector between two particles i and j, respectively. k_B is the Boltzmann constant, and T is the temperature. v_ij = v_i -v_j is the relative velocity, and θ_ij is a Gaussian random variable with the properties^[3]

(7)

(8)

where δ(t) is the Dirac delta, and δ_ij is the Kronecker delta. We choose the standard weighting functions as^[4]

(9)

Properties of DPD are determined by a priori chosen parameters, which relate microscopic, macroscopic, and mesoscopic scales of motion. We follow previous works^{[14, 31-32]} by considering wavenumber dependent macroscopic properties. The wavenumber q=2π/λ corresponds to the length scale of interest λ=L/n with n=1, 2, 3, … and L is the length of the domain. The specific dependence of macroscopic properties on q in DPD, such as the effective shear viscosity ν(q), the effective bulk viscosity ζ(q), and the effective isothermal speed of sound c_t(q), varies with length scales. With DPD, one can distinguish three different characteristic length scales: cut-off radius r_c, decorrelation length l₀ of particles, and length of interest λ. The decorrelation length l₀ of DPD particles is analogous to the mean free path^[14]. Depending on the relation of the cut-off radius r_c to the decorrelation length l₀ of DPD particles, different scaling regimes were defined^[14]: the collective regime r_c>l₀ and the particle regime r_c < l₀. Please refer to Table 1 for a classification. In analogy to the length-scale parameter l₀, the collision time t₀=l₀/v₀ was defined as time-scale parameter^[14], where v₀ is the thermal velocity. Please refer to Table 2 for the definitions of variables. In the collective regime within the collision time t₀, particles on the average do not travel far enough before interacting with neighboring particles multiple times. This situation is common for liquid molecules. In the particle regime, particles encounter less frequent interactions with the same neighboring particles and rather interact with new neighbors. The particle regime is typical for gas molecules. One can choose between collective or particle regime by setting the parameters of DPD appropriately.

For the collective regime of DPD, one can distinguish three different subregimes: the standard hydrodynamic subregime λ>r_c>l₀, the mesoscopic subregime r_c>λ>l₀, and the N-particle subregime r_c>l₀>λ. Macroscopic properties of DPD in the standard hydrodynamic subregime do not depend on wavenumber and correspond to the isothermal NS equations^[14].

For the particle regime of DPD, one can expect three different subregimes as well^[14]: the standard hydrodynamic subregime λ>l₀>r_c, the kinetic subregime l₀>λ>r_c, and the N-particle subregime l₀>r_c>λ. The functional dependence of DPD properties on q varies across the subregimes. Regimes of DPD and corresponding subregimes are summarized in Table 1.

It was shown in Ref. [33] that on the small scales, CACFs of DPD follow non-exponential laws. We investigate this property in detail in this work. For this purpose, we first consider the properties of CACFs for LLNS equations and LEs, followed by a comparison of small-scale behaviors of the LEs and of DPD.

3 CACFs

The CACF C(q_k, t) is defined as^[32]

(10)

Subscripts k, l indicate the wavenumber-vector, current-vector, and velocity-vector components in coordinate directions x, y, z, respectively. k = l results in the longitudinal CACF with C_‖(q, t), and k ≠ l results in the transverse CACF with C_⊥(q, t). is the variance of the respective Fourier mode of current w.

CACFs are used to analyze MD systems^[31-32], fluctuating hydrodynamics^[34-35], DPD^{[33, 36-37]}, and multi-particle collision dynamics solvent^[38-39], and are directly related to macroscopic properties, such as the effective shear viscosity, the effective bulk viscosity, and the effective isothermal speed of sound. CACFs may also represent the ensemble-averaged decay of an initial sinusoidal velocity field to the steady state. This property follows from the regression hypothesis of Onsager^[40] which states that microscopic thermal fluctuations at equilibrium can be modeled by non-equilibrium transport coefficients. Setting an initial sinusoidal velocity in two different directions parallel or perpendicular to the velocity direction corresponds to longitudinal and transverse CACFs, respectively^[34]. In the following subsections, we consider CACFs of two models, that is, the LLNS equations and the LEs, at thermodynamic equilibrium.

