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# Transport dynamics of complex fluids

Edited by Steve Granick, Institute for Basic Science Center for Soft and Living Matter, Ulju-gun, Ulsan, Republic of Korea, and approved May 9, 2019 (received for review January 7, 2019)

## Significance

Many disordered fluid systems exhibit anomalous transport dynamics, which do not obey Einstein's theory of Brownian motion or other currently available theories. Here, we present a new transport equation governing thermal motion of complex fluidic systems, which provides a unified, quantitative explanation of the mean-square displacement, the non-Gaussian parameter, and the displacement distribution of various complex fluids. The applicability of our theory is demonstrated for molecular dynamics simulation results of supercooled water and dense hard disc fluids and for experimental results of colloidal beads diffusing on lipid tubes. This work suggests previously unexplored directions for quantitative investigation into transport and transport-coupled processes in complex disordered media, including living cells.

## Abstract

Thermal motion in complex fluids is a complicated stochastic process but ubiquitously exhibits initial ballistic, intermediate subdiffusive, and long-time diffusive motion, unless interrupted. Despite its relevance to numerous dynamical processes of interest in modern science, a unified, quantitative understanding of thermal motion in complex fluids remains a challenging problem. Here, we present a transport equation and its solutions, which yield a unified quantitative explanation of the mean-square displacement (MSD), the non-Gaussian parameter (NGP), and the displacement distribution of complex fluids. In our approach, the environment-coupled diffusion kernel and its time correlation function (TCF) are the essential quantities that determine transport dynamics and characterize mobility fluctuation of complex fluids; their time profiles are directly extractable from a model-free analysis of the MSD and NGP or, with greater computational expense, from the two-point and four-point velocity autocorrelation functions. We construct a general, explicit model of the diffusion kernel, comprising one unbound-mode and multiple bound-mode components, which provides an excellent approximate description of transport dynamics of various complex fluidic systems such as supercooled water, colloidal beads diffusing on lipid tubes, and dense hard disk fluid. We also introduce the concepts of intrinsic disorder and extrinsic disorder that have distinct effects on transport dynamics and different dependencies on temperature and density. This work presents an unexplored direction for quantitative understanding of transport and transport-coupled processes in complex disordered media.

- complex fluids
- thermal motion
- diffusion kernel correlation
- supercooled water
- colloidal particles on lipid tube

Thermal motion in complex fluids is a complex stochastic process, which underlies a diverse range of dynamical processes of interest in modern science. Since Einstein’s seminal work on Brownian motion (1), thermal motion in condensed media has been the subject of a great deal of research. However, it is still challenging to achieve a quantitative understanding of the transport dynamics of disordered fluidic systems, including cell nuclei and cytosols (2), membranes and biological tissue (3), polymeric fluid (4), supercooled water (5, 6), ionic liquids (7, 8), and dense hard-disk fluids (9). Interestingly, thermal motion of these complex fluids commonly exhibits a mean-square displacement (MSD) with initial ballistic, intermediate subdiffusive, and then terminal diffusive behavior (10⇓–12); the associated displacement distribution is non-Gaussian except in the short- and long-time limits, and its deviation from Gaussian increases at short times but decreases at long times, as long as thermal motion is uninterrupted. These phenomena cannot be quantitatively explained by the original theory of Brownian motion (13).

To explain dynamics of anomalous thermal motion, the theory of Brownian motion has undergone a number of generalizations (14⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–27). Among these generalizations, Montroll and Weiss’s (20) continuous-time random walk (CTRW) model enables quantitative description of anomalous transport caused by a tracer particle being trapped by other objects in disordered solid media (28); the CTRW with a power-law waiting-time distribution (WTD),

While these models successfully describe subdiffusive transport occurring in disordered media, a number of disordered fluidic systems exhibit Fickian yet non-Gaussian diffusion (36⇓⇓⇓⇓⇓⇓–43); that is, the MSD is linearly proportional to time but the displacement distribution deviates from Gaussian. This issue was recently addressed by stochastic diffusivity (SD) models, in which the diffusion coefficient is treated as a stochastic variable (44⇓⇓⇓⇓⇓⇓–51). While SD models successfully demonstrate Fickian yet non-Gaussian diffusion, to the best of our knowledge, these models, too, are inconsistent with the transient subdiffusive MSD and the nonmonotonic time dependence of the non-Gaussian parameter (NGP), widely observed features of complex fluids.

