Emergence of molecular friction in liquids: bridging between the atomisticand hydrodynamic pictures

Arthur V. Straube,1, 2, a) Bartosz G. Kowalik,3 Roland R. Netz,3 and Felix Höfling1, 2, b)1)Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany2)Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany3)Freie Universität Berlin, Department of Physics, Arnimallee 14, 14195 Berlin, Germany

Friction in liquids arises from conservative forces between molecules and atoms. Althoughthe hydrodynamics at the nanoscale is subject of intense research and despite the enormousinterest in the non-Markovian dynamics of single molecules and solutes, the onset of frictionfrom the atomistic scale so far could not be demonstrated. Here, we fill this gap based onfrequency-resolved friction data from high-precision simulations of three prototypical liquids,including water. Combining with rigorous theoretical arguments, we show that friction inliquids emerges abruptly at a characteristic frequency, beyond which viscous liquids appear asnon-dissipative, elastic solids; as a consequence, its origin is non-local in time. Concomitantly,the molecules experience Brownian forces that display persistent correlations and long-lastingmemory. A critical test of the generalised Stokes–Einstein relation, mapping the friction ofsingle molecules to the viscoelastic response of the macroscopic sample, disproves the relationfor Newtonian fluids, but substantiates it exemplarily for water and a moderately supercooledliquid. The employed approach is suitable to yield novel insights into vitrification mechanismsand the intriguing mechanical properties of soft materials.

Molecular friction is a key ingredient for dynamic pro-cesses in fluids: it limits diffusion, governs dissipation, andenables the relaxation towards equilibrium. In a liquid en-vironment, the friction experienced by solvated moleculesand nanoprobes exhibits a delayed response to externalstimuli, indicating non-Markovian dynamics1–4. Suchmemory is found on sub-picosecond up to microsecondscales; it has repercussions on macromolecular transitionrates5–7 and is manifest in the visco-elastic behaviour ofsoft materials8–11. However, the origin of friction fromconservative forces between molecules and atoms remainsas one of the grand challenges of the physics of fluids12–15.

Stokes’s friction law, describing the resistance to asteadily dragged immersed sphere of radius 𝑎, links thefriction 𝜁0 to the (macroscopic) shear viscosity 𝜂0, andthe relation 𝜁0 = 6𝜋𝜂0𝑎 scales down remarkably well tosingle molecules16,17. Stokes’s hydrodynamic treatment18

from 1851 was actually more general and addressed slowoscillatory motions in viscous fluids, motivated by inac-curacies of pendulum clocks caused by air flow (fig. 1a).These predictions of a dynamic friction 𝜁(𝜔) that dependson the oscillation frequency 𝜔 have been interpreted interms of a delayed response, also referred to as hydro-dynamic memory, and have only recently been shown tobe quantitative also for micron-sized particles1,2. In thedomain of microrheology, measurements of 𝜁(𝜔) are usedto infer the mechanical properties of complex fluids19–23.

From the perspective of individual molecules or atoms,fluids are governed by conservative interactions and obeyNewton’s equations of motion, yielding smooth and time-reversible trajectories (fig. 1c). In particular, a single

a)Electronic mail: arthur.straube@fu-berlin.deb)Electronic mail: f.hoefling@fu-berlin.de

molecule is not subject to friction in this picture, andthe mechanism of the required entropy production is farfrom obvious. Macroscopic friction and other transportcoefficients have been linked to microscopic chaos and theLyapunov spectrum of the liquid24–26, yet the connectionof the latter to the corresponding Green–Kubo integrands,or equivalently, to the dynamic friction 𝜁(𝜔), is an openissue25. First-principle theories to friction are hamperedby the fact that liquids are strongly interacting systems.An insightful, formal relation between the many-particleLiouville operator and dissipation spectra was derived inthe seminal works by Zwanzig, Mori, and others27,28, butthe analytic evaluation of the resulting expressions hingeson uncontrolled approximations. For example, a rigorousshort-time expansion of the motion at all orders yieldszero friction (see Methods). Early work on dissipationspectra recognised the importance of exact sum rules29,30,the proposed ad-hoc models, however, violate the sumrules at higher orders. Theoretical progress was made forhard-sphere fluids, where billiard-like collisions generatean instantaneous, Markovian contribution to friction31,thereby rendering the frictionless regime inaccessible.

To gradually bridge between the atomistic and hydro-dynamic regimes, one would ideally like to have a mag-nifying glass that allows for zooming from the slowestto the fastest processes, thus obtaining an increasinglysharper view of the molecular details (fig. 1a). Spectralquantities such as 𝜁(𝜔) can serve this purpose with thefrequency 𝜔 as the control knob. Here, we implementedthis idea for three distinct liquids and have traced thefriction of molecules over wide frequency windows fromfully-developed dissipation all the way down to the non-dissipative regime, revealing sizable variations of 𝜁(𝜔).Such deviations of 𝜁(𝜔) from a constant friction 𝜁0 signifynon-Markovian motion that is widely cast in the gener-

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FIG. 1. Dynamic friction bridges between the hydrodynamic and atomistic pictures of liquids. Panel (a): Apendulum that oscillates in a viscous fluid with frequency 𝜔 experiences a dynamic friction 𝜁(𝜔) as calculated by Stokes (1851)for slow motion. The associated flow pattern (stream lines) leads to hydrodynamic memory of the motion. Magnifying themicroscopic scale, the fluid consists of molecules that obey Newton’s equations, which are non-dissipative and generate smoothtrajectories. Stokes’s result for the pendulum scales down to single molecules, and the function 𝜁(𝜔) provides the bridge betweenthe frictionless (microscopic) and the hydrodynamic (macroscopic) descriptions. Panels (b,c): In liquids, molecules rattle intransient cages formed by their neighbours (brown spheres); the corresponding short-time trajectories are smooth, but chaoticcurves (panel c). The onset of friction is driven by the momentum transfer to the cage, as supported by control simulations ofa single particle (red sphere) in a matrix of pinned particles (brown and yellow). Illustration for a mono-atomic fluid in twodimensions.

alised Langevin equation [Methods, eq. (1)], parametrisedby an associated memory function 𝛾(𝑡). The quantities𝜁(𝜔) and 𝛾(𝑡) are related to each other by a cosine trans-form, but the determination of either of them from datais a formidable challenge, with substantial progress inthe past years32–39. Reaching the high-frequency regimewas precluded so far by practical limitations, e.g., statis-tical noise and insufficient dynamic windows, which wehave overcome here by high-precision simulations and anadvanced data analysis that utilises physical principles.

For liquid water, for a dense Lennard-Jones (LJ) fluidrepresenting a simple, mono-atomic liquid, and for a su-percooled binary mixture serving as a model glass former,we obtained low-noise, cross-validated data for the dy-namic friction 𝜁(𝜔) and the frequency-dependent viscosity𝜂(𝜔), covering up to 3 orders in magnitude and 4 decadesin frequency. Corroborated by these data, we give rig-orous arguments that dissipative processes in molecularliquids are exponentially fast suppressed as frequencyis increased. As a consequence, the liquid’s response ispurely elastic beyond a characteristic frequency 𝜔𝑐, afeature that goes beyond popular models of friction andviscoelasticity. The rapid decay of dissipation spectrasuch as 𝜁(𝜔) is easily shadowed by approximations, andas a modelling constraint it is under-investigated in theexisting microscopic theories of liquids. We show furtherthat the high-frequency properties are reproduced by themotion of a single particle in an immobile cage, and ourwork brings up new, concrete questions on the relationbetween microscopic chaos and transport coefficients. Fi-nally, having data available for both 𝜁(𝜔) and 𝜂(𝜔), wetest the connection between microscopic friction and themacroscopic mechanical properties of complex fluids, pos-tulated by the generalised Stokes–Einstein relation. Thelatter is found to either fail completely (monoatomic liq-uid), serve as a qualitative description (water), or being

a nearly quantitative relation (supercooled liquid).

RESULTS

Instead of observing the response to an oscillatory force,we recorded the Brownian position fluctuations in equi-librium and link them to the friction, taking advantageof a fluctuation dissipation relation. For the three liquidsunder investigation, we carried out molecular dynamicssimulations to compute the mean-square displacement(MSD). Using the MSD as sole input, we estimated boththe dynamic friction 𝜁(𝜔) and the memory function 𝛾(𝑡)in an ansatz-free approach, following two complementaryroutes that allow for cross-validation (see fig. 2 and Meth-ods). The first route invokes complex analysis and isbased on the Fourier–Laplace transform of correlationfunctions, sampled on a sparse time grid (“adapted Filon

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FIG. 2. Flow chart of the data analysis. Along the upperroute, one starts from the mean-square displacement (MSD)and computes the generalised mobility 𝒴(𝜔) by numericaldifferentiation and a suitable Fourier transform (adapted Filonalgorithm); the dynamic friction 𝜁(𝜔) follows via eq. (7). AFourier backtransform then yields the memory function 𝛾(𝑡).The latter can be obtained more directly along the lower route,which is based on the velocity autocorrelation function (VACF)𝑍(𝑡) and employs a deconvolution in time domain.

