February 1, 2008 Testing Mode-Coupling Theory for a Supercooled Binary Lennard-Jones Mixture I: The van Hove Correlation Function Walter Kob∗ Institut fu¨r Physik, Johannes Gutenberg-Universita¨t, Staudinger Weg 7, D-55099 Mainz, Germany 5 9 9 1 Hans C. Andersen† n Department of Chemistry, Stanford University, Stanford, California 94305 a J 2 2 Abstract 1 v 2 We report the results of a large scale computer simulation of a binary super- 0 1 cooled Lennard-Jones liquid. We find that at low temperatures the curves for 1 themean squared displacement of atagged particle for different temperatures 0 5 fall onto a master curve when they are plotted versus rescaled time tD(T), 9 where D(T) is the diffusion constant. The time range for which these curves / t a follow the master curve is identified with the α-relaxation regime of mode- m coupling theory (MCT). This master curve is fitted well by a functional form - suggested by MCT. In accordance with idealized MCT, D(T) shows a power- d n law behavior at low temperatures. The critical temperature of this power-law o is the same for both types of particle and also the critical exponents are very c : similar. However, contrary to a prediction of MCT, these exponents are not v i equal to the ones determined previously for the divergence of the relaxation X times of the intermediate scattering function [Phys. Rev. Lett. 73, 1376 r a (1994)]. At low temperatures the van Hove correlation function (self as well asdistinctpart)showshardlyany signofrelaxation inatimeintervalthatex- tends over about three decades in time. This time interval can be interpreted as the β-relaxation regime of MCT. From the investigation of these correla- tion functions we conclude that hopping processes are not important on the time scale of the β-relaxation for this system and for the temperature range investigated. We test whether the factorization property predicted by MCT holds and find that this is indeed the case for all correlation functions inves- tigated. The distance dependence of the critical amplitudes are in qualitative accordance with the ones predicted by MCT for some other mixtures. The non-gaussian parameter for the self part of the van Hove correlation function for different temperatures follows a master curve when plotted against time 1 t. Typeset using REVTEX 2 I. INTRODUCTION About a decade has passed since two of the most seminal papers in the recent history of the field of the glass transition and supercooled liquids were published. In these papers, one by Bengtzelius, G¨otze, and Sj¨olander and the other by Leutheusser, it was proposed that the glass transition could be understood as the singular behavior of the solution of the equations of motion of the dynamic structure factor, the so-called mode-coupling equations [1]. These equations are the simplified versions of certain nonlinear equations of motion that were derived in the seventies in order to describe the dynamics of simple liquids at high densities [2]. Subsequently G¨otze, Sj¨ogren and many others analyzed these mode-coupling equations in order to work out the details of the singular behavior of their solutions [3–6]. Today the sum of all these results is known as mode-coupling theory (MCT), and a review of them can be found in some recent review articles [7,8]. The theory has stimulated a remarkable amount of experimental and computer simulation work, with various groups looking for the signs of this singularity in many different kinds of systems [9–20]. The result of all these experiments and simulations is that the theory appears to be able to rationalize the dynamical behavior of some glass forming materials in an amazingly convincing way. However, the dynamical behavior of other glass forming materials seems to be described by MCT only poorly. So far it is still not clear yet for what kind of system the prediction of the theory hold and for which systems they do not. Even if for a particular system some of the predictions of the theory hold, it is not certain that also the other predictions made by MCT will hold. Therefore it is clear that still much has to be learned about the applicability of this theory. However, it seems that there is agreement on at least one point, namely that the dynamical singularity predicted by MCT is not the same as the laboratory glass transition. The latter occurs at a temperature T that is defined as the temperature at g 13 which the viscosity of the material is 10 Poise. Below this temperature the material can no longer come to thermodynamic equilibrium because its relaxation times are longer than the time scale of typical experiments. However, it is found that if experiments see a singular behavior that can be interpreted as the singularity predicted by MCT, then the temperature atwhich thissingularbehavior isobserved isabout30-50KaboveT . Furthermoreithasalso g been found empirically that the predictions of MCT seems to work better for fragile glass formers than for strong glass formers. For fragile glass formers, a plot of the logarithm of the viscosity vs. 1/T is curved, and the temperature of the MCT singularity is often near the temperature at which the plot shows a pronounced bend. At this temperature the viscosity changes