The NGP has a long history in transport theory. Rahman, Singwi, and Sjölander first noted that the NGP is an observable in neutron scattering experiments (52), and Rahman recognized it as the first-order coefficient in the Hermite-polynomial expansion of the displacement distribution around Gaussian (53). Nieuwenhuizen and Ernst (54) showed that the NGP, or the fourth cumulant of displacement, is related to the time-correlation function (TCF) of the diffusion coefficient fluctuation and the Burnett correlation function (BCF), a functional of velocity autocorrelation functions (VAFs), for a system of independent charged particles hopping on a one-dimensional lattice with static disorder. Later, the NGP and BCF were investigated for interacting gas and fluid systems (55, 56) and, more recently, also for glassy systems (57, 58) for which the authors suggested the NGP as a measure of the diffusion coefficient fluctuation and dynamic heterogeneity. However, for complex fluid systems commonly exhibiting initial ballistic and intermediate subdiffusive thermal motion before terminal diffusion, the TCF of the diffusion coefficient is neither well defined nor a good measure of mobility fluctuation before terminal diffusion emerges. For complex fluid systems exhibiting non-Fickian diffusion, there is no precise definition of mobility fluctuation or exact relationship of mobility fluctuation with the NGP and VAFs. Of course, it remains a challenge to achieve a unified, quantitative understanding of the anomalous MSD, nonmonotonic NGP, and non-Gaussian displacement distribution of various complex fluid systems.

## Transport Equation of Complex Fluids

Here, to address these issues, we present a transport equation that provides a quantitative description of thermal motion for various complex fluids. This equation can be derived by considering the continuum limit of a random walk model with a general sojourn time distribution, *m*th site and the environmental state is at Γ at time *t* can be written as (49, 59)**1**, *m*th site, given that it has undergone *N* jumps, which is well known: *N* and the environmental state is at Γ at time *t*. **1** in the continuum limit (*SI Appendix*, Text S1):*t*. This joint probability density satisfies the following normalization condition: **2**, **2**,

For the sake of generality, we do not assume a particular model of environmental state dynamics, nor do we assume a particular form of mathematical operator

Eq. **2** encompasses the CTRW model and the diffusing diffusivity models (45⇓⇓⇓⇓⇓–51) (see *Discussion* for more details). A further generalization of Eq. **2** for complex fluidic systems under a spatially heterogeneous external potential is presented in *SI Appendix*, Text S2.

## Analytic Expressions of the Moments

From Eq. **2**, we obtain the exact analytical expressions of the first two nonvanishing moments, **3a** and **3b**, we assume that the hidden environment is initially in a stationary state, such as the equilibrium state or the nonequilibrium steady state (see *SI Appendix*, Text S3 for details). The bracket notation, **3a** and **3b** to the case where the initial state of the hidden environment is a nonstationary state, as is the case for glass, for which the analytic expressions of **3a** and **3b** (*SI Appendix*, Text S4). However, transport dynamics of various complex fluids can be quantitatively explained by Eqs. **3a** and **3b**, as demonstrated in this work. Hereafter, we focus on quantitative analysis of transport dynamics of complex fluidic systems with use of Eqs. **3a** and **3b**. We leave a quantitative explanation of transport dynamics in glass to future research.

