Numerical implementation

In this section, the basic requirements of a 3d implementation of the EQM (2.57) are discussed, i.e. isotropic resemblance of the quiescent, isotropic non-ergodicity parameters f_q^cof Ref. [71] and the choice of aq-grid discretization, which approximates an asymptotic solution of EQM (2.57) (or identifying that there is an asymptotically high grid resolution, which minimizes discretization errors).

Computing a fully anisotropic solution of the MCT-ITT EQM (2.57) in three dimen-sions (D= 3) remains challenging, because computation time and required access mem-ory increase exponentially with 2 ·D, cf. Sec. 2.3.3. With state-of-the-art hardware (OpenMP parallelization on 8 CPUs with ≃ 3.4MHz), the quiescent (D = 1) solution can be computed up to any desired precision, while the 3d case can still easily exceed computational limits; hence, a compromise in precision and computation time must be accepted. The key obstacle is the calculation of the memory kernel m_q(t), Eq. (2.58), which needs a k integration with the vertex V_qkp(t) being one factor of the integrand.

This vertex function is, due to shear-advection from time t0 = 0 to t, an anisotropic function ofq(t)and k(t) and thus2·Ddimensional in independent vector components.

Hence theqgrid is limited to a relatively small resolution compared to the 2d calculations of Ref. [55].

The numerical evaluation has been performed on a q grid with spherical symmetry,

3.1 Numerical implementation

i.e. the grid is discretized in polar coordinates

q=



 q_x q_y qz



=q





cosϕ_qsinϑ_q sinϕ_qsinϑ_q

cosϑq



, (3.1)

with modulus q, azimuthal angle ϕ_q, and inclination angle ϑ_q. Specific qs will be ad-dressed in this work with either notation,(q_x, q_y, q_z)or(q, ϕ_q, ϑ_q), which is unambiguous, because angles are given in degrees or ratios of π; see also Fig.3.2for an illustration of spherical coordinates. Moduli q are given in units of an inverse hard-core diameter [q] = d⁻¹, unless explicitly noted differently. Spherical coordinates have been also cho-sen for the 2d calculations behind [23, 42, 50]. The motivation to choose a spherical discretization, in contrary to the 2d Cartesian grid of Refs. [22, 55, 64], is, that the quiescent state is isotropic. Therefore the necessarily coarse discretization has at least the symmetry of the quiescent state and approximates the isotropic k integration in Eq. (2.58) better than a Cartesian grid. As a disadvantage, the density of grid points for largeq moduli is decreased compared to that of smallq moduli. Fortunately, the correla-torsΦ_q(t) are small for largeqand play a minor role in the mode-coupling equations, cf.

Fig.3.1. Another motivation is to identify spherical symmetries in the structure, viz. in the SF distortionsδSq(t). This might help in the future to expand the MCT-EQM (2.57) in a spherical-harmonics approximation. Then, an approximation of V_qk(t), Eq. (2.55), must be found to gain a useful formulation in spherical harmonics, which is however yet unknown due to its complete and time-dependent anisotropy. A projection of spherical harmonics on the memory kernelm_q(t)in each time and iteration step must be avoided;

see App. A.3.2 for the numerical iteration equations. Otherwise the computation time will be even increased, accompanied with a decrease in information compared to evalu-ating Φ_q directly on the 3d q grid. Hence, evaluating the MCT-ITT equations directly on a 3d anisotropically-sheared qgrid has been chosen as the method of this chapter.

In App. A.3, an explanation of the usually used iteration schemes for the one-time MCT-EQM (2.57) (momentum algorithm and Brader-Voigtmann (BV) algorithm) can be found. The BV algorithm has been chosen rigorously throughout this work, because it approaches an asymptotic solution of the MCT-EQM (2.57) for smaller time-grid resolutions and thus saves computing time. It is also usable for a two-times MCT, e.g.

in Ref. [59], which therefore increases the congruence of this work’s results to two-times MCT. Throughout this chapter, the Percus Yevick (PY) SF was used for calculations.

To achieve isotropic non-ergodicity parameters f_q^c even on a rather coarse q-grid dis-cretization, the integration overkwas done withqchosen ask_zaxis. Then, all correlators Φq(t)evaluate to be isotropic in the computation of the quiescent/equilibrium state. As a second advantage, p=q−k becomes effectively D+ 1dimensional in k integrations (three components of k and one from q, which points always in k_z direction), which saves a lot of access memory and computation time. However, V_qkp(t) remains 2·D dimensional, becauseq(t) is shear advected and does in general not point into the k_z(t) direction anymore. The calculation of k(t) becomes even more complicated, because k(t) is shear advected and depends on q(t= 0), which has been chosen as kz axis; see

3.1 Numerical implementation

App.A.1.2 for an explanation of the underlying coordinate rotation transformation.

Figure 3.1 shows the f_q^cs calculated with the memory-kernel iteration Eq. (2.59) to-gether with thef_q^cs of the one-dimensionally discretized Bengtzelius grid used in Ref. [71].

One verifies that for increasing grid resolution, the Bengtzelius f_q^cs are approximated asymptotically. Becauseq is chosen as k_z axis, the azimuthal angular resolution has no effect on this approximation in the quiescent, isotropic calculations, only the inclination discretization does, see App. A.1.1. The azimuthal discretization however has an effect on the shear stress calculations, cf. Fig. A.2, because under shear Vqkp(t), Φq(t), and thus the integrand in the shear stress k integral, Eq. (2.41), are completely anisotropic.

It has been checked (not shown here) that choosing q as another axis, e.g. k_x, changes thef_q^cs, but approximates the same asymptote.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 5 10 15 20 25 30 35 40

Bengtzelius

∆q= 0.4: π/∆ϑ_q= 8 2π/∆ϕ_q= 16

16 8

16 32

24 16

32 16

∆q= 0.2: 8 16

10 11 12 13 14

0.4 0.44 0.48 0.52

q d⁻¹

fc q

Fig. 3.1:Isotropic non-ergodicity parameters f_q^c for different q-grid discretizations. The 1d Bengtzelius grid [71] is shown as red crosses. All grids, despite the one of the blue dashed curve, have ∆q = 0.4, like the Bengtzelius grid. The comparison with ∆q = 0.2 shows that

∆q = 0.4 is sufficient. The legend gives the inclination angle resolution π/∆ϑq and azimuthal angle resolution2π/∆ϕq. Thef_q^cs depend only on∆ϑq, cf. magenta dotted andturquois dash-dotted curves. All sphericalq grids do not exhibit the small-qdecrease of the Bengtzelius grid.

Thedetailshows the second-peak region, where variations are biggest.

Table 3.1 shows the critical packing fraction φ_c of the hard-sphere MCT glass and its dependence on the q-grid discretization of∆q and ∆ϑ_q. Concluding from Tab. 3.1, Fig.3.1, and a more detailed discussion in App. A.2aboutq-grid discretization effects, a discretization of the inclination angle in 24 equidistant steps on[0;π]was chosen as best trade-off. The azimuthal angle was discretized in 24equidistant steps on [0; 2π]and the modulus in 100 steps on[0.2; 39.8], like on the Bengtzelius grid. This discretization was