Density Mapping between MCT and BD

Solving MCT equations requires first and foremost an input for the static structure factors.

While previous investigations of the ABP-MCT carried out in [47] have been obtained by using the hard-disk structure factors predicted by the theory of Baus and Colot [94], the hard-disk structure factors used here were obtained from a density functional theory approach proposed by Thorneywork et. al [95]. The latter theory additionally provides the tracer bath direct correlation function for different size ratios of the tracer compared to the bath particle and allow to calculate structure factors for arbitrary compositions. With this theory as an input, the mode-coupling equations were solved with a time-decimation algorithm that is specially adapted to the present structure of the Mori-Zwanzig equation with a hopping term and provides solutions for the ISF and SISF over many decades in time. Details of the algorithm are presented in the appendix of [47] and are not repeated here. Having successively determined both the ISF and SISF, the low-qlimits of the tagged particle memory-kernel can be computed and a similar time-decimation scheme can be used to solve the equation of motion forδr²(t). All results presented in this chapter were obtained on a uniform wavevector-grid with qmin = 2.5 and qmax = 40 withN_q= 128grid points, were the wavevector integration has been carried out with a higher-order open Newton-Cotes formula to account for the divergence at q → 0 that occurs in the

wavenumber integration in 2D. Moreover, the rotational cutoff was chosen as Λ_L = 1. This set of parameters has provided numerically stable solution for self-propulsion velocities up to v₀ ∼8D_t/σ.

By using a bisection method that searches for the root off_0,0(q)as a function ofφin the passive system, where f_0,0(q) is determined through iteratively solving the algebraic equation (3.2.1) for D_r = 0, the passive glass transition point was found at φ_c ≈0.6986, which is close to that reported by Bayer et. for the same 2D system under usage of a structure factor taken from a hypernetted chain approximation, bringing φ_c ≈ 0.6968 [107]. Besides the difference in the structure factor, the deviation can also be attributed to the varying discretization scheme for the wavenumber grid and different integration schemes. Either way, a very precise determination of the transition point is not required in a comparative study with simulations since an equal density mapping is not expected in the first place for several reasons. First of all, it is well-known from former simulations in 2D [107] and 3D [50] that structure factors found in simulations generally differ from those predicted by approximate theories. Most theories are capable to reproduce the right peak position that stems from the hard-core volume exclusion and consistently reveals an increasing local ordering in the form of sharper peaks when increasing the density. Correctly, these peaks are also shifted to higherq values when increasing the density, but the effect of local ordering shows to be overestimated resulting in a lower second peak as observed in simulations.

This peak height influences the outcome of the wavevector integration carried out in the MCT calculations and results in an overestimation of the glass-formation tendency in MCT compared to what is observed in simulations [51]. But even when incorporating the structure factors obtained from the BD simulation as input for the MCT calculation as it was carried out in [51], this still leads to discrepancies owing to the approximate nature of MCT whose degree of quality is hard to quantify such that a parameter-free comparison is not possible even in that case and requires a density-mapping between both methods. For this work, the deviations from such a density-mapping should be minimized for an optimal investigation of activity effects. To achieve such an optimal mapping, least-square fits of the SISFs from the passive BD at fixed wavenumber q = 7.5 to the SISFs from the passive MCT have been carried out for several densities. Results from these fits for different φ_BD are depicted in figure 4.2.1 (a) and show a very good agreement in the whole range of fitted densities. The fit reveals a linear relation between the densities from MCT and BD given by φ_BD = 1.651·φ_{M CT} −0.376, which is additionally presented in the inset of figure 4.2.1 (a). With higher densities though, it becomes apparent that the MCT underestimates the SISFs in an intermediate time window after the short-time diffusion. This effect has also been reported for the ISF in a similar comparative study between BD and MCT in 2D [52] and reveals the general weak point of MCT in the description of the intermediate time regime. Still, the quantitative description of the final α-relaxation phase and the resulting structural α-relaxation time whose prediction is the major strength of MCT, is throughout satisfactory over the whole range of presented densities. Going beyond this density range reveals a breakdown of the linear density-mapping is found, which can be attributed to ergodicity-restoring effects in the simulations that are not captured by the MCT.

These effects stem from various origins and are reflected in deviations of the generalized Stokes-Einstein relation in terms of a decoupling of self-diffusion and structural relaxation caused by

cage-hopping effects [108] or dynamical heterogeneities [109]. To that reason, the investigations of this thesis are largely limited to the density range where a linear mapping was still possible, offering a good compromise between manageable simulation times and the emergence of a slow structural dynamics.

Figure 4.2.1.: (a) S_0,0^s (þq, t) of a passive tracer particle in the passive hard-disk system for a fixed wavenumber q = 7.5 and different packing fractions φBD as indicated. Crosses show BD simulation results, lines are least-square fits to the MCT predictions with resulting optimalφ_{M CT} described by the linear fit φ_BD= 1.651φ_{M CT} −0.376, shown as black line in the inset. (b)-(c) S_0,0^s (þq, t) as in the main figure, but for fixed densities φ_BD = 0.73 and φ_BD = 0.77 each for varying q.

To estimate the quality of the density mapping for the description of the SISF in a full range of wavenumbers, additional correlation functions are shown in the figures 4.2.1 (b) and (c).

These correspond to the different maxima of the structure factor at q = 7.5, q = 12.5, q = 17.5 and the lowest accessible wavenumber q = 2.5 for MCT, each for the densities φ_BD = 0.73 and φBD = 0.77 in the respective figures. For the density, φBD = 0.77 the deviations in the intermediate time range become visible again and do increase with higher wavenum-bers, which can be attributed to the erroneous treatment of the short-time collision regime of MCT.

The performed MCT fits for the SISF could additionally be used as a basis for a comparison between the MSDs. Although the fit for the SISF to the lowest wavenumber is well matched between BD and MCT, systematic deviations in the limiting case q → 0, that is necesarry to resolve the MSD, are to be expected, as documented in detail in [50] and [51]. Since this work primarily aims for an investigation of the influence of the activity, it is important to minimize these systematic deviations for an optimal comparison of the respective active systems. A separate density adjustment for the comparison of MSDs between BD and MCT was therefore performed, as shown in figure 4.2.2, that forms the basis for later comparisons of MSDs of active particles. This comparison reveals a quantitatively satisfactory agreement over the entire density range.

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10²

t !

σ

/D

"

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10²

δr

( t )

! σ

"

φBD= 0.50 φBD= 0.60 φ_BD= 0.65 φBD= 0.70 φBD= 0.73

φBD= 0.74 φBD= 0.75 φ_BD= 0.76 φBD= 0.77 M CT

0.51 0.56 0.61 0.66 0.71

φ^{M CT}

0.5 0.6 0.7 0.8

φBD

Figure 4.2.2.: MSD of a passive tracer particle in the passive hard-disk system for different packing fractionsφBD as indicated. Crosses show BD simulation results, lines are MCT fits with resulting optimal φ_{M CT} described by the linear fit Φ_BD = 1.661φ_{M CT} −0.337, shown as black line in the inset.