its behavior from a weak dependence on temperature to a strong dependence on temperature as the temperature is lowered. Thus it can be that this bend in the viscosity is a signature of the singularity predicted by MCT. This bend occurs at viscosities which are about 1-100 Poise or relaxation times that are around 10−11 − 10−9 seconds. Thus the singularity predicted by MCT is not the laboratory glass transition but rather is an anomalous dynamic behavior in supercooled liquids for fragile glass forming materials that takes place at temperatures above the glass transition temperature. Computer simulations are particularly well suited to test the predictions of MCT, since they allow access to the full informationon the system at any time of the simulation. This in turn permits the calculation of many different types of correlation functions, some of which are not experimentally measurable but about which the theory makes definite predictions, 3 and thus very stringent tests of the theory become possible. Furthermore the measurement of these correlationfunctions is very straightforward in that they are computed directly from the positions and velocities of the particles. Thus no theoretical model or assumption, as has to be used to explain, e.g., the scattering mechanism in light scattering experiments, is needed. A severe drawback of most computer simulations is the limited range of times over which simulations can be performed. This in effect makes the cooling rates in simulations much larger than those in laboratory experiments. As a result the simulated material falls out of equilibrium at a higher temperature than would the corresponding real material in the laboratory [21]. In other words, the glass transition temperature in the simulation T is higher than the laboratory glass transition T . If T is too high, the lack of g−sim g g−sim equilibration in the simulation can obscure not only the MCT singularity at T but also c the higher temperature signatures of the onset of the singularity. Thus, it is important in simulations to have a range of temperatures, extending down as close as possible to T , c at which the system can be thoroughly equilibrated and the slow dynamics studied. This requires a significant amount of computation to achieve. Inthe present paper we will present data for thoroughly equilibrated systems in such a temperature range. This paper presents the results of a molecular dynamics computer simulation in order to makecarefultestsofwhetherthepredictionsofMCTholdforthesystemunderinvestigation. In two previous papers we investigated for the same system the scaling behavior of the intermediate scattering function in the β-relaxation regime [15,16]. We found that this correlation function shows many of the features predicted by the theory. In the present paperwe willfocusonthebehavior ofthevanHovecorrelationfunctionandtest whether the predictions of MCT hold for these kind of correlation functions as well. In a following paper [22] we will investigate in detail the time and wave-vector dependence of the intermediate scattering function in the α-and β-relaxation regime (defined in the next section) and also the frequency dependence of the dynamic susceptibility. Thus the sum of the results of these investigations will allow us to make a stringent test on whether or not MCT is able to rationalize the dynamical behavior of the system studied. The present paper is organized as follows: In Sec. II we will summarize those predictions of MCT that are relevant to understand the results of this work. In Sec. III we introduce our model and give some details on the computation. In Sec. IV we present our results and in Sec. V we summarize and discuss these results. II. MODE COUPLING THEORY In this section we give a short summary of those predictions of MCT that are relevant for the interpretation of the results presented in this paper. An extensive review of the theory can be found in references [7,8]. In its simplest version, also called the idealized version, mode-coupling theory predicts the existence of a critical temperature T above which the system shows ergodic behavior c and below which the system is no longer ergodic. All the predictions of the theory which are considered in this paper are of an asymptotic nature in the sense that they are valid only in the vicinity of T . For temperatures close to T , MCT makes precise predictions c c about the dynamical behavior of time correlation functions φ(t) = hX(0)Y(t)i between 4 those dynamical variables X and Y that have a nonzero overlap with δρ(q), the fluctuations of the Fourier component of the density for wave vector q, i.e. for which hδρ(q)Xi and hδρ(q)Yi is nonvanishing. Here hi stands for the canonical average. In particular the theory predicts that for T > T φ(t) should show a two step relaxation behavior, i.e. the correlation c function plotted as a function of the logarithm of time should show a decay to a plateau value, for intermediate times, followed by a decay to zero at longer times. The time interval in which the correlation functions are close to this plateau is called the β-relaxation region. Despite a similar name this region should not be confused with the β-relaxation process as described