The mean diffusion kernel, **3a** is nothing but the two-point VAF; i.e., **3a** and the Laplace transform of the well-known relation, *s* limit; i.e., *s* limit, the value of *s* limit asymptotic behavior of **3a**, we recover the well-known asymptotic behavior of the MSD:

While the second moment, **3b**, *SI Appendix*, Eq. **S3-12** for the precise definition). At long times, where the MSD is linear in time, the diffusion kernel becomes the diffusion coefficient; i.e., *q*, *SI Appendix*, Text S5). Given that the initial speed distribution obeys the Maxwell–Boltzmann distribution, we obtain *D* and *SI Appendix*, Fig. S1*D*).

In our theory,

Using Eqs. **3a** and **3b** and the definition of the NGP, *Methods*). In Fig. 2, we demonstrate our quantitative analysis of the molecular dynamics (MD) simulation results of the MSD and NGP for 4-point transferable intermolecular potential/2005 (TIP4P/2005) water (64), assuming specific analytic forms of the MSD and NGP but without assuming a particular model of the hidden environment or its influence on the diffusion kernel. As shown in this work, this information is useful in constructing an explicit model of transport dynamics of complex fluid systems; from this explicit model, we can predict or quantitatively understand the time dependence of the displacement distribution. Meanwhile, a model-free analysis of the MSD and NGP based on Eqs. **3a** and **3b** yields accurate and robust quantitative information about the time profiles of

## Diffusion Kernel Correlation and Velocity Autocorrelation Functions

The diffusion kernel correlation, **5**, **v**, and **4**, we obtain **3a** and **3b** into the latter equation, we obtain an exact relation of the DKC to the BCF and the normalized VAF (see *SI Appendix*, Text S6 for the derivation):**6** along with Eq. **5** and *B* for supercooled water at 193 K.

## Ergodicity and Long-Time Limit of Diffusion Kernel Correlation and Non-Gaussian Parameter

For fluidic systems showing terminal Fickian diffusion, **3a** and **3b** and the definition of the NGP (*SI Appendix*, Text S7). This result indicates that the long-time limit NGP value, *SI Appendix*, Fig. S2), which was noted by Odagaki (65) for the glass formation process. There exist transport systems with anomalous diffusion, *SI Appendix*, Text S7). This result was previously reported by Odagaki and Hiwatari (66) on the basis of the so-called coherent-medium approximation, which corresponds to a vanishingly small DKC; i.e., *SI Appendix*, Text S7). These results suggest that the finite value of

## Intrinsic and Extrinsic Disorder

For ergodic fluid systems, the long-time limit value of the product between the MSD and NGP serves as a measure of disorder strength. This disorder strength measure is decomposable into intrinsic and extrinsic disorder. To show this, consider the long-time asymptotic behavior of the MSD and NGP:**7a** can be obtained by substituting the Maclaurin series of **3a** and by taking the inverse Laplace transform of the resulting equation. We present the derivation of Eqs. **7a** and **7b** in *SI Appendix*, Text S7. In Eq. **7a**, *SI Appendix*, Text S8). On the other hand, in Eq. **7b**,

From Eqs. **7a** and **7b**, we obtain the long-time limit value of the product of the MSD and NGP as**8** tells us that disorder strength has two different components, *E* and *SI Appendix*, Fig. S1*E*. On the other hand, we designate

Intrinsic disorder is far less sensitive to the temperature and density of media than extrinsic disorder and can be easily estimated from Eq. **7a**, the asymptotic long-time behavior of the MSD. Extrinsic disorder can be estimated by a direct numerical calculation of *E*); i.e.,

Disorder strength, defined in Eq. **8**, is directly related to the whole-time integration of the BCF; that is, *s* limit of **4**, and from Eqs. **3a** and **3b** (*SI Appendix*, Text S6).

## Model-Based Quantitative Analysis of the MSD and NGP

Intrinsic disorder causes the MSD to deviate from *SI Appendix*, Text S9):*A* and *SI Appendix*, Fig. S1*A*).