3.1 CACFs of the LLNS equations

The LLNS equations are an extension of the NS equations to smaller scales where thermal fluctuations are important^[1]. A stochastic term is introduced to account for the spontaneous fluctuations of stresses in a finite thermodynamic system. The dissipative term of the LLNS is the same as that of the NS equations, and the stochastic term in the LLNS together with the dissipative term satisfies the fluctuation-dissipation theorem. For simplicity, in this paper, we consider only isothermal LLNS in the linearized form. For the linearized equations in a periodic domain, one can derive the transverse and longitudinal CACFs as well as density auto-correlation function^[41] as

(11)

(12)

(13)

respectively, where

(14)

3.2 CACFs of the LEs

The LEs were introduced to describe the Brownian motion of particles^[42]. In this section, we consider one particle driven by the LE

(15)

(16)

Here, we take a constant dissipation coefficient β and a delta-correlated in time Gaussian random variable θ_{k, i},

(17)

(18)

where k_B is the Boltzmann constant, T is the temperature, and m is the mass of the particle.

The CACFs can be derived from simulations or analytically. An analytical derivation is provided in the Appendix for reference. The CACFs of the LEs are

(19)

(20)

(21)

We compare the analytical expressions with simulations. In the simulations, for one particle, we take^[33]

(22)

where k, l=x, y, z are direction indices. Equation (22) can be used to compute CACFs according to Eq. (10). Equation (10) for N_r independent realizations gives

(23)

where angular parentheses 〈·〉_i denote averaging in time of the ith realization. In order to determine the CACFs for LEs, we perform a numerical simulation with one particle moving according to the LEs (15) and (16) with k_BT = 0.25 and β=0.1. The particle moves in a three-dimensional box of size 250 × 250 × 250 with periodic boundary conditions. For the simulation, we use the velocity-verlet algorithm for integration in time. The number of independent realizations is N_r=128, and the time interval of each simulation spans T=5 × 10⁵ time units. The comparison of numerical results with theoretical predictions for CACFs is presented in Fig. 1 for different wavenumbers. The statistical errors in Fig. 1 are smaller than the symbol size. The results show that the numerical simulations agree well with the analytical predictions for CACFs of LEs, which verifies the numerical approach.

3.3 Effective shear viscosity

It is convenient to operate with the nondimensional DPD parameters which were introduced in Ref. [14] and are listed in Table 2. Dynamic overlapping Ω₀ is introduced as the ratio between cut-off radius and decorrelation length,

(24)

where ρ is the number density. For the particle regime of DPD, Ω₀ < 1, and for the collective regime, Ω₀>1. Analytical estimations of macroscopic DPD properties for large scales and for mesoscopic scales have been derived in Refs. [12]-[14]. For the standard hydrodynamic subregime, the nondimensional effective shear viscosity is estimated as

(25)

where the coefficient a₂ is

(26)

For the weighting function of Eq. (9) and . In the mesoscopic subregime, the estimation is

(27)

whereas in the N-particle subregime of DPD, the estimation is ^[14].

The common macroscopic definition of viscosity is the coefficient relating the stress tensor to the deformation rate of the fluid^[1]. In DPD literature, viscosity is usually defined as a measure of energy dissipation, which is a wavenumber dependent property^[12-14]. The energy dissipation in DPD has dissipative and kinematic contributions. The energy dissipation can be derived directly from CACFs. In this paper, we compare CACFs which are derived from Langevin dynamics of a single particle with CACFs derived from DPD simulations. For such a comparison, it is convenient to define a quantity of interest that resembles the definition of DPD viscosity which is used in the literature. However, the definition of this quantity of interest as kinematic viscosity for LEs may be misleading. For this reason, we employ the effective viscosity. In this paper, the effective shear viscosity is defined from transverse CACF as

(28)

This definition allows us to determine the effective shear viscosities for systems without momentum conservation, such as described by the LEs. The effective shear viscosity has the same dimension as the usual kinematic viscosity. Note that the effective shear viscosity defined by Eq. (28) may have properties different from the kinematic shear viscosity in a continuum description. For example, in the limit of q → 0, the effective shear viscosity diverges for Langevin dynamics, but reaches a plateau value for DPD fluids.

The effective shear viscosity ν_LE of LE can be derived from the transverse CACF. Upon inserting Eq. (19) into Eq. (28), we obtain

(29)

where is the upper incomplete gamma function, is the gamma function, and is a coefficient.

We emphasize that Eq. (29) does not go beyond what has been covered in the previous studies^[43], but it provides a convenient tool to compare properties of Langevin dynamics with that of DPD.