by Johari and Goldstein [23]. Furthermore the theory predicts that in the vicinity of the plateau the so-called “factorization-property” holds. This means that the correlator φ(t) can be written in the form φ(t) = fc +hG(t) , (1) where fc is the height of the plateau at T , h is some amplitude that depends on the c correlator but not on time and the function G(t) depends on time and temperature but not on φ. Thus for a given system G(t) is a universal function for all correlators satisfying the above mentioned condition. The details of the function G(t) depend on a system specific parameter λ, the so-called exponent parameter. In principle λ can be calculated if the structure factor of the system is known with sufficient precision, but since this is rarely the case for real experiments, it is in most cases treated as a fitting parameter. For all values of λ the theory predicts that for certain time regions the functional form of G(t) is well approximated by two power-laws. In particular it is found that for those times for which the correlator is still close to the plateau but has started to deviate from it, G(t) is given by the so-called von Schweidler law: G(t) = −B(t/τ)b . (2) where B is a constant that can be computed from λ. The relaxation time τ is the relaxation time of the so-called α-relaxation, i.e. the relaxation at very long times where the correlator decays to zero. The exponent b, often called the von Schweidler exponent, can be computed if the value of λ is known and is therefore not an additional fitting parameter. In the α-relaxation regime MCT predicts that the correlation functions obey the time temperature superposition principle, i.e. φ(t) = F(t/τ) , (3) where the overwhelming part of the temperature dependence of the right hand side is given by the temperature dependence of τ. Equation (3) says that if the correlation function for different temperatures are plotted versus t/τ(T) they will fall onto the master curve F(t/τ). Note that the time range in which the von Schweidler law is observed belongs to the late β-relaxation regime as well as to the early α-relaxation regime. In addition MCT predicts that the diffusion constant D shows a power-law behavior as a function of temperature with the critical temperature T : c D ∝ (T −T )γ , (4) c 5 where γ can also be computed once λ is known. Note that some of these predictions of MCT are valid only for the simplest (or ideal- ized) version of the theory in which the so-called hopping processes are neglected. If these processes are present some of the statements made above have to be modified. However, below we will give evidence that for the system under investigation hopping processes are not important in the temperature range investigated and that therefore the idealized version of the theory should be applicable. Furthermoreit hastobeemphasized thatMCT assumes thatthesystem under investiga- tion is in equilibrium. Thus great care should be taken to equilibrate the system properly. A recent computer simulation of a supercooled polymer system has shown that nonequilibrium effects can completely change the behavior of the correlation functions [14]. Thus a com- parison of the predictions of MCT with the results of a simulation in which nonequilibrium effects are still present becomes doubtful at best. III. MODEL AND COMPUTATIONAL PROCEDURES In this section we introduce the model we investigated and give some of the details of the molecular dynamics simulation. The system we are studying in this work is a binary mixture of classical particles. Both types of particles (A and B) have the same mass m and all particles interact by means of 12 6 a Lennard-Jones potential, i.e. V (r) = 4ǫ [(σ /r) − (σ /r) ] with α,β ∈ {A,B}. αβ αβ αβ αβ The reason for our choice of a mixture was to prevent the crystallization of the system at low temperatures. However, as we found out in the course of our work, choosing a binary mixture is by no means sufficient to prevent crystallization, if the system is cooled slowly. In particular we found that a model that has previously been used to investigate the glass transition [24], namely a mixture of 80% A particles and 20% B particles with ǫ = ǫ = ǫ , and σ = 0.8σ , and σ = 0.9σ , crystallizes at low temperatures, AA AB BB BB AA AB AA as evidenced by a sudden drop in the pressure. In order to obtain a model system that is less prone to crystallization, we adjusted the parameters in the Lennard-Jones potential in such a way that the resulting potential is similar to one that was proposed by Weber and Stillinger to describe amorphous Ni80P20 [25]. Thus we chose ǫAA = 1.0, σAA = 1.0, ǫ = 1.5, σ = 0.8, ǫ = 0.5, and σ = 0.88. The number of particles of type A and AB AB BB BB B were 800 and 200, respectively. The length of the cubic box was 9.4 σ and periodic AA boundary conditions were applied. In order to lower the computational burden we truncated and shifted the potential at a cut-off distance of 2.5σ . In the following all the results will αβ be given in reduced units, i.e. length in units of σ , energy in units of ǫ and time in AA AA units of (mσ2 /48ǫ )1/2. For Argon these units correspond to a length of 3.4˚A, an energy AA AA of 120Kk and a time of 3·10−13s. B