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FIG. 3. Dynamic friction and generalised mobility of three prototypical liquids. Columns show simulations resultsfor a mono-atomic (LJ) fluid, liquid water, and a supercooled liquid, obtained from the data analysis depicted in fig. 2, withthe MSD (not shown) as initial input. Panels (a)–(c): The dynamic friction (red) is the real part of the memory kernel 𝛾(𝜔),as computed from eq. (6). It interpolates between the hydrodynamic value 𝜁0/𝑚 (horizontal line) and a rapid decrease tozero at high frequencies (𝜔 ≫ 𝜔𝑐); thin red lines mark exponential decays, ∼ e−𝜔/𝜔𝑐 , to guide the eye. In case of the LJ fluid(panel a), 𝜁(𝜔) is consistent with Stokes’s small-𝜔 asymptote [grey line, eq. (18)], with parameters taken from a fit to thelong-time tail of 𝑍(𝑡) (inset of fig. 4a). The elastic response Im 𝛾(𝜔) (turquoise) exhibits a local maximum at high frequencies,defining the characteristic frequency 𝜔𝑐 (vertical lines), and follows the high-frequency asymptote [eq. (15), grey line], with allparameters fixed by short-time fits to the MSD data. The frequency 𝜔𝑐 is close to, but different from the Einstein frequency 𝜔0.Panels (d)–(f): Numerical results for the generalised mobility 𝒴(𝜔), which describes the response to an applied small, oscillatoryforce (see Methods). The real part Re 𝒴(𝜔) (green) approaches the hydrodynamic mobility 1/𝜁0 (horizontal line) as 𝜔 → 0 andvanishes rapidly at large frequencies. The imaginary part (yellow) encodes the elastic response, which has a resonance near 𝜔𝑐

(vertical line); at larger frequencies, the data match with theoretical predictions [grey lines, eq. (14)].

algorithm”). Second and independently, we computedthe antiderivative of 𝛾(𝑡) using a stable deconvolutiontechnique for uniform time grids.

Molecular friction in liquids emerges rapidly.Although all three liquids display rather different dy-namics, leaving distinct fingerprints in their friction spec-tra, their high-frequency behaviours of 𝜁(𝜔) share signifi-cant similarities (fig. 3a–c). Most remarkably, the datademonstrate that beyond a liquid-characteristic frequency,𝜔 & 𝜔𝑐, the friction 𝜁(𝜔) goes exponentially fast to zero.Such a rapid spectral variation has to be contrasted tothe typical algebraic peaks, i.e., the Lorentz–Debye shape,and we argue in the following that our finding is genericfor molecular fluids. Upon decreasing frequency, the on-set of friction is followed by liquid-specific behaviour overseveral decades in time until the hydrodynamic value 𝜁0is established: in water, our results for the friction 𝜁(𝜔)exhibit a local maximum at 𝜔/2𝜋 ≈ 5 THz, followed bya slow increase towards the limiting value 𝜁0, which isreached near frequencies of 0.1 THz. For the LJ fluid,𝜁(𝜔) varies more smoothly with a global maximum at an

intermediate frequency, and 𝜁0 is approached slowly fromabove in accord with the hydrodynamic square-root singu-larity [eq. (18)], essentially calculated by Stokes already18.On theoretical grounds, this feature of 𝜁(𝜔) is genericfor all liquids, yet it is suppressed in our data for theother two liquids due to a small prefactor. In the super-cooled liquid, we observe a scale separation by 3 dynamicdecades of (i) the rapid onset of friction and (ii) the slowemergence of the hydrodynamic limit. The second processis associated with cage relaxation, strongly delayed in theglassy state, which suggests that the driving mechanismbehind the onset of friction is not in the structural re-laxation of the fluid. Close to the glass transition, thesmall-frequency friction 𝜁0 is governed by self-similar re-laxation processes and obeys asymptotic scaling laws40,41;the magnitude (prefactor) of these laws, however, is setat high frequencies.

Friction depends on the coupling of fast andslow processes. The obtained data of 𝜁(𝜔) cover thefull range of the dynamic response, thereby connectingphysics at different scales. Key features will be ratio-

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FIG. 4. Velocity autocorrelators and corresponding memory functions for the three investigated liquids. Columnsshow results for a mono-atomic (LJ) fluid, liquid water, and a supercooled liquid. Panels (a)–(c): simulation results for the VACF(brown symbols) display a parabolic decrease at short times (grey lines) and power law decays at long times (insets, negativevalues are dotted). The VACF is obtained from the second derivative of the MSD with respect to lag time. Panels (d)–(f):The memory function 𝛾(𝑡) encodes the autocorrelation of Brownian forces on the molecules [eq. (2)]. For each liquid, 𝛾(𝑡) wascomputed from a cosine transform of 𝜁(𝜔) [blue line, eq. (9)] with input data from fig. 3a–c] and is compared to the deconvolutionresults in time domain [orange line, eq. (23)]. The data follow the predicted short-time decay eq. (16) (grey line) and exhibitpower-law decays consistent with eq. (19) (insets), preceded by an ultra-slow decay in case of the supercooled liquid [panel (f)].

nalised by tracing their origins to the dynamics of thefluid particles at short and long times (going backwards infig. 2). The relevant properties are prominently visible inthe second derivative of the MSD, the velocity autocorrela-tion function (VACF), 𝑍(𝑡) := 𝜕2

𝑡 MSD(𝑡)/6, after numeri-cal differentiation with respect to the lag time 𝑡 (fig. 4a–c).The following should be contrasted to Ornstein’s model forBrownian motion, employing a single exponential decayof velocity correlations, 𝑍(𝑡) ≈ 𝑣2

th exp(−𝜁0𝑡/𝑚), whichimplies a constant (Markovian) friction, 𝜁(𝜔) ≈ 𝜁0; by 𝑣thwe denote the thermal velocity, and 𝑚 is the molecularmass. As a distinct feature of molecular fluids, obeyingNewton’s equations, the VACF’s true short-time decayis parabola-shaped17, 𝑍(𝑡 → 0) ≃ 𝑣2

th(︀1 − 𝜔2

0𝑡2/2)︀, intro-

ducing the Einstein frequency 𝜔0. For dense liquids, theVACF, after a sign change, generically displays a regimeof anti-correlations, which reflect the transient caging byneighbouring molecules (fig. 1b). For water and the super-cooled liquid, these anti-correlations relax slowly with anintermediate power-law decay, 𝑍(𝑡) ∼ −𝑡−5/2 (insets offig. 4b,c); such a decay was observed earlier in supercooledliquids42,43 and it is a well-established long-time tail incolloidal suspensions44–46 and for diffusion in an arrested,disordered environment47,48.

The famous long-time tail encoding hydrodynamicmemory17,49,50, 𝑍(𝑡 → ∞) ∼ 𝑡−3/2, is clearly developedin our data for the LJ fluid after another sign change

(inset of fig. 4a), and in this situation, Stokes’s hydro-dynamics describes the slow motion of single molecules(fig. 3a). For the other two liquids studied, the tail isnot visible in our data due to a small prefactor, whichfollowing mode-coupling arguments decreases as eitherviscosity or diffusivity increases17.

The dynamic friction is closely linked to the complex-valued, generalised mobility 𝒴(𝜔) via 𝜁(𝜔) = Re[𝒴(𝜔)−1].This mobility encodes the response to a small, oscillatoryforce and is accessible in, e.g., scattering experiments51.Here, we computed 𝒴(𝜔) from the VACF upon employ-ing a fluctuation dissipation relation (see Methods andfig. 3d–f). Our data reveal a generic, rapid decrease ofthe dissipative part, 𝒴 ′(𝜔) := Re 𝒴(𝜔), upon increasingfrequency towards the microscopic regime, 𝜔 ≫ 𝜔𝑐; con-comitantly, the elastic response, 𝒴 ′′(𝜔) := Im 𝒴(𝜔), has aresonance near 𝜔𝑐 due to vibrational motion of moleculesin their cages. In the low-frequency limit, the reciprocalof the macroscopic friction is recovered, 𝒴(𝜔 → 0) = 1/𝜁0.In the examples studied, both regimes are separated byat least two decades in time, which show material-specificfeatures: the mobility of water molecules is 2.5-fold en-hanced over its macroscopic value near 𝜔/2𝜋 ≈ 1.3 THz;similarly, a factor of 30 is observed for the supercooled liq-uid. At variance, the hydrodynamic long-time tail, as forthe LJ liquid, demands 𝒴 ′(𝜔 → 0) to be approached frombelow. The interplay of slow and fast processes enters the

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FIG. 5. Theoretical model justifies exponentially fast onset of friction and persistent memory. Exact results (greysolid lines) for the VACF model of eq. (20), combining the smoothness and time-reversal symmetry of molecular autocorrelationfunctions with a long-time tail. As a sensitive test of our numerical procedures, symbols show numerical results from the MSDas input, sparsely sampled on a geometrically spaced grid. The panels show (a) the complex-valued, generalised mobility 𝒴(𝜔),(b) the dynamic friction 𝜁(𝜔) and its elastic counterpart Im 𝛾(𝜔), and (c) the memory function 𝛾(𝑡) in time domain. The latterinherits the long-time tail ∼ 𝑡−2 from the VACF, but of opposite sign (inset). The tail induces non-analytic behaviour of 𝜁(𝜔)at small frequencies, which crosses over to ∼ 𝜔2 due to the smooth, exponential cutoff of the Fourier integrals (inset of panel b).Same colour code as in figs. 3 and 4.

response 𝒴(𝜔), and thus the dynamic friction 𝜁(𝜔), at allfrequencies, which is borne out by the Kramers–Kronigrelations. In particular, singular behaviour of 𝒴 ′′(𝜔) as𝜔 → 0 influences the detailed onset of friction at largefrequencies.

