Inherent stress correlations in a quiescent two-dimensional liquid: Static analysis including finite-size effects

After constructing a formalism to analyze spatial stress correlations in two-dimensional equilibrated liquids, we show that the sole conjunction of mechanical balance and material isotropy demands all anisotropic components of the inherent state (IS) stress autocorrelation matrix to decay at long range as $1/r^2$ in the large system size limit. Furthermore, analyzing numerical simulation data for an equilibrated supercooled liquid, we bring evidence that, in finite-sized periodic systems, the autocorrelations of pressure and shear stresses present uniform backgrounds of amplitudes proportional to the inverse cell area. These backgrounds bring relevant contributions to macroscopic IS stress fluctuations, with the consequence that the latter scale as inverse area, yet in an anomalous way, inconsistent with viewing an IS as equivalent, in the thermodynamic limit, to an ensemble of independent finite-sized subsystems. In that sense, ISs are not spatially ergodic.

Figure 1

Coarse-grained fields—pressure (a) ${\sigma }_1$, (b) ${\sigma }_2$, and (c) ${\sigma }_3$—measured in one IS of a $L=80$ system equilibrated at $T=0.28$, using $r_c=2$ for better visibility.

Figure 2

The IS stress autocorrelation measured in $L=160$ systems equilibrated at $T=0.28$ and presented as a matrix of pictures corresponding to the fields $C_{ab}$ with $a,b=1...3$. In each frame, the origin of space lies at the center. The matrix is clearly symmetric: $C_{ab}=C_{ba}$. Note that $C_{11}$ resembles a $\delta$ function, and that $C_{22}$ and $C_{33}$ have opposite signs.

Figure 3

Cuts of the spatial autocorrelation functions from Fig. 2 ($L=160$, $T=0.28$) along the $x$ and the $\theta =\pi /4$ axes. (a) Comparing $C_{22}$ and $C_{33}$ along both axes with $C_{12}$ along $x$ and $C_{13}$ along $\theta =\pi /4$: the central peak is displayed by the self-correlations ($C_{22}$ and $C_{33}$) but not by the cross correlations ($C_{12}$ and $C_{13}$); its width $\simeq 1.5$ compares with the coarse-graining radius $r_c=1$. (b) log-log plots of $C_{22}$ and $C_{33}$ cuts modulo a minus sign to make the tail data positive; the dashed line is $\propto 1/r^2$. (c) $C_{33}$, $\theta =\pi /4$ cuts obtained with different values of the coarse-graining radius $r_c$: the peak width grows with $r_c$, but the tail is unchanged.

Figure 4

The RS inherent stress autocorrelation in $L=160$ systems, from the data of Fig. 2, using Eq. (15). The data are presented as a matrix of pictures corresponding to the fields ${\mathring{C}}_{ab}$ with $a,b=1...3$. In each case, the origin of space lies at the center. The color map is the same as in Fig. 2. Note that only the fields corresponding to the non-zero matrix components in Eq. (19) are visible, as predicted. The white spots at the centers of ${\mathring{C}}_{11}$ and ${\mathring{C}}_{33}$ mark the presence of positive correlation peaks.

Figure 5

Angle-averaged values of the RS correlation fields ${\mathring{C}}_{ab}$ in $L=160$ systems. (a) ${\mathring{C}}_{ab}$ vs $r$. (b) $r^2{\mathring{C}}_{ab}$ vs $r$. The fields ${\mathring{C}}_{13}$ and ${\mathring{C}}_{23}$ are shown on both graphs using thin black lines: they uniformly vanish as expected from isotropy.

Figure 6

The real part of ${{\displaystyle \underset{\approx }{\mathring{\widehat{C}}}}}_{\underline{k}ab}$, for the same conditions as in Figs. 2 and 4, is presented as a matrix of pictures, each of which is centered around $\underline{k}=\underline{0}$. The structure of the fields corresponds precisely to the form expected in Eq. (28).

Figure 7

Scatter plot of ${\mathring{\widehat{C}}}_{\underline{k}ab}$ and ${\mathring{\widehat{C}}}_{\underline{k}ab}^{\delta }$ vs $k$, for all $a,b\in \{1,2\}$, all measured $\underline{k}$ values, and different system sizes. Only $L=40$ and $L=80$ data are provided for ${\mathring{\widehat{C}}}_{\underline{k}ab}^{\delta }$ due the difficulty of its computation in large systems.

Figure 8

$r^2{\mathring{C}}_{ab}^{\square }$ vs $r$, for three system sizes: $L=40$ (open circles), 80 (filled circles), and 160 (lines).

Figure 9

(a) Integrals $I_{ab}\left(R\right)$ vs $R$, using the same color convention as in previous graphs. The purple line is the average $I_{*}=[I_{22}\left(R\right)+I_{33}\left(R\right)]/2$. The black and cyan lines tracking $I_{11}$ and $I_{*}$ (respectively), are the functions of the form $A+B\mathcal{A}$ as discussed in the text. (b) $I_{22}-I_{*}$ and $I_{33}-I_{*}$.