Themoleculardynamicssimulationwasperformedbyintegratingtheequationsofmotion usingthevelocityformoftheVerletalgorithmwithastepsize0.01and0.02athigh(T ≥ 1.0) and low (T ≤ 0.8) temperatures, respectively. These step sizes were sufficiently small to reduce the fluctuation of the total energy to a negligible fraction of k T. The system was B equilibrated at high temperature (T = 5.0) where the relaxation times are short. Changing the temperature of the system to a temperature T was performed by coupling it to a f stochastic heat bath, i.e. every 50 steps the velocities of the particles were replaced with 6 velocities that were drawn from a Boltzmann distribution corresponding to the temperature T . This was done for a time period of length t , which was chosen to be larger than the f equi relaxation time of the system at the temperature T . After this change of temperature we f let the system propagate with constant energy, i.e. without the heat bath, for a time that was also equal to t , in order to see that there was no drift in temperature, pressure, or equi potential energy. If no drift was observed, we considered the final state to correspond to an equilibrium state of the system at the temperature T , and we used this final state as the f initial state for a molecular dynamics trajectory. In this trajectory, there was no coupling to a heat bath; it was a constant energy trajectory, and the results were used to provide the correlation function data discussed below. The temperatures we studied were T = 5.0, 4.0, 3.0, 2.0, 1.0, 0.8, 0.6, 0.55, 0.50, 0.475, and 0.466. At the lowest temperatures the length 5 of the run was 10 time units. Thus, again assuming Argon units, the present data covers the time range from 3 · 10−15s to 3 · 10−8s. At the lowest temperature the equilibration 5 time t was 0.6·10 time units. Thus if we define the cooling rate to be the difference of equi the starting temperature and the final temperature divided by the time for the quench, the smallest cooling rate (used to go from the second lowest temperature to the lowest one) is 1.5 ·10−7. In the case of Argon this smallest cooling rate would correspond to 6 ·107K/s. Thus, although this cooling rate is still very fast it is smaller than the fastest cooling rate achievable in experiments and about an order of magnitude smaller that the one used in previous computer simulations. In order to improve the statistics, we performed eight different runs at each tempera- ture, each of which was equilibrated separately in the above described way, and averaged the results. Each of these runs originated from a different point in configuration space. The thermal history of these starting points differed significantly from one to another. In particular this history sometimes included periods in which we reheated the system after having cooled it to low temperatures and had it equilibrated at these temperatures or, in some other cases, included periods in which we cooled it with twice the normal cooling rate. Despite these different thermal histories the results we obtained from these eight different runs were the same to within statistical fluctuations. Thus we have good evidence that the results reported in this work are all equilibrium properties of the system and are not dependent on the way we prepared the system at a given temperature. Most of the results presented in this paper deal with the van Hove correlation function (self and distinct part), which are defined below. This space-time correlation function is the one that is most easily obtained from molecular dynamics computer simulations. As mentioned in the previous paragraph we averaged our results over at least eight different runs. Since the resulting correlation functions φ(r,t) still showed some short wavelength noise in r even after we did this averaging, we smoothed the data in space by means of a spline under tension [26]. No smoothing was done in time, since the data was so smooth that such a treatment seemed not necessary. IV. RESULTS In this section we present the results of our simulation. In the first part we will deal with time independent quantities and in the second part with time dependent quantities. In many computer simulations dealing with the glass transition it is observed that there 7 exists a temperature T at which certain thermodynamic quantities, like e.g. the po- g−sim tential energy per particle, show some sort of discontinuity (e.g. a rapid change of slope) when plotted versus temperature (see, e.g., [17]). The temperature where these features are observed have nothing to do with the laboratory glass transition temperature T , the tem- g 13 perature where the viscosity of the material has a value of 10 Poise, since T is usually g−sim significantly higher than T . The physical significance of T is that at this temperature g g−sim the system under investigation has fallenout of equilibrium, or inother words hasundergone a glass transition, on the time scale of the computer simulation. Obviously T depends g−sim on the cooling rate with which the system was cooled and on the thermal history of the system. InFig.1 weshow thepressure, thetotalenergy, andthepotentialenergy ofour system as a functionoftemperature. Inorder to expand