Friction is non-local in time. The rapid decreaseof 𝒴 ′(𝜔) is mathematically justified from the short-timeproperties of the VACF. Physical molecular trajectories,being solutions to Newton’s equations, are smooth andyield a smooth function 𝑍(𝑡); in particular, all derivatives𝑍(𝑛)(𝑡) exist at 𝑡 = 0 and are finite. Thus, invokingexact sum rules [eq. (10)], all moments of the spectrum𝒴 ′(𝜔) are finite, which requires an exponentially fastdecay as 𝜔 → ∞. (This is a special case of a more gen-eral characterisation of exponentially decaying probabilitymeasures52.) Combining with the large-𝜔 asymptote ofthe imaginary part, 𝒴 ′′(𝜔) ≃ 1/𝑚𝜔 ≫ 𝒴 ′(𝜔) (see Meth-ods) and using 𝜁(𝜔) = 𝒴 ′(𝜔)/

[︀𝒴 ′(𝜔)2 + 𝒴 ′′(𝜔)2]︀

provesthat 𝜁(𝜔) ≃ (𝑚𝜔)2 𝒴 ′(𝜔) as 𝜔 → ∞ and thus an exponen-tially fast suppression of the friction. We stress furtherthat such behaviour is not contained in representationsof 𝜁(𝜔) as a truncated continued fraction17,53.

It is tempting to use a systematic short-time expansionof 𝑍(𝑡) to predict the large-frequency behaviour of thefriction. However, 𝑍(𝑡) being an even function due totime-reversal symmetry in equilibrium renders the large-𝜔asymptotes zero, 𝒴 ′(𝜔) ≡ 0 and thus 𝜁(𝜔) ≡ 0, even ifthe complete Taylor series of 𝑍(𝑡) in 𝑡 = 0 was known (seeMethods). Note that 𝒴 ′′(𝜔) and the elastic counterpartof 𝜁(𝜔) are well captured by such an expansion (fig. 3d–i).This observation underlines that, on all scales, frictionemerges as a phenomenon that is non-local in time, i.e.,it cannot be anticipated from the local behaviour of themolecular trajectories.

Our numerical and theoretical findings are supportedby an analytically solvable example. The choice 𝑍(𝑡) =

𝑣2th

[︀1 + (𝑡/𝜏)2]︀−1 combines the smoothness and time

reversal symmetry of molecular autocorrelation func-tions with a long-time tail. It yields an exponentialdecay of the mobility, 𝒴 ′(𝜔) = (𝜋𝜏/2𝑚) e−|𝜔𝜏 | (dissi-pative part), and hence a rapid suppression of friction,𝜁(𝜔 → ∞) ∼ (𝜔𝜏)2e−𝜔𝜏 , as demanded above (see Meth-ods and fig. 5).

Irreversible momentum transfer drives the on-set of friction. A pressing question is about the physi-cal mechanism that generates the onset of friction. Moti-vated by our results for the supercooled liquid (fig. 3c),we performed a control simulation for the LJ fluid withthe structural relaxation switched off by pinning all par-ticles but one. The rattling motion in such a frozen-incage (fig. 2) experiences a dynamic friction that closelyresembles our generic findings for 𝜁(𝜔) at high frequencies,𝜔 & 𝜔𝑐 (fig. 6). Upon decreasing frequency from 𝜔𝑐 tozero, the two dynamics deviate strongly as is most evidentin the elastic response: whereas Im 𝛾(𝜔) decreases for theunconstrained fluid and exhibits a zero crossing enforcedby hydrodynamics [eq. (17) and fig. 3a], it remains positivefor the pinned case and grows as Im 𝛾(𝜔 → 0) ∼ 1/𝜔, rem-iniscent of what one obtains for an Ornstein–Uhlenbeck(OU) particle in a harmonic trap1. For even smallerfrequencies, 𝜔 . 𝜏−1

LJ , the dissipation diverges too, ap-proximately as 𝜁(𝜔 → 0) ∼ 𝜔1/2, which we attribute tothe irregular shape of the confining cages; for the OUmodel with harmonic confinement, 𝜁(𝜔) ≃ const. At highfrequencies, however, the confinement is not relevant, andwe conclude that it is the fast, yet irreversible momentumtransfer to neighbouring molecules that drives the onsetof friction. This is corroborated by the observation thatinstantaneous momentum exchange implies a non-zerolimit, 𝜁(𝜔 → ∞) > 0, as in the case of hard spheres31.The fact that dissipation is linked to trajectories for whichthe time-reversed path is extremely improbable leads usto speculate that the onset frequency 𝜔𝑐 is intimately

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FIG. 6. Friction emerges due to rattling motion inimmobile cages. Dynamic friction 𝜁(𝜔) (red symbols) ob-tained in a control simulation of a single particle moving ina frozen-in cage formed by neighbouring particles (fig. 1b).The setup was created by pinning the particles of the LJ fluidexcept for one; results correspond to an ensemble average over106 typical cages. The imaginary part of the memory kernel𝛾(𝜔) (turquoise) ties in with the high-frequency prediction(gray). Dashed lines show the results for the unconstrainedfluid at the same conditions for comparison (fig. 3a).

related to the largest Lyapunov exponent 𝜆max of thefluid, which is close to, but different from the Einsteinfrequency 𝜔0 (ref. 26).

Dynamic friction implies intricate memory ofBrownian motion. Within the framework of the gen-eralised Langevin equation [eq. (1)], momentum relaxationis governed by a memory function 𝛾(𝑡) that is fully deter-mined by 𝜁(𝜔) [eqs. (7) and (9)]. At the same time, 𝛾(𝑡) isalso the autocorrelator of Brownian, random forces on themolecules, up to a constant prefactor, and 𝜁(𝜔) encodesthe corresponding spectrum [eq. (2)]. Within Ornstein’sidealised model of Brownian motion, one assumes inde-pendent Brownian forces, implying a flat, white spectrum,𝜁(𝜔) ≈ 𝜁0, and a delta-peaked memory function 𝛾(𝑡). Formolecular liquids, however, the memory functions displaya universal parabola-shaped short-time decay [fig. 4d–fand eq. (16)]. For water and the LJ fluid, the short-time regime of 𝛾(𝑡) is followed by oscillatory behaviourincluding sign changes and, finally, different power-lawdecays for the two liquids encoding different physics (in-sets of fig. 4d,e, also see fig. 5c); generally, power-lawtails of the memory function are directly inherited fromthe VACF without a change of exponent [eq. (19)]. Forthe supercooled liquid, 𝛾(𝑡) remains positive and exhibitsthe onset of a plateau (near 𝑡 ≈ 0.3𝜏LJ), which decayslogarithmically slowly over 2 decades in time (fig. 4f).From the modelling perspective, it is desirable to approx-imate the memory functions such that the initial decay,the typical persistence time, and the integral of 𝛾(𝑡) arereproduced, the latter yielding 𝜁0 [eq. (13)]. For all three

liquids, the complexity and the long-lived nature of thememory, however, preclude simple models of 𝛾(𝑡) such asthe superposition of few exponential decays. The quan-titative knowledge of 𝜁(𝜔), as obtained here, paves theway for more favourable approximations of memory in thefrequency domain, which can yield mathematically andphysically consistent interpolations of Brownian motionfrom the fastest to the slowest scales.

DISCUSSION

Relation to viscosity and elasticity of liquids.Going beyond the dynamics of single molecules and theirfriction, the relation to the visco-elastic properties of com-plex fluids is of ongoing interest for the physics of poly-mers, living cells, and the glass transition. The potentiallytight coupling between single-particle and collective re-sponses was phrased as an ad hoc extension of Stokes’s fric-tion law to the frequency domain, 𝜁(𝜔) = 6𝜋 Re[𝜂(𝜔)]𝑎, re-ferred to as generalised Stokes–Einstein relation (GSER),which has found wide applications in the context of mi-crorheology experiments19–22. It links the dynamic fric-tion 𝜁(𝜔) of a probe particle to the dynamic shear modu-lus, �̂�(𝜔) = −i𝜔𝜂(𝜔), a complex-valued function encodingthe stress response of the macroscopic fluid sample to asmall, oscillatory shear strain. The generalised viscos-ity 𝜂(𝜔) tends to the hydrodynamic shear viscosity 𝜂0as 𝜔 → 0, with Re 𝜂(𝜔) representing the spectrum ofshear stress fluctuations (up to a constant factor) by afluctuation dissipation relation. Thus, if the GSER holdsthe single-particle memory 𝛾(𝑡) is proportional to theautocorrelator of shear stresses, which means that theBrownian force on the particle and the fluctuation of theshear stress are statistically equivalent variables.

A critical assessment of the validity of the GSER ispermitted by comparing our data for 𝜁(𝜔) to results for𝜂(𝜔), calculated within the same simulations (see fig. 7and Methods). Generically, the dissipative part, Re 𝜂(𝜔),decays exponentially fast as 𝜔 → ∞, which is requiredby analogous arguments as given for 𝜁(𝜔) and 𝒴 ′(𝜔). Forhigh frequencies, only the imaginary part survives due toits slow decay, 𝜂(𝜔) ≃ 𝐺∞/(−i𝜔), inducing a non-zero andreal-valued modulus, �̂�(𝜔 ≫ 𝜔𝑐) ≈ 𝐺∞ > 0. Therefore,our data clearly demonstrate that, indeed, liquids respondto high-frequency shear like a non-dissipative, elastic solidas put forward by the classical work of Frenkel54. However,the attempt to predict the elastic modulus 𝐺∞ fromthe vibrational motion of molecules in their cages, byvirtue of the GSER, would considerably overestimate themodulus by factors of ≈ 2 for the three liquids studied(fig. 7e–f). In passing, we note that Maxwell’s model forviscoelasticity17,53,54, 𝜂(𝜔) = 𝐺∞𝜏/(1 − i𝜔𝜏𝑀 ) with somerelaxation time 𝜏𝑀 , breaks down at high frequencies asit implies a slowly decaying real part, Re 𝜂(𝜔 → ∞) ∼𝜔−2, in sharp contrast to exact sum rules29 and to theexponentially fast decay for molecular liquids. Therefore,treatments of sound-like, elastic waves on the footing of