4 CACFs of DPD

In the previous section, we have derived analytical expressions for C_LLNS and C_LE. In order to measure CACFs of DPD, we consider numerical simulations for different wavenumbers q. We calculate CACFs for DPD using the same numerical technique as for LEs. For N DPD particles, Eq. (22) is

(30)

and the corresponding CACFs can be evaluated according to Eq. (23). The number of independent realizations is N_r=16, and the time interval of each simulation spans T=4 × 10⁶ time units. Simulation parameters are given in Table 3. For DPD simulations, periodic boundary conditions are used. The statistical errors in Fig. 2 and Fig. 3 are smaller than the symbol size. In Fig. 2, the current characteristics of DPD are shown for different wavenumbers and different subregimes. We compare these characteristics with the corresponding analytical predictions derived from the LE theory, Eqs. (19), (20), and (21). The decorrelation length of a particle governed by LE is Parameters for the analytical predictions are chosen in such a way that the particle in the LE has the decorrelation length equal to the one of DPD l₀ and the relaxation time in the LE is equal to the collision time t₀ of DPD particles.

For this purpose, the temperature in the LE is set equal to the temperature in DPD, and the dissipation coefficient in the LE is set the same as the collision frequency in DPD β = ω₀. Nondimensional wavenumbers in the LE predictions correspond to the ones in DPD. For the N-particle subregime, which is depicted with green color in Fig. 2, we find that the CACFs of DPD are in good agreement with those predicted by the LE theory. The correspondence of CACFs of DPD and LE is independent of the DPD parameters. This observation suggests that the large wavenumbers of DPD have a common, universal behavior. We compare CACFs on small wavenumbers of DPD with CACFs of the LLNS equation. Lines with triangles in Fig. 2 present a fit of CACFs of DPD in the standard hydrodynamic regime with Eqs. (11)-(13). For large dynamic overlapping Ω₀=20, DPD agrees with LLNS. Transverse CACFs derived from DPD for Ω₀=5 differ from the ones of LLNS. This demonstrates that the transition between different dynamic subregimes occurs gradually. An alternative way to look at the subregimes of DPD is to analyze the nondimensional effective shear viscosity.

Figure 3(a) presents a comparison of the nondimensional effective shear viscosity of DPD and LE. Every data point in Fig. 3(a) corresponds to the respective C_⊥ of DPD and is computed according to the definition of the effective shear viscosity in Eq. (28). One can observe that the dependency of the nondimensional effective shear viscosity on differs for different values of dynamic overlapping Ω₀. Nondimensional wavenumbers that correspond to the cut-off radius and decorrelation length are and , respectively. We choose the decorrelation length as length unit. For that reason, the nondimensional parameter coincides for different values of Ω₀ in Fig. 3(a). However, differs for different Ω₀. For wavenumbers larger than the cut-off wavenumber , the nondimensional effective shear viscosities for different Ω₀ follow the same universal trend. For the large wavenumbers , the trend follows a scaling law and coincides with the LE effective shear viscosity. However, for small wavenumbers , the trend differs from the LE effective shear viscosity and differs from the scaling law. Note that the scaling law corresponds to the ballistic regime of LE^[44] and corresponds to the case where the CACFs of LE are exponential functions. The effective shear viscosities of DPD and LE are similar in the N-particle subregime and are different in the mesoscopic subregime. This fact deviates from the estimates for the mesoscopic and N-particle regimes given by Eq. (27), represented by a brown line in Fig. 3(a), previously derived in Ref. [14]. According to Eq. (27) in the N-particle subregime, the effective shear viscosity of DPD is constant, and in the mesoscopic subregime, it follows a scaling law. To improve the prediction given by Eq. (27), we approximate of DPD for the mesoscopic subregime empirically as

(31)

An LE-based model was applied to rarefied gas flows in Refs. [30] and [45]. The Knudsen number is defined as the ratio of the mean free path to the length scale of interest n_K :=l₀/λ. For high Knudsen numbers, length scales of interest are comparable to the mean free path of gas molecules. In the proposed model^{[30, 45]}, Lagrangian particles are governed by a modified version of LE and the length scales of interest are comparable with the mean free path of Lagrangian particles. In Fig. 3(a), we demonstrate the similarity of the LE and DPD on the N-particle subregime. However, it is also possible to compare of a rarefied gas with that of DPD. The velocity auto-correlation of a Brownian particle in gas is^[46-47]