thetemperature scale atlowtemperatures, we plot these quantities versus T−1. We recognize from this figure that there is no temperature at which one of these quantities shows any sign of an anomalous behavior. Thus we can conclude that in this simulation the system is not undergoing a glass transition, i.e. that for the effective cooling rates used in this work, T is less than the lowest temperature g−sim investigated. Thus this is evidence that we are able to equilibrate the system even at the lowest temperatures. Stronger evidence for equilibration will be presented below. One of the simplest time dependent quantities to measure in a molecular dynamics 2 2 simulationisthemeansquareddisplacement (MSD)hr (t)iofataggedparticle,i.e. hr (t)i = h|r(t) −r(0)|2i. In Fig. 2 we show this quantity for the particles of type A versus time in a double logarithmic plot. The corresponding plot for the B particles is very similar. The curvestotheleftcorrespondtohightemperaturesandthosetotherighttolowtemperatures. We recognize that for short times all the curves show a power-law behavior with an exponent of two . Thus this is the ballistic motion of the particles. At high temperatures this ballistic motion goes over immediately into a diffusive behavior (power-law with exponent one). For low temperatures these two regimes are separated by a time regime where the motion of the particles seems to be almost frozen in that the MSD is almost constant and thus shows a plateau. At the lowest temperature this regime extends from about one time unit to about 3 10 time units. Only for much longer time (note the logarithmic time scale!) the curves show a power-law again, this time with unit slope indicating again that the particles have a diffusive behavior on this time scale. The fact that the length of our simulation is long enough in order to see this diffusive behavior even at the lowest temperature is a further indication that we are able to equilibrate the system at all temperatures. The reader should note that for the discussion of these different time regimes it is most helpful to plot the curves of the MSD with a logarithmic time axis. Only in this way it is possible to recognize that the dynamics of the system is very different on the various time scales. Note, that the value of the MSD in the vicinity of the plateau is about 0.04, thus corresponding to a distance of about 0.2. We therefore recognize that on this time scale the tagged particle has moved only over a distance that is significantly shorter than the next nearest neighbor distance (which is close to one, see below). Thus it is still trapped in the cage of particles that surrounded it at time zero, and it takes the particle a long time to get out of this cage. The initial stages of this slow breakup of the cage is exactly the type of process MCT predicts to happen during the β-relaxation. (We will later elaborate more on this point in the discussion of the self part of the van Hove correlation function.) Thus we 8 can identify the time range where we observe the plateau in the MSD with the β-relaxation regime of MCT. Fromthe MSD it is now easy to compute the self diffusion constant D(T) ofthe particles. (Using a plot such as Fig. 2, a straight line, with unit slope, fit to the long time behavior of the data intersects a vertical line at log t = 0 at a height of log 6D.) Since MCT 10 10 predicts that diffusion constants should have a power-law dependence on temperature at low temperatures (see Eq. (4)), we tried to make a three parameter fit with such a functional form. In Fig. 3 we show the result of this fit by plotting D versus T − T in a double c logarithmic way. We clearly observe, that, in accordance with MCT, for temperatures T ≤ 1.0 the diffusion constants follow a power-law behavior. The value of T is 0.435, c independent of the type of particle. This independence of T of the type of particles is in c accordance with the prediction of MCT. From the value of T we now can compute the small c parameter of the theory, i.e. ǫ = |T −T |/T . At the lowest temperature ǫ is 0.07, thus quite c c small and therefore it is not unreasonable to assume that we are already in the temperature range where the asymptotic results of the theory hold. At T = 1.0 the value of ǫ is 1.3, which seems rather too large for the asymptotic expansion to apply. However, it has been found in experiments that for some systems the predictions of MCT hold for values of ǫ of at least 0.5 [19]. Therefore our finding is not that astonishing. Also, by investigating the relaxation time of the intermediate scattering function we found that the asymptotic behavior at low temperatures is obtained for this quantity only for T ≤ 0.6 [15,22]. This corresponds to a value of ǫ of 0.4, which is comparable to the values found in experiments. Thus we see that the upper temperature for which the asymptotic behavior can be observed clearly depends on the quantity investigated. Note that this observation is in accordance with MCT since the theory predicts that the magnitude of the corrections to the asymptotic behavior will depend on the quantity considered. The exponent γ of the power-law for D(T) (see Eq. (4)) is 2.0 for the A particles and 1.7 for the B particles. Although MCT predicts these two exponents to be the same, a 10% deviationfromanasymptotic result isnotsurprising andthereforenota severe contradiction to this prediction of the theory. However, in a different work [15] we have analyzed the temperature dependence of the α-relaxation time τ (see Eq.