7

10�2

10�1

100

O�.!/=�0

Im O�.!/

Re O�.!/

amono-atomic fluid

10�4

10�3

10�2

10�1

100

Im O�.!/=4

Re O�.!/bwater

10�4

10�3

10�2

10�1

100Im O�.!/ � 3

Re O�.!/

csupercooled liquid

10�2 10�1 100 101 102

!�LJ

0

1

2

3

GS

ER

ratio

d

10�2 10�1 100 101 102

!=2� [THz]

012345e

10�2 10�1 100 101 102

!�LJ

0

1

2

3

f

FIG. 7. Test of the generalised Stokes–Einstein relation (GSER). Panels (a–c): The generalised viscosity Re 𝜂(𝜔)(violet symbols) of the three liquids under investigation (columns) is compared to the GSER prediction 𝜁(𝜔)/6𝜋𝑎 (red dashedlines), based on the dynamic friction data of single molecules (fig. 3a,c), and correspondingly for the imaginary counterparts ofthe elastic responses (orange symbols and turquoise dashed lines). The effective particle radius 𝑎 for each liquid is chosen suchthat the viscosity and friction curves coincide at 𝜔 = 0. The pink solid line in panel (a) is an empirical fit of a compressedexponential, Re 𝜂(𝜔) ≃ 𝜂0 exp(−(𝜔/𝜔𝜂)𝛽) with 𝛽 = 1.29 and 𝜔𝜂 = 1.19 𝜔0. In panels (b,c), the data for the elastic responses areshifted by the indicated factors for clarity. Panels (d–f): The GSER is tested by plotting the ratios 𝜁(𝜔)/6𝜋[Re 𝜂(𝜔)]𝑎 (violet)and 𝑚[Im 𝛾(𝜔)]/6𝜋[Im 𝜂(𝜔)]𝑎 (orange); deviations from unity quantify the GSER violation.

this and similar models appear incomplete.The passage from the elastic to the viscous limit oc-

curs upon decreasing frequency, leading in case of the LJfluid to a monotonic increase of Re 𝜂(𝜔), which, empiri-cally, follows a compressed exponential over almost thefull frequency domain (fig. 7a). In particular, 𝜂(𝜔) ≈ 𝜂0is constant for 𝜔 . 2𝜏−1

LJ , which defines the hydrody-namic regime. For these frequencies, the single-moleculeresponse is very well described by Stokes’s dynamic fric-tion [eq. (18)], making the GSER violation apparent forNewtonian fluids, for which 𝜂(𝜔) = 𝜂0. It is evidencedfurther by the dissimilarity of the elastic parts, Im 𝛾(𝜔)and Im 𝜂(𝜔).

For water, the viscosity and friction spectra share sim-ilar features and coincide fairly well (fig. 7b), includingthe elastic parts. Thus, the GSER serves as a reasonablequalitative description, in particular for frequencies be-low ≈ 2 THz, i.e., slower than the vibrations of the firsthydration shell. A detailed analysis of the visco-elasticspectrum of water can be found in ref. 55.

In supercooled liquids, the Stokes–Einstein relation formolecules (i.e., the GSER for 𝜔 → 0) holds in the pres-ence of a huge variation of the viscosity. In particular, theratio 𝜁0/𝜂0 is observed to be constant over a wide tem-perature range—in line with the mode-coupling theory ofthe idealised glass transition41. At very low temperatures,

however, marked deviations from the Stokes–Einstein rela-tion (mostly studied for 𝜔 → 0) have received considerableattention as they signify the opening of additional relax-ation channels not included in the standard theory56–60.For the moderately supercooled liquid studied here ex-emplarily, both viscous and elastic responses satisfy theGSER over a wide frequency window (fig. 7c). Notably,the elastic components collapse almost perfectly in thiscase, Im 𝛾(𝜔) ∼ Im 𝜂(𝜔), which we attribute to the same(apparent) power law scaling, ≈ 𝜔−0.75, at intermedi-ate frequencies. Yet, the collective response lacks theelastic peak of Im 𝛾(𝜔) at 𝜔𝑐, causing the breakdown ofthe GSER at large frequencies. This suggests that a fu-ture frequency-resolved study of systematic deviationsfrom the GSER upon further supercooling can clarify theseparate contributions of fast and slow processes to thedecoupling of diffusion and viscosity (“Stokes–Einsteinviolation”) close to the glass transition temperature.

Further perspectives. Molecular friction in liquidsarises from a complex interplay of processes on disparatetime scales, and the large variability of 𝜁(𝜔) over ordersof magnitude reveals the strongly non-Markovian natureof Brownian motion in liquid environments, with far-reaching implications for nanoscale processes. Examplesare as diverse as reaction rates and barrier crossings in

8

macromolecular dynamics5–7 and flows near liquid–solidinterfaces45,61–63; the ability to quantify the correspondingmemory is vital for their realistic descriptions.

Beyond that, the finding of a generic drop of 𝜁(𝜔)at a large, liquid-specific frequency 𝜔𝑐 marks the rapidonset of friction, which we attribute to the momentumtransfer to neighbouring molecules. These results refinethe fundamental question on a quantitative link betweenfriction and microscopic chaos: Whether and how doesthe frequency-dependence of transport coefficients relateto the Lyapunov spectrum of the liquid?24,25

From a numerical point of view, our ansatz-free ap-proach has immediate applications to and extends cur-rent methods38,39 for the analysis of high-resolution mi-crorheology data1–3,22, which involves deducing frequency-dependent moduli from the displacement statistics alongthe same lines as done here for the dynamic friction. Rely-ing merely on the existence of a steady state [cf. eq. (12)],the developed methodology is not limited to friction, butcan be transferred to the analysis of non-Markovian timeseries from simulation and experiment. It finds novel usesin, e.g., the anomalous diffusion within living cells11 andthe kinetics of chemical reactions64. It opens a promisingavenue for research on the migration of malignant cells intissue65 and on predictive stochastic models of financialmarket66 and geographic67 data.

1T. Franosch, M. Grimm, M. Belushkin, F. Mor, G. Foffi, L. Forró,and S. Jeney, “Resonances arising from hydrodynamic memoryin Brownian motion,” Nature 478, 8–11 (2011).

2S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen,“Observation of Brownian motion in liquids at short times: In-stantaneous velocity and memory loss,” Science 343, 1493–1496(2014).

3J. Berner, B. Müller, J. R. Gomez-Solano, M. Krüger, andC. Bechinger, “Oscillating modes of driven colloids in overdampedsystems,” Nat. Commun. 9, 999 (2018).

4J. O. Daldrop, B. G. Kowalik, and R. R. Netz, “External potentialmodifies friction of molecular solutes in water,” Phys. Rev. X 7,041065 (2017).

5T. Guérin, N. Levernier, O. Bénichou, and R. Voituriez, “Meanfirst-passage times of non-Markovian random walkers in confine-ment,” Nature 534, 356 (2016).

6D. de Sancho, A. Sirur, and R. B. Best, “Molecular origins ofinternal friction effects on protein-folding rates,” Nat. Commun.5, 4307 (2014).

7J. O. Daldrop, J. Kappler, F. N. Brünig, and R. R. Netz, “Butanedihedral angle dynamics in water is dominated by internal friction,”Proc. Natl. Acad. Sci. U. S. A. 115, 5169–5174 (2018).

8P. Sollich, F. Lequeux, P. Hébraud, and M. E. Cates, “Rheologyof soft glassy materials,” Phys. Rev. Lett. 78, 2020–2023 (1997).

9D. Mizuno, C. Tardin, C. F. Schmidt, and F. C. MacKintosh,“Nonequilibrium mechanics of active cytoskeletal networks,” Sci-ence 315, 370–373 (2007).

10D. Winter, J. Horbach, P. Virnau, and K. Binder, “Activenonlinear microrheology in a glass-forming Yukawa fluid,” Phys.Rev. Lett. 108, 028303 (2012).

11F. Höfling and T. Franosch, “Anomalous transport in the crowdedworld of biological cells,” Rep. Prog. Phys. 76, 046602 (2013).

12E. Secchi, S. Marbach, A. Niguès, D. Stein, A. Siria, and L. Boc-quet, “Massive radius-dependent flow slippage in carbon nan-otubes,” Nature 537, 210–213 (2016).

13F. Perakis, G. Camisasca, T. J. Lane, A. Späh, K. T. Wikfeldt,J. A. Sellberg, F. Lehmkühler, H. Pathak, K. H. Kim, K. Amann-Winkel, S. Schreck, S. Song, T. Sato, M. Sikorski, A. Eilert,

T. McQueen, H. Ogasawara, D. Nordlund, W. Roseker, J. Koralek,S. Nelson, P. Hart, R. Alonso-Mori, Y. Feng, D. Zhu, A. Robert,G. Grübel, L. G. M. Pettersson, and A. Nilsson, “Coherent X-raysreveal the influence of cage effects on ultrafast water dynamics,”Nat. Commun. 9, 1917 (2018).

14A. Bylinskii, D. Gangloff, and V. Vuletić, “Tuning friction atom-by-atom in an ion-crystal simulator,” Science 348, 1115–1118(2015).

15P. Gaspard, M. E. Briggs, M. K. Francis, J. V. Sengers, R. W.Gammon, J. R. Dorfman, and R. V. Calabrese, “Experimentalevidence for microscopic chaos,” Nature 394, 865–868 (1998).

16A. Bartsch, K. Rätzke, A. Meyer, and F. Faupel, “Dynamicarrest in multicomponent glass-forming alloys,” Phys. Rev. Lett.104, 195901 (2010).

17J.-P. Hansen and I. McDonald, Theory of Simple Liquids, 3rd ed.(Academic Press, Amsterdam, 2006).

18G. G. Stokes, “On the effect of the internal friction of fluids onthe motion of a pendulum,” Trans. Cambridge Philos. Soc. 9, 8(1851).