The applicability of Eq. **9** to various disordered fluid systems implies a universality in the MSD time profile of disordered fluids, which is decomposable into one unbound-mode dynamic and multiple bound-mode dynamics, comparable to viscoelastic motion of a bead in a polymer network. At short times, a tracer molecule is trapped by the surrounding molecules. This bound state consists of multiple bound modes, each with their own characteristic frequencies. Meanwhile, at long times, a tracer molecule escapes the cage of the surrounding molecules and moves around in the media, repeatedly being caged and escaping the cage. The first term on the right-hand side (R.H.S.) of Eq. **9** accounts for the unbound mode, and the second term accounts for the bound modes. In Eq. **9**, *i*th mode *i*th bound mode and is related to

At all times, temperatures, and densities investigated, Eq. **9** with only two bound modes *A* and *SI Appendix*, Fig. S1*A*) and the experimental results for colloidal beads moving along lipid tubes (*SI Appendix*, Fig. S3) (36). According to Eq. **3a**, the analytic expression of the mean diffusion kernel yielding the MSD given in Eq. **9** can be obtained by*C* shows the mean diffusion kernel, or the VAF, calculated from Eq. **10** with parameter values optimized against MSD data from MD simulation shown in Fig. 2*A* for supercooled water.

The NGP is dependent not only on the mean transport dynamics, **3a** and **3b** yield *SI Appendix*, Fig. S4).

The NGP of disordered fluids is a nonmonotonic function of time with a single peak. According to our model, the NGP quadratically increases with time, *SI Appendix*, Text S10 and Eq. **S10-9**), but decreases with time, **7b**. As shown in Fig. 2*A*, it is only after the NGP peak time that Fickian diffusion emerges. These properties of the NGP are not specific to supercooled water but common across various disordered fluids (9, 12).

It was recently shown that diffusion coefficient fluctuation strongly correlates with string-like cooperative motion in dense fluids (57, 58), which is reportedly related to the NGP peak height (58). We find that the NGP peak height, *SI Appendix*, Fig. S5*D*). This is not a coincidence. We find the NGP peak height has the same value as the relative variance of the diffusion coefficient at the Fickian diffusion onset time or the NGP peak time, *SI Appendix*, Text S11). Both the NGP peak height and the NGP peak time increase with inverse temperature and density (Fig. 2*B* and *SI Appendix*, Fig. S1*B*).

## Explicit Model for Diffusion Kernel

In the previous section, we demonstrated that the time profile of **3a** and **3b** using Eq. **9** for the MSD. To achieve a physical understanding of the time profile of **10**, *i*th bound-mode dynamics, given by *SI Appendix*, Text S12),**11** in the small-*s* regime, *B* and *C*), the bound-mode terms are negligible compared with unbound-mode term, so that the first term on the R.H.S. of Eq. **12** contributes the most to the relaxation of diffusion kernel fluctuation, leaving us with *SI Appendix*, Fig. S5*D*), we can then extract

We note here that the whole-time integration of the diffusion kernel, *SI Appendix*, Fig. S6). The bound-mode contribution to *D*).

## Model-Free Quantitative Analysis of the MSD and NGP

By analyzing the numerical data of the time-dependent MSD and NGP using Eqs. **3a** and **3b**, we can extract the time profiles of the mean diffusion kernel and the DKC without assuming a physical model, such as Eq. **9**. In Fig. 3, we demonstrate this model-free analysis of the MSD and NGP data for supercooled water at 193 K (*Methods*).