(32)

where the damping coefficient Γ₀ is

(33)

(34)

An alternative way to represent the motion of a Brownian particle in gas is to consider that the effective viscosity depends on the Knudsen number

(35)

where is the constant effective shear viscosity of the gas for n_K ≪ 1. Define the nondimensional wavenumber as so that . We take , without restricting the validity of the results in Fig. 3(b). Different colors in Fig. 3(b) correspond to different subregimes of DPD. The N-particle subregime (green lines) is in good agreement with the LE-theory (29). The kinetic subregime (yellow lines) deviates slightly from the LE theory. The mesoscopic subregime of DPD (red lines) exhibits a universal scaling range, which differs from the LE theory and is in good agreement with the empirical function given by Eq. (31). For wavenumbers smaller than the cut-off wavenumber and the decorrelation-length wavenumber, we observe that DPD recovers the LLNS viscosity dependence which is characterized by the constant nondimensional effective shear viscosity (blue lines).

We conclude that the nondimensional effective shear viscosity of a rarefied gas and of DPD have common properties. For large wavenumbers, they follow the power law, corresponding to the N-particle subregime of DPD (green lines). In the limit of small wavenumbers, a plateau of the effective shear viscosity corresponds to the standard hydrodynamic subregime (blue lines). A smooth transition from N-particle to standard hydrodynamic subregime corresponds to the kinetic subregime of DPD (yellow lines).

5 Summary and discussions

In this paper, we review properties of the CACFs for two different systems, LLNS and LE. For these systems, expressions for CACFs can be derived analytically. The analytical solutions are well reproduced by numerical simulations for the CACFs of the LE.

The effective shear viscosity is defined through an integration of the transverse current auto-correlation function (28). This definition allows to introduce the effective shear viscosity for systems without momentum conservation, such as the LE. We determine the wavenumber dependence of the nondimensional effective shear viscosity of DPD without repulsive potential with the method introduced in a previous study^[33].

The plateau exhibited by the DPD model is associated with macroscopic properties. For the standard hydrodynamic subregime, isothermal laws of motion in liquids and gases can be described with the LLNS equations, and the CACFs of DPD and LLNS are in good agreement, as was shown in Fig. 2 and in Ref. [33]. We show that the CACFs of the N-particle subregime of DPD and LE coincide. We also demonstrate that the effective shear viscosity of the DPD mesoscopic subregime differs from the LE effective shear viscosity. We introduce an empirical function to match the dependency for the mesoscopic subregime of DPD.

We extend previous findings in Ref. [14] where it was suggested that the effective shear viscosity is constant both in the limit of large wavenumbers and small wavenumbers. For the range between small and large wavenumbers, it was estimated that the effective shear viscosity of DPD follows a 1+q^-2 scaling law. In the current paper, we confirm that on small wavenumbers, the effective viscosity is constant. However, for large wavenumbers, the effective viscosity exhibits a similar behavior as the effective viscosity for LEs (29), which corresponds to a q^-1 scaling law. The correspondence of DPD and Langevin equations for this wavenumber range is also shown by comparison of the non-trivial form of CACFs of the LEs and of DPD. Moreover, we show that the 1+q^-2 scaling law can be improved by the introduction of the empirically derived (31). The difference from previous reports may be attributed to the fact that in such previous studies, e.g., Ref. [14], an exponential approximation for the transverse CACF was used. Note that in another previous study^[33], it was shown that on small scales, the transverse CACF in DPD is not exponential.

The concept of DPD particle is loosely defined and lacks physical meaning^[11]. Scales smaller than the cut-off radius are underresolved. However, in this work, we show that such scales exhibit a consistent dissipation law. We suggest that the small-scale behavior of DPD may be employed as an implicit fluid model, e.g., for high-Knudsen number flows. Properties of small, numerically underresolved scales are used for instance in turbulence modeling. For so-called implicit large-eddy simulation models, numerical dissipation on underresolved scales can be tuned to reproduce established energy-transfer mechanisms of turbulence^[48-49].

We compare properties of Langevin and DPD thermostats. For that purpose, we consider DPD without conservative force. In principle, the effective shear viscosity can be measured for Kolmogorov flow^[22]. The estimate from Kolmogorov flow coincides with the estimate of effective viscosity from CACF analysis^[33]. However, for purely dissipative systems, such as the LE, it is challenging to set up Kolmogorov flow simulation, and a CACF analysis is preferable.

