Theory of length-scale dependent relaxation moduli and stress fluctuations in glass-forming and viscoelastic liquids

K_T(t), K_A(t) are universal functions by their definition, implying a perfect temperature adjustment to keep constant either T or the entropy S. Therefore, K_T(t) or K_A(t) cannot be directly obtained by, say, simulations with imperfect thermostats (allowing for some undefined T-variations). Still, they can be calculated even in this case based on the non-universal correlation functions, but it requires application of certain transformations to the correlation data, as described in Ref. 10. By contrast, this problem is absent for G(q, t) since shear stress fluctuations are insensitive to small T-variations.

An alternative assumption is that the periodic boundary conditions (PBCs) are applied.

Note that condition (15) does not imply any jump of $σαβ(q̲,t)$ at t = 0⁺ if the center-of-mass velocity of the system is kept equal to 0.

We do not use subscript “L” in K(q, t) since a q-dependent compression can be only longitudinal; it can be isotropic only for q = 0.

Here and below, the brackets 〈⋯〉 mean ensemble average supplemented with the averaging over t′, and $δX≡X−X$ for any X = T(t), p(t), etc.

Simultaneous correlations corresponding to the time-shift t = 0 are not altered by the thermostat due to its canonical nature.

Note that here we deal with nondimensional specific heat per particle, c_v, which is equal to the total heat capacity of the system divided by Nk_B.

The glass-transition temperature T_g ≈ 0.26 was determined in prior Monte Carlo (MC) and MD studies using a continuous cooling protocol with a (standard choice for the) finite cooling rate.¹⁰ Therefore, T_g corresponds to the temperature where the α-relaxation time becomes comparable to the cooling time (in the MC simulations with local moves or in MD simulations), and the system falls out of equilibrium. However, thanks to the particle-swap MC technique,⁵⁸ we succeed in equilibrating our 2D pLJ system down to T = 0.16.¹⁰ Starting from these equilibrated configurations, we performed MD simulations to explore the equilibrium dynamics of the supercooled liquid, also below T_g ≈ 0.26 (cf. Ref. 10 for a detailed discussion). Here, we continue to report T_g as a relevant temperature for our model because for T ≲ T_g, dynamic correlation functions show pronounced plateaux for times beyond the initial short-time relaxation. Upon cooling, these plateaux increase in length and extend over the entire sampling time of the MD simulation at low T.

The importance of interdiffusion processes for glass-forming systems with size polydispersity was discussed in Ref. 61 and for binary mixtures in the framework of mode-coupling theory in Ref. 62. However, the general form of the magnitude of the interdiffusion term [cf. prefactor in Eq. (52)] was not recognized there.

Macroscopic isotropy of the studied systems was demonstrated in Ref. 10.

Note that the agreement at long times slightly deteriorates at low temperatures, T ≲ T_g, which is a natural effect of the poorer statistics in the regime of long structural relaxation time τ_α.

Note that time-dependence of heat capacity c_v(t) is weak at t ≳ 50.⁴¹ 

The affine moduli μ_A and η_A are discussed in Ref. 10.

More precisely, the rate 1/τ_s is equal to the real part of the smallest roots of equation ρs²/q² + G(q, − s) = 0; cf. the denominator in the rhs of Eq. (70): $1/τs=R(s)$⁠. In a similar way, 1/τ_L is defined by ρs²/q² + K(q, −s) = 0; cf. the denominator in the rhs of Eq. (71).

Both relations (95) and (96) assume a virtual stationarity of a well-equilibrated system within metabasin.

Note that the region t ≲ 2τ_L provides a negligible contribution to the integral in Eq. (97) since Δt_max ≫ τ_L.

Note that while N_q = 2 for all q’s we consider (with n_q2 = 1, 8, 32, 64), it typically gets higher for larger q.