According to our simulation, shown in Fig. 3*A*, the MSD of supercooled water exhibits oscillatory behavior with a slight bump and dip between 0.1 ps and 1 ps. This mysterious nonmonotonic oscillation in the MSD time profile of supercooled water was previously reported in the literature (5, 68, 69). We find the nonmonotonic MSD time dependence is unrelated to the finite-size effect and emerges not only under a constant temperature condition but also under the constant energy condition (*SI Appendix*, Fig. S7). The origin of the slight oscillation in the MSD time profile may be attributable to the intermolecular hydrogen-bond stretching vibration in supercooled water, which was previously identified in the quenched normal mode spectrum of the TIP4P/2005 water model at low temperatures (70) (*SI Appendix*, Fig. S8) and may also be the origin of the small oscillatory behavior in the NGP time profile between 0.1 ps and 1 ps (Fig. 3*B*). We find that these oscillations in the MSD and NGP time profiles are absent in liquid water above the melting temperature (Fig. 2*A*) and hard disk fluids (*SI Appendix*, Fig. S1*A*) at any density and cannot be accurately represented by Eqs. **9** and **18**, used for the model-based analysis of the MSD and NGP in the previous section.

The mean diffusion kernel and the diffusion kernel correlation extracted from the model-free analysis of the MSD and NGP transiently display oscillatory behaviors at times around 0.1 ps, which are more complicated than the behavior of their counterparts extracted using Eqs. **9** and **18** of the MSD and NGP (Fig. 3 *C* and *D*). However, at times shorter than 0.01 ps or longer than 0.4 ps, the model-free analysis yields essentially the same results as the model-based analysis. As shown in Fig. 3*D*, *Inset*, both the model-free and model-based methods yield essentially the same result for the long-time DKC or the TCF of the diffusion coefficient fluctuation, quantitatively explainable by the unbound-mode component, or the first term on the R.H.S. of Eq. **12**, only. Consequently, both methods yield the same value for the whole-time integration of the DKC, or extrinsic disorder,

## Microscopic Measurement of the Bound- and Unbound-Mode Components of Diffusion Kernel Correlation

To test the correctness of our results in the previous sections, we perform an alternative, microscopic measurement of the mean diffusion kernel and DKC using MD simulation and compare the results with those obtained in the previous sections. We then show that the bound-mode and unbound-mode transport dynamics, separately embodied in our transport model, clearly manifest, respectively, on the short-time and long-time dependence of the displacement distribution and the spatial volume spanned by the MD simulation trajectories for supercooled water at 193 K.

The mean diffusion kernel and DKC calculated from the MD simulation results of the two- and four-point TCF are found to be in excellent agreement with those extracted from the MSD and NGP. This is demonstrated for an example of supercooled water at 193 K in Figs. 3 and 4.

The short-time DKC, dominantly contributed from the bound-mode component, can be directly calculated using Eq. **6** and the direct MD simulation results of the VAFs; the BCF appearing in Eq. **6** can be calculated from its definition, Eq. **5**. Note that the BCF can also be calculated from the MSD and NGP data, with use of the following relation: *A* for supercooled water at 193 K. When substituted into Eq. **6**, these two BCF time profiles yield essentially the same results for the DKC, which is also in perfect agreement with the DKC extracted from our analysis of the MSD and NGP in the previous section (Fig. 4*D*).

We find that, at long times, the DKC is linearly proportional to the BCF (*SI Appendix*, Text S6):**13** as *D*, *Inset*. It is not easy to calculate the long-time profile of the BCF directly from its definition, Eq. **5**, because of the large computational expense involved in accurately estimating the multipoint VAFs from MD simulation and the 2D integral appearing in Eq. **5**.

An independent estimation of the long-time DKC can be made by using MD simulations to measure the diffusion coefficient fluctuation along each time trace and calculating the TCF of the diffusion coefficient (*Methods*). This is because the DKC becomes the diffusion coefficient at long times, as mentioned earlier. In Fig. 4*C*, for supercooled water at 193 K, we show that **12**, as shown in Fig. 4*C*.