19T. G. Mason and D. A. Weitz, “Optical measurements offrequency-dependent linear viscoelastic moduli of complex fluids,”Phys. Rev. Lett. 74, 1250–1253 (1995).

20F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, andC. F. Schmidt, “Microscopic viscoelasticity: Shear moduli of softmaterials determined from thermal fluctuations,” Phys. Rev. Lett.79, 3286–3289 (1997).

21T. M. Squires and T. G. Mason, “Fluid mechanics of microrheol-ogy,” Annu. Rev. Fluid Mech. 42, 413–438 (2010).

22T. A. Waigh, “Advances in the microrheology of complex fluids,”Rep. Prog. Phys. 79, 074601 (2016).

23A. Rigato, A. Miyagi, S. Scheuring, and F. Rico, “High-frequencymicrorheology reveals cytoskeleton dynamics in living cells,” Nat.Phys. 13, 771–775 (2017).

24R. Dorfman, An Introduction to Chaos in Nonequilibrium Statis-tical Mechanics (Cambridge University Press, 2003).

25E. G. D. Cohen, “Transport coefficients and Lyapunov exponents,”Physica A 213, 293–314 (1995).

26H. A. Posch and W. G. Hoover, “Lyapunov instability of denseLennard-Jones fluids,” Phys. Rev. A 38, 473–482 (1988).

27R. Zwanzig, “Time-correlation functions and transport coefficientsin statistical mechanics,” Annu. Rev. Phys. Chem. 16, 67–102(1965).

28H. Mori, “Transport, collective motion, and Brownian motion,”Prog. Theor. Phys. 33, 423–455 (1965).

29D. Forster, P. C. Martin, and S. Yip, “Moment method approxi-mation for the viscosity of simple liquids: Application to argon,”Phys. Rev. 170, 160–163 (1968).

30N. K. Ailawadi, A. Rahman, and R. Zwanzig, “Generalizedhydrodynamics and analysis of current correlation functions,”Phys. Rev. A 4, 1616–1625 (1971).

31L. Bocquet, J. Piasecki, and J.-P. Hansen, “On the Brownianmotion of a massive sphere suspended in a hard-sphere fluid. I.Multiple-time-scale analysis and microscopic expression for thefriction coefficient,” J. Stat. Phys. 76, 505–526 (1994).

32H. K. Shin, C. Kim, P. Talkner, and E. K. Lee, “Brownian motionfrom molecular dynamics,” Chem. Phys. 375, 316 – 326 (2010).

33F. Gottwald, S. Karsten, S. D. Ivanov, and O. Kühn, “Parametriz-ing linear generalized Langevin dynamics from explicit moleculardynamics simulations,” J. Chem. Phys. 142, 244110 (2015).

34D. Lesnicki, R. Vuilleumier, A. Carof, and B. Rotenberg, “Molec-ular hydrodynamics from memory kernels,” Phys. Rev. Lett. 116,147804 (2016).

35G. Jung, M. Hanke, and F. Schmid, “Iterative reconstructionof memory kernels,” J. Chem. Theory Comput. 13, 2481–2488(2017).

36H. Meyer, T. Voigtmann, and T. Schilling, “On the non-stationarygeneralized Langevin equation,” J. Chem. Phys. 147, 214110(2017).

37B. Kowalik, J. O. Daldrop, J. Kappler, J. C. F. Schulz, A. Schlaich,and R. R. Netz, “Memory-kernel extraction for different molecular

9

solutes in solvents of varying viscosity in confinement,” Phys. Rev.E 100, 012126 (2019).

38M. Tassieri, R. M. L. Evans, R. L. Warren, N. J. Bailey, and J. M.Cooper, “Microrheology with optical tweezers: data analysis,”New J. Phys. 14, 115032 (2012).

39K. Nishi, M. L. Kilfoil, C. F. Schmidt, and F. C. MacKintosh, “Asymmetrical method to obtain shear moduli from microrheology,”Soft Matter 14, 3716–3723 (2018).

40M. Fuchs, W. Götze, and M. R. Mayr, “Asymptotic laws fortagged-particle motion in glassy systems,” Phys. Rev. E 58, 3384–3399 (1998).

41W. Götze, Complex Dynamics of Glass-Forming Liquids: AMode-Coupling Theory, International Series of Monographs onPhysics (Oxford University Press, Oxford, 2009).

42S. R. Williams, G. Bryant, I. K. Snook, and W. van Megen,“Velocity autocorrelation functions of hard-sphere fluids: Long-time tails upon undercooling,” Phys. Rev. Lett. 96, 087801 (2006).

43H. L. Peng, H. R. Schober, and T. Voigtmann, “Velocity auto-correlation function in supercooled liquids: Long-time tails andanomalous shear-wave propagation,” Phys. Rev. E 94, 060601(2016).

44B. J. Ackerson and L. Fleishman, “Correlations for dilute hardcore suspensions,” J. Chem. Phys. 76, 2675–2679 (1982).

45M. Fuchs and K. Kroy, “Statistical mechanics derivation of hydro-dynamic boundary conditions: the diffusion equation,” J. Phys.:Condens. Matter 14, 9223 (2002).

46S. Mandal, L. Schrack, H. Löwen, M. Sperl, and T. Franosch,“Persistent anti-correlations in Brownian dynamics simulations ofdense colloidal suspensions revealed by noise suppression,” Phys.Rev. Lett. 123, 168001 (2019).

47H. van Beijeren, “Transport properties of stochastic Lorentzmodels,” Rev. Mod. Phys. 54, 195–234 (1982).

48F. Höfling and T. Franosch, “Crossover in the slow decay ofdynamic correlations in the Lorentz model,” Phys. Rev. Lett. 98,140601 (2007).

49B. J. Alder and T. E. Wainwright, “Velocity autocorrelations forhard spheres,” Phys. Rev. Lett. 18, 988–990 (1967).

50M. H. Ernst, E. H. Hauge, and J. M. J. van Leeuwen, “Asymptotictime behavior of correlation functions,” Phys. Rev. Lett. 25, 1254–1256 (1970).

51A. Arbe, P. Malo de Molina, F. Alvarez, B. Frick, and J. Colmen-ero, “Dielectric susceptibility of liquid water: Microscopic insightsfrom coherent and incoherent neutron scattering,” Phys. Rev. Lett.117, 185501 (2016).

52A. Mimica, “Exponential decay of measures and Tauberian theo-rems,” J. Math. Anal. Appl. 440, 266–285 (2016).

53J. P. Boon and S. Yip, Molecular Hydrodynamics (Dover Publi-cations, Inc., New York, 1991) reprint.

54J. Frenkel, Kinetic Theory of Liquids, edited by R. H. Fowler,P. Kapitza, and N. F. Mott (Oxford Univ. Press, London, 1946)pp. 188–249.

55J. C. F. Schulz, A. Schlaich, M. Heyden, R. R. Netz, and J. Kap-pler, “Molecular interpretation of the non-Newtonian viscoelasticbehavior of liquid water at high frequencies,” arXiv:2003.08309[physics.flu-dyn].

56G. Tarjus and D. Kivelson, “Breakdown of the Stokes-–Einsteinrelation in supercooled liquids,” J. Chem. Phys. 103, 3071–3073(1995).

57S. K. Kumar, G. Szamel, and J. F. Douglas, “Nature of thebreakdown in the Stokes–Einstein relationship in a hard spherefluid,” J. Chem. Phys. 124, 214501 (2006).

58S. Gupta, J. Stellbrink, E. Zaccarelli, C. N. Likos, M. Camargo,P. Holmqvist, J. Allgaier, L. Willner, and D. Richter, “Validityof the Stokes–Einstein relation in soft colloids up to the glasstransition,” Phys. Rev. Lett. 115, 128302 (2015).

59A. Dehaoui, B. Issenmann, and F. Caupin, “Viscosity of deeplysupercooled water and its coupling to molecular diffusion,” Proc.Natl. Acad. Sci. U. S. A. 112, 12020–12025 (2015).

60A. D. S. Parmar, S. Sengupta, and S. Sastry, “Length-scaledependence of the Stokes–Einstein and Adam–Gibbs relations in

model glass formers,” Phys. Rev. Lett. 119, 056001 (2017).61J. Mo, A. Simha, and M. G. Raizen, “Brownian motion as a new

probe of wettability,” J. Chem. Phys. 146, 134707 (2017).62K. Huang and I. Szlufarska, “Effect of interfaces on the nearby

Brownian motion,” Nat. Commun. 6, 8558 (2015).63L. Bocquet and J.-L. Barrat, “Flow boundary conditions from

nano- to micro-scales,” Soft Matter 3, 685 (2007).64E. Herrera-Delgado, R. Perez-Carrasco, J. Briscoe, and P. Sollich,

“Memory functions reveal structural properties of gene regulatorynetworks,” PLoS Comput. Biol. 14, e1006003 (2018).

65V. Hakim and P. Silberzan, “Collective cell migration: a physicsperspective,” Rep. Prog. Phys. 80, 076601 (2017).

66K. Kanazawa, T. Sueshige, H. Takayasu, and M. Takayasu,“Derivation of the Boltzmann equation for financial Brownian mo-tion: Direct observation of the collective motion of high-frequencytraders,” Phys. Rev. Lett. 120, 138301 (2018).