We use the analytical derivation of CACFs of a particle governed by the LEs. The same results are valid for a system of non-interacting particles governed by the LE. However, in the case of an additional force between the particles, e.g., due to a Lennard-Jones potential, the analytical expressions for CACFs are much more challenging to derive. In Fig. 1, it is shown that CACFs of LE can be accurately measured in simulations. For that reason, the analysis provided here can be extended to systems with repulsive potential. One can compare simulations of a Langevin thermostat with a certain interaction between particles against a DPD thermostat with the same interaction. From preliminary results, we can find that for large wavenumbers, such comparison reveals similar behavior of CACFs. Moreover, it is plausible that for DPD with a classical potential^[4], the same scaling law as for DPD without potential will be observed.

Although we compare only the effective shear viscosity of DPD with that derived from the LE, an extension of the analysis for other macroscopic parameters, such as the effective isothermal speed of sound and the effective bulk viscosity, is straightforward. For this purpose, one needs to introduce a definition of effective bulk viscosity and effective isothermal speed of sound that is based on the longitudinal CACF. However, with any definition chosen, the isothermal speed of sound and the effective bulk viscosity of the N-particle subregime of DPD and LE would be similar due to the similarity of the longitudinal CACFs for both systems, which is presented in Figs. 2(d)-2(f). Another possibility is to extend the analysis to types of non-Markovian DPD (NM-DPD)^[8]. The NM-DPD model on small scales can be compared with the colored noise LEs^[50]. A comparison of the q-dependency of the effective shear viscosity of DPD with that of a rarefied gas reveals that DPD has the potential to model flows at high Knudsen number. For the correct representation of complex high Knudsen number flows, one needs to introduce a proper treatment for rarefied-gas wall boundaries in DPD. Practical rarefied gas modeling computations of DPD are beyond the scope of the current paper. We consider CACFs of DPD without the presence of mean shear. However, it is possible to extend the findings of the paper to the case of moderate shear rates following^[36]. From the modeling perspective, the presented analysis of DPD small scales extends the application of DPD to new problems, such as the modeling of turbulent phenomena on the mesoscale level or modeling of flows with high Knudsen number.

An extended consideration of current memory functions of the LE as well as DPD enables to use grid-based models for both systems. Numerical algorithms for the generalized Langevin model with repulsive potential were described in a previous study^[51], where a repulsive potential was used for stability of the numerical algorithm and enters as a model parametrization. By tuning the form and the value of the repulsive potential and by using correlated form in time noise, current characteristics of grid-based models can be matched with Lagrangian ones.

Appendix A

The LEs are characterized by the velocity auto-correlation function and the mean-squared displacement (MSD)^{[44, 47]}

(A1)

(A2)

Another general way to describe the properties of the LE is through its conditional joint probability distribution function (PDF) P(v, x, t|v₀), which is the probability of the particle with the initial velocity v(t=0, x=0)=v₀ and location x₀ to be at time t in x with velocity v. The expression for the conditional joint-PDF for the LEs in d-dimensions is^[43]

(A3)

(A4)

(A5)

(A6)

As the LE is an isotropic model, it is sufficient to consider only one direction. The corresponding equations for other directions are equivalent. The marginal PDF results by definition from integration of the joint-PDF,

The one-dimensional Maxwell-Boltzmann distribution results as equilibrium PDF,

(A7)

We further derive the analytical expressions for the transverse CACF, longitudinal CACF, and density auto-correlation functions, respectively, as

(A8)

(A9)

After integration, we obtain Eqs. (19)-(21),

To our knowledge, Eqs. (19)-(21) have not been explicitly stated before, although the Laplace-Fourier transformed version of Eqs. (19)-(21) were provided in Ref. [52].

Note that the first factor of Eq. (19) on the right originates from the exponential auto-correlation of Lagrangian particle velocity, Eq. (A1). The other factor originates from the diffusion of the particles in space, Eq. (A2), and coincides with the density auto-correlation function C_LEρ.

Acknowledgements The first author acknowledges travel support from the Technical University of Munich (TUM) Graduate School. We thank the Munich Center of Advanced Computing for providing computational resources. We thank V. BOGDANOV for proofreading.