An alternative estimation of the long-time DKC can be made from the time profile of the NGP. The long-time DKC is directly related to the NGP and its time derivatives as follows:**13**. Eq. **14** tells us that the NGP carries the complete information about the long-time relaxation of mobility fluctuation of complex fluids. As shown in Fig. 4*C* for supercooled water at 193 K, the long-time DKC estimated by Eq. **14** quantitatively agrees with the long-time DKC obtained from three other methods, namely, extraction from the MSD and NGP data, using Eq. **6** and MD simulation results of the VAF, and MD simulation of the diffusion coefficient fluctuation and calculation of its TCF.

Bound-mode (unbound-mode) transport dynamics are reflected on the time dependence of the displacement distribution at short (long) times. For supercooled water at 193 K, the displacement distribution broadens rapidly before 1 ps but after this does not greatly change for several picoseconds (Fig. 4*D*). This bound-mode feature of transport dynamics also manifests itself on the time-dependent volume spanned by the MD simulation trajectories of a water molecule. As shown in Fig. 4*E*, the spatial volume spanned by the simulation trajectories rapidly increases with time before 1 ps, but afterward, this trajectory volume tends to saturate to a certain critical value over several picoseconds while the trajectory length continues increasing with time. In contrast, at long-time scales, the displacement distribution and the trajectory volume exhibit unbound-mode dynamics, as demonstrated in Fig. 4 *F* and *G*.

## Quantitative Explanation of Fickian Yet Non-Gaussian Displacement Distribution

Disordered fluids exhibit non-Gaussian diffusion; that is, the displacement distribution is non-Gaussian even at long times where the MSD linearly increases with time (9, 12, 36⇓–38). The displacement distribution of disordered fluids starts as Gaussian with variance given by

To understand the time-dependent relaxation of the non-Gaussian displacement distribution in the Fickian diffusion regime, we need an explicit model of the diffusion coefficient fluctuation for the fluid system in question. In the literature, the diffusion coefficient is often modeled as

The first model assumes that the diffusion coefficient is given by *SI Appendix*, Text S13) (74). For this model, exact analytic expressions of *SI Appendix*, Eq. **S13-3**), where the adjustable parameters are optimized against the diffusion coefficient value, the NGP peak value, and the long-time DKC data (*SI Appendix*, Text S13). The optimized parameter values are presented in Table 1. Using the first model with optimized parameter values, we can now predict the time-dependent relaxation of the non-Gaussian displacement distribution in the Fickian diffusion regime (*SI Appendix*, Text S13). The prediction of this model is in excellent agreement with the MD simulation results for the displacement distribution of supercooled water, as shown in Fig. 5*A*.

In the second model, we model the diffusion coefficient as *SI Appendix*, Eq. **S14-3**). We find this expression provides an excellent quantitative explanation of the experimentally measured displacement distribution of colloidal beads diffusing on lipid tubes reported in ref. 36 (Fig. 5*B*). The optimized parameters of the second model are presented in Table 2. Eq. **12** with optimized parameter values allows us to calculate the time profiles of the DKC for the colloidal bead system (Table 2 legend and *SI Appendix*, Fig. S3).

The displacement distribution approaches a Gaussian distribution only after individual displacement trajectories become statistically equivalent. If individual displacement trajectories are statistically equivalent, the EB parameter proposed by He, Burov, Metzler, and Barkai (67) is linear in *SI Appendix*, Fig. S9). Deviation of the displacement distribution from Gaussian becomes negligible only at times much longer than *SI Appendix*, Fig. S9*B* for supercooled water. On the other hand, *SI Appendix*, Fig. S9*B*). This is due to the fact that the long-time relaxation of the NGP is contributed not only from extrinsic disorder leading to the trajectory-to-trajectory variation in the transport dynamics, but also from intrinsic disorder, whose effects persist even for homogeneous systems with statistically equivalent displacement trajectories (Eq. **7b**). This analysis shows that the long-time tail of