67C. L. E. Franzke, T. J. O’Kane, J. Berner, P. D. Williams, andV. Lucarini, “Stochastic climate theory and modeling,” WIREsClim Change 6, 63–78 (2014)

10

METHODS

Generalised Langevin equation. A labelled fluidparticle of mass 𝑚, position 𝑟(𝑡), and momentum 𝑝(𝑡) =𝑚�̇�(𝑡) obeys the generalised Langevin equation (GLE)68:

�̇�(𝑡) = −∫︁ 𝑡

0d𝑠 𝛾(𝑡 − 𝑠) 𝑝(𝑠) + 𝜉(𝑡), (1)

where the Brownian force 𝜉(𝑡) is a stochastic process withzero mean and covariance

⟨𝜉(𝑡) ⊗ 𝜉(𝑡′)⟩ = 𝑚𝑘B𝑇𝛾(|𝑡 − 𝑡′|) 1 (2)

to satisfy the fluctuation–dissipation theorem. Rewrit-ing eq. (1) for the the velocity auto-correlation function(VACF), 𝑍(𝑡) = ⟨𝑝(𝑡) · 𝑝(0)⟩/(𝑚2𝑑), describing momen-tum relaxation, yields

�̇�(𝑡) = −∫︁ 𝑡

0d𝑠 𝛾(𝑡 − 𝑠)𝑍(𝑠) , 𝑍(0) = 𝑘B𝑇

𝑚. (3)

Its Fourier–Laplace transform [eq. (8)] provides the linkto and the definition of the (complex-valued) memorykernel 𝛾(𝜔),

𝑍(𝜔) = 𝑘B𝑇/𝑚

−i𝜔 + 𝛾(𝜔) . (4)

Linear response. For a mass 𝑚 driven by a periodicforce 𝐹 (𝑡) = 𝐹𝜔 cos(𝜔𝑡) with frequency 𝜔 and amplitude𝐹𝜔, the stationary response 𝑣(𝑡), averaged over manyrealisations of the experiment, obeys68

𝑚dd𝑡

𝑣(𝑡) = 𝐹 (𝑡) −∫︁ 𝑡

−∞𝑚𝛾(𝑡 − 𝑠) 𝑣(𝑠) d𝑠 , (5)

corresponding to eq. (1) after shifting the lower inte-gral boundary to −∞ to ensure relaxation of transients.The upper boundary can be shifted to +∞ with theconvention that the response function 𝛾(𝑡 < 0) = 0.By linearity of the equation, the solution is of the form𝑣(𝑡) = Re

[︀𝑣𝜔e−i𝜔𝑡

]︀with complex amplitude 𝑣𝜔, and in-

serting into eq. (5) yields 𝑣𝜔 = 𝒴(𝜔) 𝐹𝜔 in terms of thegeneralised mobility,

𝒴(𝜔) := [−i𝜔𝑚 + 𝑚𝛾(𝜔)]−1 , (6)

also referred to as complex-valued admittance. Its cen-tral ingredient is the one-sided Fourier transform of theresponse function, 𝛾(𝜔) :=

∫︀ ∞0 ei𝜔𝑡𝛾(𝑡) d𝑡. Comparing to

eq. (4), which describes equilibrium correlations, yieldsthe fluctuation–dissipation relation: 𝑍(𝜔) = 𝑘B𝑇 𝒴(𝜔).

Friction describes the resistance to a prescribed ve-locity, as in Stokes’s pendulum experiments18. Thus,inverting the above argument, the force response to anoscillatory velocity 𝑣(𝑡) = 𝑣𝜔 cos(𝜔𝑡) has complex am-plitude 𝐹𝜔 = 𝒴(𝜔)−1 𝑣𝜔, and we identify 𝒴(𝜔)−1 asa generalised friction. However, merely the real part

describes dissipation and deserves to be called a fric-tion, which is seen from the mean dissipated power:𝑇 −1

𝑝

∫︀ 𝑇𝑝

0 𝑣(𝑡) · 𝐹 (𝑡) d𝑡 = Re[︀𝒴(𝜔)−1]︀

|𝑣𝜔|2/2, averagedover a full cycle of length 𝑇𝑝 = 2𝜋/𝜔. Thus, we set thedynamic friction as

𝜁(𝜔) := Re[︀𝒴(𝜔)−1]︀

= 𝑚 Re 𝛾(𝜔) ; (7)

in particular, 𝜁(𝜔) > 0. This is in line with theconventional (Markovian) Langevin equation, �̇�(𝑡) =−(𝜁0/𝑚) 𝑝(𝑡) + 𝜉(𝑡). There, the response is governedby 𝒴(𝜔) = [−i𝜔𝑚 + 𝜁0]−1, implying a static friction,𝜁(𝜔) = 𝜁0.

Equations (6) and (7) (and variants thereof) are thebasis of (passive) microrheology experiments19–22, whichuse observations of Brownian motion to infer the fric-tion 𝜁(𝜔) and Im 𝛾(𝜔) of a probe particle in a com-plex medium and relate it via the GSER to the localvisco-elastic properties. The macroscopic shear viscosity,𝜂0 = (𝑘B𝑇 )−1 ∫︀ ∞

0 𝐶Π(𝑡) d𝑡, is the Green–Kubo integralof the autocorrelation, 𝐶Π(𝑡) = ⟨𝛿Π⊥(𝑡) 𝛿Π⊥(0)⟩/𝑉 , ofshear stress fluctuations 𝛿Π⊥(𝑡), given as an off-diagonalelement of the stress tensor17; 𝑉 denotes the sample vol-ume. Similarly by a fluctuation–dissipation relation, thefrequency-dependent response coefficient 𝜂(𝜔) to oscilla-tory shear is the Fourier–Laplace transform [eq. (8)] of𝐶Π(𝑡)/𝑘B𝑇 , and thus 𝜂(𝜔 → 0) = 𝜂0.

Mathematical framework. For the harmonic anal-ysis of the autocorrelation function 𝐶(𝑡) of a stationarytime series, we use the Fourier–Laplace transform

𝐶(𝑧) =∫︁ ∞

0ei𝑧𝑡𝐶(𝑡) d𝑡 , (8)

which is well-defined for all frequencies 𝑧 in the uppercomplex plane, C+ = {𝑧| Im 𝑧 > 0}. Along the imaginaryaxis, 𝑧 = i𝑦, it recovers the conventional Laplace trans-form. For real frequencies 𝜔, the real and imaginary partsof 𝐶(𝜔) describe physically accessible spectra, which arerelated to each other by Kramers–Kronig integrals68; forexample, Re 𝐶(𝜔) for fixed 𝜔 is determined by the fullfunction Im 𝐶(𝜔). The real part is positive, Re 𝐶(𝜔) > 0,and most importantly, we have the unique Fourier back-transform:

𝐶(𝑡) = 1𝜋

∫︁ ∞

−∞e−i𝜔𝑡 Re 𝐶(𝜔) d𝜔 . (9)

If 𝐶(𝑡) is 𝑛-times continuously differentiable at 𝑡 = 0, thisimplies sum rules for the spectrum (𝑘 = 0, 1, . . . , 𝑛):

1𝜋

∫︁ ∞

−∞(−i𝜔)𝑘 Re[𝐶(𝜔)] d𝜔 = 𝐶(𝑘)(0) < ∞ . (10)

In equilibrium, only positive frequencies are needed asRe 𝐶(𝜔) = Re 𝐶(−𝜔), and the integrals are real-valued.

Next, we introduce a memory function of 𝐶(𝑡) solelyby invoking results from complex analysis69,70. 𝐶(𝑧) as

11

above is a holomorphic function with Re 𝐶(𝑧) > 0, i.e.,i𝐶(𝑧) is of Herglotz–Nevanlinna type, and (Im 𝑧) |𝐶(𝑧)|bounded in C+. Suppose that 𝐶(𝑡) has a regular short-time expansion, 𝐶(𝑡 → 0) ≃ 𝐶0

[︀1 − 𝜈𝑡 − 1

2 𝑎𝑡2]︀, which

implies

𝐶(𝑧) ≃ 𝐶0(−i𝑧)−1 − 𝜈𝐶0(−i𝑧)−2 − 𝑎𝐶0(−i𝑧)−3 (11)

for large frequencies, |𝑧| → ∞ with |arg 𝑧| > 𝛿 for some𝛿 > 0. Under these mild requirements, one shows70: Forgiven 𝐶(𝑧) there is a unique memory kernel �̂�(𝑧) suchthat

𝐶(𝑧) = 𝐶0

−i𝑧 + �̂�(𝑧)(12)

with i�̂�(𝑧) of Herglotz–Nevanlinna type and �̂�(𝑧) ≃𝜈 + 𝑎/(−i𝑧) as |𝑧| → ∞. In particular, �̂�(𝑧) correspondsto the autocorrelation function of another (a priori un-known) observable. Iterating the argument yields thecontinued-fraction representation of 𝐶(𝑧), well-knownfrom the Zwanzig–Mori projection formalism17.

In the context of the VACF, one puts 𝐶0 = 𝑣2th, 𝜈 = 0,

and 𝑎 = 𝜔20 and infers for the memory kernel �̂�(𝑧) =: 𝛾(𝑧)

that Re 𝛾(𝑧) > 0 and 𝛾(𝑧) ≃ 𝜔20/(−i𝑧) as |𝑧| → ∞. This

justifies eq. (4) independently of the notion of a GLE,after taking 𝑧 along the real line.