(2)) and found, in accordance with the prediction of MCT, that at low temperatures τ shows a power-law behavior. MCT predicts that the exponent of the power-law for τ(T) and the exponent in the power-law for D(T) should be the same. Since we found that the former is about 2.6 [15] and we now find that the latter is around 1.9 we conclude that this prediction of the theory is not correct for our system. Since the connection proposed by MCT between the von Schweidler exponent b and the critical exponent γ [7] would imply γ=2.8 (using b = 0.49, which we determined in Ref. [15]) we tested whether a plot of D1/γ with γ = 2.8 versus T gives a straight line in some temperature interval. This would imply that in this temperature interval a power-law with exponent 2.8 would fit the data well. We found that for the A particles the data points for T ≤ 0.6 lie reasonably well on a straight line. This is not the case for the B particles in any range of temperatures. Furthermore, the critical temperature that is obtained for the A particles is around 0.40. This is significantly smaller that the critical temperatures we determined by other means and which were all around 0.435 [15,22]. Thus we think that a power-law with an exponent of 2.8 and a temperature around 0.435 is inconsistent with our 9 dataforthediffusionconstant. Fromthetheoreticalpointofviewitisofcourseinteresting to find that the relaxation times of the intermediate scattering function for nonzero values of q behavethewayMCTpredicts[15,16,22]whereasthediffusionconstants, relatedtoquantities at q = 0, do not follow these predictions as closely. To understand this observation it probablywillbenecessary toincreaseourunderstanding onthecorrectionsoftheasymptotic expressions ofthetheoryforsmallvaluesofq andwehopethatsomeprogresswillbepossible in this direction in the future. Also included in Fig. 3 is the result of a fit to the diffusion constants with a Vogel-Fulcher law, i.e. D ∝ exp(−B/(T −T0)). The Vogel-Fulcher temperature T0 is 0.268 and 0.289 for the A and B particles, respectively. These two temperatures are significantly lower than the critical temperatures found for other quantities, which were all around 0.435 [15,22]. Also, as can be seen from Fig. 3 in our case the quality of the Vogel-Fulcher fit is inferior to the one with a power-law. This shows that our data is good enough to distinguish between the two functional forms and that therefore the power-law we found is really significant. Note that this finding is not in contradiction with the situation often encountered in experiments, where the viscosity, or a relaxation time, is fitted well by a Vogel-Fulcher law over many orders of magnitude. The temperatures for which these fits are done are usually closer to the laboratory glass transition temperature T than the temperatures we deal with here. Thus g the viscosity is much larger than the viscosity one would obtain at the lowest temperature investigated in this work. Hence our statement is that in the temperature region investigated here the diffusion constant is better fitted by a power-law than by a Vogel-Fulcher law and at present nothing can be said about its behavior at lower temperatures. Since the inverse of the constant of diffusion gives a time scale, we plotted the MSD versus tD(T). The resulting plot is shown in Fig. 4. The curves to the left correspond to low temperatures and those to the right to high temperatures. We recognize from this figure that for intermediate and low temperatures the curves fall onto a master curve. A comparison with Fig. 2 shows that this master curve is present for those times that fall into the α-relaxation regime. Thus it can be expected that the master curve has something to do with the time temperature superposition principle (see Eq. (3)). We thus used as an ansatz a functional form that is an interpolation between the von Schweidler behavior at short rescaled times and a diffusive behavior at long rescaled times, e.g.: hr2(t)i = A(r2 +(Dt)b)+Dt . (5) c Here r ,A and b are fit parameters. The best fit for the A particles is included in Fig. 4 c as a dashed line. We recognize that the functional form given by Eq. (5) leads to a quite satisfactory fit in the time region where the master curve is observed. For the best value of b we obtained 0.48 and 0.43 for the A and B particles, respectively. These values are in accordance with the value for the von Schweidler exponent which we found for this system to be around b = 0.49 [15,16,22]. Thus MCT is able to rationalize this master curve quite convincingly. We now turn our attention to a closer examination of the motion of the particles. This is done conveniently with the help of Gα(r,t) and Gαβ(r,t) (α,β ∈ {A,B}), the self and s d distinct part of the van Hove correlation function [27]. Here and in the following we assume thatthesystemisisotropicandthereforeonlythemodulusofr enterstheequations. Gα(r,t) s is defined as 10