## Discussion

### Main Findings.

We derived a transport equation, Eq. **2**, describing stochastic thermal motion of various complex fluids, which yields exact analytic results, Eqs. **3a** and **3b**, that enable a unified, quantitative explanation of not only the MSD but also the NGP time profiles of various complex fluids (Fig. 2 and *SI Appendix*, Fig. S1). The central dynamic quantity governing transport dynamics of complex fluids is the environment-dependent diffusion kernel. The mean diffusion kernel (MDK) and DKC can be unambiguously extracted from the MSD and NGP time profiles (Fig. 3 and *SI Appendix*, Fig. S1). We also established an exact relationship of the MDK and DKC with the two-point and four-point VAFs (Eqs. **4**–**6**), allowing for alternative, microscopic measurements of the MDK and DKC using MD simulation (Fig. 4 *A*–*C*). DKC is an ideal measure of mobility fluctuation of complex fluids exhibiting non-Fickian diffusion and is simply related to the NGP by Eq. **14**, at long times (Fig. 4*C*).

We constructed a physical model of the environment-coupled diffusion kernel (Eqs. **10**–**12**), composed of one unbound mode and multiple bound modes. This model provides a quantitative explanation of the MSD, NGP, and displacement distribution for various complex fluidic systems (Figs. 2 and 5 and *SI Appendix*, Fig. S1). Our model-based analysis of the frequency spectrum of the VAF suggests that the slight oscillation in supercooled water’s MSD originates from intramolecular hydrogen bond stretching motion. We introduced the notion of intrinsic disorder and extrinsic disorder for complex fluid systems in Eq. **8**, which originate from a finite relaxation time of the mean diffusion kernel and the environment-coupled fluctuation of the diffusion kernel, respectively. We demonstrated a separate estimation of intrinsic and extrinsic disorder for supercooled water (Fig. 2*E*) and dense hard disk fluids (*SI Appendix*, Fig. S1*E*). Extrinsic disorder is more sensitive to temperature and density of complex fluids than intrinsic disorder; extrinsic disorder increases with inverse temperature and density, unless the complex fluids enter a solid-like phase.

### Comparison with Previous Models.

Our model-based analysis of DKC is reminiscent of the memory kernel analyses by Berne, Boon, and Rice (77) and Douglas and Hubbard (78). Our random walk model is a generalization of the CTRW model, to account for environment-coupled fluctuation of transport dynamics. We refer to Shlesinger’s review (79) on the origins and applications of the CTRW model. Our model reduces to the CTRW when *SI Appendix*, Figs. S2*E* and S4). For the CTRW model, the NGP is given by *SI Appendix*, Eq. **S10-4**); *SI Appendix*, Fig. S10 and Text S15).

Our transport equation, Eq. **2**, encompasses the SD model, which accounts for extrinsic disorder but neglects intrinsic disorder. Eq. **2** reduces to the transport equation of the SD model when the diffusion kernel is replaced by the diffusion coefficient. The MSD and NGP of the SD model are given by *SI Appendix*, Text S11), which cannot describe complex fluids with a non-Fickian MSD and initially vanishing NGP.

### Potential Applications and Limitation.

Our transport equation can be extended for complex fluids under an external potential field or for nonergodic fluids such as glass (*SI Appendix*, Texts S2 and S4). It can be further generalized for dynamics of transport-coupled reactions in complex fluids (80⇓–82), which we leave for future research. Our MSD model in Eq. **9** is only approximate in the sense that it cannot capture weak oscillation in the MSD of supercooled water and the asymptotic long-time power-law relaxation of the two-point VAF of dense fluids (83⇓⇓–86). Improving our model to capture these phenomena is another future research topic.

In our transport equation, the diffusion kernel is independent of the absolute position of the tracer particle. However, by applying the projection operator technique (19, 87) to the Liouville equation, one can obtain a similar transport equation (*SI Appendix*, Text S16) but with the diffusion kernel dependent on the absolute position of the tracer particle, potentially important for a system with position-dependent transport dynamics. In most fluid systems, however, thermal motion is independent of the absolute position of the tracer particle.