By means of eq. (9), 𝛾(𝑧) specifies the memory function𝛾(𝑡), which has a physical interpretation as the autocor-relator of the fluctuating acceleration 𝜉(𝑡)/𝑚 in eq. (1),divided by 𝑣2

th. At low frequencies, 𝑚𝛾(𝑧 → 0) = 𝜁0implies a Green–Kubo relation for the hydrodynamic fric-tion:

𝜁0 = 𝑚

∫︁ ∞

0𝛾(𝑡) d𝑡 . (13)

Short-time expansion. The smoothness of physi-cal molecular trajectories, being solutions to Newton’sequations, allows for a rigorous short-time expansionof the VACF. Combining with the time-reversal sym-metry in equilibrium, 𝑍(𝑡) = 𝑍(−𝑡), only even pow-ers in 𝑡 contribute and one obtains 𝑍(𝑡 → 0) ≃𝑘B𝑇

∑︀∞𝑘=0 𝑐𝑘𝑡2𝑘/(2𝑘)! with Taylor coefficients 𝑐𝑘 given

from equilibrium matrix elements of powers of the under-lying Liouville operator53. To connect with the notationof the main text, 𝑐0 = 1/𝑚, 𝑐1/𝑐0 = −𝜔2

0 , and we put𝑐2/𝑐0 =: Ω4. Fourier–Laplace transforming term by termyields the high-frequency expansion of 𝑍(𝜔) and thus of𝒴(𝜔) = (𝑘B𝑇 )−1𝑍(𝜔), which is purely imaginary: 𝒴(𝜔 →∞) ≃

∑︀∞𝑘=0 𝑐𝑘 (−i𝜔)−1−2𝑘 = 𝑐0/(−i𝜔) + 𝑐1/(−i𝜔)3 + . . . .

Using eq. (7), we have 𝜁(𝜔) = |𝒴(𝜔)|−2 Re 𝒴(𝜔), whichimplies that for high frequencies the friction vanishes,𝜁(𝜔) ≡ 0, at all orders in 𝜔 → ∞. A similar situationis familiar from calculus text books: 𝑓(𝑥) = e−1/𝑥 has aTaylor series 𝑓(𝑥) ≡ 0 at 𝑥 = 0; the radius of convergenceis 0.

The expansion of 𝒴(𝜔) can be represented as a contin-ued fraction that has the same large-𝜔 asymptotics up toterms of order 𝜔−5:

𝒴(𝜔) ≃ 1⧸︀{︀

−i𝜔𝑚+𝑚𝜔20⧸︀[︀

−i𝜔+𝜔21/(−i𝜔+. . . )

]︀}︀, (14)

introducing 𝜔21 :=

(︀Ω4 − 𝜔4

0)︀⧸︀

𝜔20 for brevity. This trun-

cation is an excellent description of our data for 𝒴 ′′(𝜔)at high frequencies, with 𝜔0 and Ω obtained from fits to𝑍(𝑡), see fig. 3d–f. For the memory kernel 𝛾(𝜔), one readsoff

𝛾(𝜔 → ∞) = 𝜔20/(−i𝜔) − 𝜔2

0𝜔21/(−i𝜔)3 + 𝒪

(︀𝜔−5)︀

, (15)

using eq. (6), implying for the memory function in timedomain:

𝛾(𝑡 → 0) = 𝜔20

[︀1 − 𝜔2

1𝑡2/2]︀

+ 𝑂(︀𝑡4)︀

. (16)

Long-time tails. In an unbounded fluid, momen-tum conservation leads to persistent velocity correlations,𝑍(𝑡 → ∞) ≃ 𝑣2

th(𝑡/𝜏∞)−3/2, which was explained in termsof hydrodynamic backflow and diffusion of a momentumvortex17,49,50. The tail induces a small-𝜔 singularity inthe frequency domain71, which reads for the memory ker-nel: 𝑚𝛾(𝜔 → 0) ≃ 𝜁0

[︀1 + (𝜏∞𝜁0/𝑚)

√−4𝜋i𝜔𝜏∞)

]︀, using

eq. (4) and the hydrodynamic friction 𝜁0 := 𝑚𝛾(0) =𝑘B𝑇

⧸︀ ∫︀ ∞0 𝑍(𝑠) d𝑠.

In the framework of the creeping flow equations, Stokesfound18

𝑚𝛾(𝜔 → 0) ≃ 6𝜋𝜂0𝑎(1 +√

−i𝜔𝜏fl) − i𝜔𝑚fl/2 (17)

in terms of the vorticity diffusion time 𝜏fl and an ef-fective particle mass 𝑚fl; matching with the previousexpression for the asymptote of 𝛾(𝜔), one identifies𝜏fl = 4𝜋(𝜁0/𝑚)2𝜏3

∞. The real part yields the dynamicfriction,

𝜁(𝜔 → 0) ≃ 6𝜋𝜂0𝑎(1 +√︀

𝜔𝜏fl/2) , (18)

showing that its macroscopic limit, 𝜁0 = 6𝜋𝜂0𝑎, is ap-proached from above as 𝜔 → 0 (see fig. 3b). For waterand the supercooled liquid, a different type of power lawdecay, 𝑍(𝑡) ∼ −𝑡−5/2, was found (fig. 3e,f).

For a general long-time tail of the VACF, 𝑍(𝑡) ∼ 𝑡−𝜎

with 𝜎 > 1, the memory function 𝛾(𝑡) asymptoticallyinherits a power-law decay with the same exponent, butof opposite sign72:

𝛾(𝑡) ≃ −𝛾(0)2𝑍(𝑡)/𝑍(0) , 𝑡 → ∞ ; (19)

which follows from eq. (3) and by invoking a Taubertheorem71. Without any adjustable parameter, the pre-diction is in excellent agreement with our data for 𝛾(𝑡)in case of the LJ fluid. Inspection of a few examples(figs. 4d,e and 5c) suggests that, in order to accommodatethe sign change of the tail, the number of zero crossings(knots) in 𝛾(𝑡) is increased by one relative to 𝑍(𝑡).

12

Analytically solvable example. Consider the fol-lowing analytically tractable model for the VACF:

𝑍(𝑡) = 𝑣2th

1 + (𝑡/𝜏)2 , 𝑣2th = 𝑘B𝑇/𝑚 , (20)

with relaxation time 𝜏 and thermal velocity 𝑣th (sup-plemental fig. S1d). It favourably combines the physi-cally required smoothness at 𝑡 = 0 and time-reversibility,𝑍(𝑡) = 𝑍(−𝑡), with a power-law decay at long times,𝑍(𝑡 → ∞) ≃ 𝑣2

th(𝑡/𝜏)−2; in particular, only even pow-ers of 𝑡 contribute to the the short-time expansion:𝑍(𝑡 → 0) ≃ 𝑣2

th[︀1 − (𝑡/𝜏)2 + 𝑂(𝑡4)

]︀. From the one-

sided Fourier transform of 𝑍(𝑡), we obtain the real andimaginary parts of 𝑍(𝜔) as Re 𝑍(𝜔) = 𝐷∞e−|𝜔𝜏 | andIm 𝑍(𝜔) = 𝐷∞[e−𝜔𝜏 Ei(𝜔𝜏)−e𝜔𝜏 Ei(−𝜔𝜏)]/𝜋, being evenand odd functions in 𝜔, respectively (fig. 5a). Here, Ei(·)denotes the exponential integral, and 𝐷∞ = 𝑣2

th𝜏𝜋/2 isthe long-time limit of the diffusivity, 𝐷(𝑡) =

∫︀ 𝑡

0 𝑍(𝑠) d𝑠 =𝑣2

th𝜏 arctan(𝑡/𝜏). In application of theorem 1.2 of ref. 52,we confirm that

lim𝜔→∞

(−𝜔)−1 log(Re 𝑍(𝜔)) =: 𝜏rc (21)

yields the radius of convergence, 𝜏rc = 𝜏 , of the short-timeexpansion of 𝑍(𝑡); in particular, 𝜏rc > 0.

Given 𝑍(𝜔), the explicit expression for the complexmemory kernel 𝛾(𝜔) and the friction 𝜁(𝜔) follow fromeqs. (4) and (7), see fig. 5b. The low- and high-frequencyasymptotes correspond to 𝜁(𝜔 → 0) ≃ 𝜁0(1 + |𝜔𝜏 |) and𝜁(𝜔 → ∞) ≃ 𝜁0(𝜋𝜔𝜏/2)2 e−|𝜔𝜏 |, respectively, with 𝜁0 =𝑘B𝑇/𝐷∞. The friction attains its maximum ≈ 1.2𝜁0near 𝜔max ≈ 0.892𝜏−1 and falls off rapidly for larger 𝜔;the position of the maximum of Im 𝛾(𝜔) sets the onsetfrequency 𝜔𝑐 ≈ 4.01𝜏−1. The memory function 𝛾(𝑡) isobtained numerically from 𝜁(𝜔) using eq. (9), with theshort- and long-time asymptotes 𝛾(𝑡 → 0) ≃ 2𝜏−2[1 −5(𝑡/𝜏)2] and 𝛾(𝑡 → ∞) ≃ −(𝜁0/𝑚)2(𝑡/𝜏)−2, respectively(fig. 5c).

Adapted Filon algorithm. The computation of thefrequency-dependent friction requires a robust numeri-cal Fourier transform, for which we developed a physics-enriched version of Filon’s quadrature formula. The goalis to evaluate 𝑓(𝜔) =

∫︀ ∞0 ei𝜔𝑡𝑓(𝑡) d𝑡 for a function 𝑓(𝑡)

sparsely sampled on an irregular grid 𝑡0 = 0, 𝑡1, . . . , 𝑡𝑛 foran arbitrary set of frequencies (𝜔𝑗). The idea of Filon’salgorithm is to interpolate 𝑓(𝑡) by elementary functionsbetween the grid points (usually parabolas), thereby re-ducing the Fourier integral to a finite sum of integrals,for which analytic expressions exist. Anticipating thatthe normal physical decay of correlation functions is ex-ponential, we approximate 𝑓(𝑡) ≈ 𝑎𝑘e−𝛾𝑘𝑡 in the interval

[𝑡𝑘, 𝑡𝑘+1] with 𝑎𝑘 and 𝛾𝑘 fixed by 𝑓(𝑡𝑘) and 𝑓(𝑡𝑘+1). Then,

𝑓(𝜔) ≈∫︁ 𝑡1

0ei𝜔𝑡𝑓(𝑡) d𝑡 +

𝑛−1∑︁𝑘=1

∫︁ 𝑡𝑘+1

𝑡𝑘

𝑎𝑘e(i𝜔−𝛾𝑘)𝑡 d𝑡

+∫︁ ∞

𝑡𝑛

𝑎𝑛e(i𝜔−𝛾𝑛)𝑡 d𝑡 . (22)

Spurious low-frequency oscillations of the transform areremoved by smoothly truncating the integral at 𝑡𝑛, hereby assuming a terminal exponential decay of 𝑓(𝑡), whichleads to the last term on the r.h.s. of eq. (22). In orderto preserve the short-time properties of 𝑓(𝑡) we fit apolynomial in 𝑡2 to the first few data points and solvethe integral analytically; this improves the high-frequencybehaviour of 𝑓(𝜔).

The dynamic friction 𝜁(𝜔) and the memory function𝛾(𝑡) are obtained from MSD data as follows (fig. 2):The timescale-dependent diffusion coefficient, 𝐷(𝑡) :=𝜕𝑡MSD(𝑡)/6, is obtained from numerical differentiation.In all cases studied, it grows out from zero, has amaximum, and converges slowly towards the long-timediffusion constant 𝐷∞ = 𝐷(𝑡 → ∞), see supplemen-tal fig. S1. Using the above algorithm, we compute73

𝑍(𝜔) = 𝐷∞ − i𝜔∫︀ ∞

0 d𝑡 ei𝜔𝑡[𝐷(𝑡) − 𝐷∞]. Then, 𝜁(𝜔) isgiven by eq. (4) and is transformed back to the timedomain with the same algorithm [eq. (9)]; in particular,we use again a smooth, exponential cutoff. In fig. 5, thenumerical procedure is successfully tested against theanalytical model with high accuracy.

Deconvolution in time domain. Inversion of theconvolution in eq. (3) yields the memory function 𝛾(𝑡)directly74, without resorting to the frequency domain.Numerically, it is not easy to obtain accurate and robustresults, and a variety of algorithms have been developed,see ref. 37 for a comparative study. The presence of�̇�(𝑡) in eq. (3) is removed by integration, yielding 𝑍(𝑡) =𝑍(0) −

∫︀ 𝑡

0 d𝑠 𝐺(𝑡 − 𝑠)𝑍(𝑠) with the integrated memory𝐺(𝑡) :=

∫︀ 𝑡

0 d𝑠 𝛾(𝑠). Discretising on a uniform time grid,𝑡𝑖 = 𝑖Δ𝑡 (𝑖 = 0, 1, . . . ), and employing the trapezoidalrule for the integral, a recursion relation for 𝐺𝑖 := 𝐺(𝑡𝑖)with the initial value 𝐺0 = 0 follows37:

𝐺𝑖 = 1 − 𝑍𝑖/𝑍0

Δ𝑡/2 − 2𝑖−1∑︁𝑗=1

𝐺𝑗𝑍𝑖−𝑗/𝑍0 , 𝑖 > 1 . (23)

Going beyond ref. 37, we introduce a predictor–correctorscheme to stabilise the numerical solutions: In the predic-tor step, one evaluates 𝐺*

𝑖 and 𝐺*𝑖+1 from eq. (23). After-

wards, the weighted average 𝐺𝑖 := (𝐺𝑖−1 +3𝐺*𝑖 +𝐺*

𝑖+1)/5manifests itself as the corrector step. Results for 𝐺(𝑡) canbe found in the supplemental fig. S2. Finally, the memoryfunction 𝛾(𝑡) = 𝜕𝑡𝐺(𝑡) is obtained by central differences.If one starts from MSD data on a sparse (e.g., geomet-rically spaced) time grid, a cubic spline interpolation ofthe MSD in the variable 𝑡2 is suitable to sample 𝑍(𝑡) ona uniform grid of up to 105 points.

13

Molecular dynamics simulations. Simulations ofliquid water were performed with the GROMACS 5.1package75 using the SPC/E water model, which wasshown to accurately reproduce the linear absorption spec-tra of water from experiments and ab-initio MD simula-tions up to frequencies of about 30 THz76. The system of3,007 molecules in a cubic box of size 4.49 nm was equili-brated at 300 K and 1 bar following standard procedures37.Correlation functions were obtained from an NVE simu-lation over 275 ps with the velocity-Verlet integrator anda time step of 1 fs, using double floating-point precisionto achieve good energy conservation. The frequency-dependent viscosity was computed from additional NVEruns, totalling to 49 ns.

For the other two liquids, we used the massively parallelsoftware HAL’s MD package77 (version 1.0-𝛼6), permit-ting the sampling of dynamic correlations on a sparsetime grid and featuring smoothly truncated potentials tovirtually eliminate any energy drift. The mono-atomicfluid consists of 105 particles interacting pairwise via theLJ potential, 𝑈(𝑟) = 4𝜀

[︀(𝑟/𝜎)−12 − (𝑟/𝜎)−6]︀

, truncatedfor 𝑟 > 2.5𝜎; a unit of time is defined by 𝜏LJ :=

√︀𝑚𝜎2/𝜀.

Equilibration in the NVE ensemble at number density𝜚 = 0.8𝜎3 and thermal energy 𝑘B𝑇 = 1.3𝜀 followed theprotocol in ref.78. The supercooled liquid was realisedby a Kob–Andersen 80:20 binary mixture79 of 64,000 LJbeads at 𝜚 = 1.2𝜎−3 and 𝑇 * := 𝑘B𝑇/𝜀 = 0.6, equilibratedover a time span of 9,000 𝜏LJ, and we traced the speciesof the larger particles. The chosen temperature is wellbelow the melting point80, 𝑇 * ≈ 1.03, and is on the onsetof universal scaling behaviour according to mode-couplingtheory41; here, the value of the critical temperature is𝑇 *

MCT ≈ 0.43.The simulations generate trajectories 𝑟(𝑡) of an en-

semble of labelled particles for each fluid; our main ob-servable is the mean-square displacement MSD(𝑡) :=⟨︀|𝑟(𝑡) − 𝑟(0)|2

⟩︀for lag time 𝑡. For both liquids, single-

particle MSDs were averaged from 10 production runs,each over 108 integration steps of length 0.001 𝜏LJ.

Control simulations of a single particle in its pinned cageare based on an equilibrated sample of the LJ fluid with106 particles. MSDs were recorded after equilibration ofthe mobile particle in its static environment over 100 𝜏LJand were averaged over 106 different cages, computedin parallel, to remove spurious oscillations. Technically,the setup was realised by making two initially identicalcopies of the fluid interact with each other: the first copycontains the immobile matrix, the second one the tracers(not interacting with each other).

Acknowledgements. FH is indebted to Thomas Fra-nosch for introducing him to the field of complex trans-port. We benefited from discussions with Lydéric Boc-quet, Matthias Fuchs, Julian Kappler, and Klaus Kroy.

This research has been funded by Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) throughthe grant SFB 1114 (projects B03 and C01) and underGermany’s Excellence Strategy – MATH+ : The BerlinMathematics Research Center (EXC-2046/1) – projectID: 390685689 (subproject EF4-4). Further funding bythe European Research Council (ERC Advanced H 2020Grant “NoMaMemo”) is gratefully acknowledged. Someof the data were produced with the supercomputer “Lise”(HLRN-IV) of the North-German Supercomputing Al-liance.

Author contributions. AS, RN, and FH conceivedthe project and wrote the manuscript. BK and FH per-formed the simulations, and AS and FH analysed thedata. AS carried out the analytical work. All authorsdiscussed the results and implications and commented onthe manuscript at all stages.

Data availability. The datasets generated and anal-ysed during the current study are available from thecorresponding author upon reasonable request.

Code availability. Primary data were generatedwith open source software as indicated in the Methodssection. The source code used to analyse the data for thecurrent study is available from the corresponding authorupon reasonable request.

REFERENCES

68R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II:Nonequilibrium Statistical Mechanics (Springer, Berlin, Heidel-berg, 1991).

69G. Teschl, Mathematical methods in quantum mechanics: withapplications to Schrödinger operators, 2nd ed. (Am. Math. Soc.,Providence, RI, 2014).

70T. Franosch, J. Phys. A: Math. Theor. 47, 325004 (2014).71J. Karamata, J. reine angew. Math. 164, 27 (1931).72N. Corngold, Phys. Rev. A 6, 1570 (1972).73T. Franosch, M. Spanner, T. Bauer, G. E. Schröder-Turk, and

F. Höfling, J. Non-Cryst. Solids 357, 472 (2011).74B. J. Berne, M. E. Tuckerman, J. E. Straub, and A. L. R. Bug,

J. Chem. Phys. 93, 5084 (1990).75B. Hess, C. Kutzner, D. Van Der Spoel, and E. Lindahl, J. Chem.

Theory Comput. 4, 435 (2008).76S. Carlson, F. N. Brünig, P. Loche, D. J. Bonthuis, and R. R.

Netz, “Exploring the absorption spectrum of simulated waterfrom MHz to the infrared,” arXiv:2003.07260 [physics.chem-ph].

77P. H. Colberg and F. Höfling, Comput. Phys. Commun. 182, 1120(2011); “Highly Accelerated Large-scale Molecular Dynamicspackage,” (2007–2019), https://halmd.org.

78S. Roy, S. Dietrich, and F. Höfling, J. Chem. Phys. 145, 134505(2016).

79W. Kob and H. C. Andersen, Phys. Rev. Lett. 73, 1376 (1994).80U. R. Pedersen, T. B. Schrøder, and J. C. Dyre, Phys. Rev. Lett.

120, 165501 (2018)