For a more extensive discussion, see *SI Appendix*, Text S17, which we present at the request of an anonymous reviewer.

## Outlook

The essential feature of our approach to transport dynamics of complex fluids is hidden environmental variables that represent the entire set of dynamic variables affecting transport dynamics of our tracer particles. By accounting for their effects without using an a priori explicit model, this approach enables the extraction of robust, quantitative information about the transport dynamics of complex fluid systems. This information can then be used to construct a more explicit model of the environment-coupled diffusion kernel,

## Methods

### Extraction of Diffusion Kernel Correlation from MSD and NGP.

Here, we present the procedure for extracting the DKC, *SI Appendix*, Eq. **S3-12** in the Laplace domain. From *SI Appendix*, Eq. **S3-12**, we can represent

On the other hand, the NGP,

which can be rearranged with respect to the fourth moment,

The simulation results for the MSD and NGP are well represented by Eq. **9** with two bound modes (*n* = 2) and a linear combination of three or four Gaussian-shaped functions given by

respectively. We perform the best fits of Eqs. **9** and **18** to the simulation results for the MSD and NGP. By substituting the optimized results into Eq. **17**, we obtain the analytic expression of **15**, we obtain the Laplace transform of *t*, we perform the numerical Laplace inversion of Eq. **15** using the Stehfest algorithm (90).

### Generation of Time Traces of Fluctuating Diffusion Coefficient.

Along the *i*th particle trajectory, *C*, *Inset*, the MSD of a TIP4P/2005 water molecule at 193 K becomes fully linear in time beyond roughly 10 ns. From the selected displacements, the time-local MSD, *i*th particle trajectory is calculated. *N* = 20 and several elapsed times around 10 ns are well superimposed on each other (Fig. 4*C*). Here, the elapsed time, *t* as noted in ref. 71. It is also verified that the mean-scaled TCF obtained with *N* = 10 or 30 is essentially the same as the result with *N* = 20. When ^{−6} m^{2}⋅s^{−1} and 1.2, respectively, consistent with ^{−6} m^{2}⋅s^{−1} and

## Acknowledgments

We gratefully acknowledge Professors Mike Shlesinger, Eli Barkai, Ralf Metzler, YounJoon Jung, and Jae-Hyung Jeon for their helpful comments and Mr. Luke Bates for his careful reading of our manuscript. This work was supported by the Creative Research Initiative Project program (2015R1A3A2066497) and the National Research Foundation of Korea Grant (2015R1A2A1A15055664) funded by the Korean government. S.S. also acknowledges the Chung-Ang University Graduate Research Scholarship in 2016.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: jaeyoung{at}cau.ac.kr or jihyunkim{at}cau.ac.kr.

Author contributions: J.S. designed research; S.S., S.J.P., J.S.K., B.J.S., S.L., J.-H.K., and J.S. performed research; S.S., S.J.P., M.K., and J.-H.K. analyzed data; and S.S., J.S.K., B.J.S., S.L., J.-H.K., and J.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1900239116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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- Article
- Abstract
- Transport Equation of Complex Fluids
- Analytic Expressions of the Moments
- Diffusion Kernel Correlation and Velocity Autocorrelation Functions
- Ergodicity and Long-Time Limit of Diffusion Kernel Correlation and Non-Gaussian Parameter
- Intrinsic and Extrinsic Disorder
- Model-Based Quantitative Analysis of the MSD and NGP
- Explicit Model for Diffusion Kernel
- Model-Free Quantitative Analysis of the MSD and NGP
- Microscopic Measurement of the Bound- and Unbound-Mode Components of Diffusion Kernel Correlation
- Quantitative Explanation of Fickian Yet Non-Gaussian Displacement Distribution
- Discussion
- Outlook
- Methods
- Acknowledgments
- Footnotes
- References

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