11institutetext: Institut Charles Sadron, Université de Strasbourg & CNRS, 23
rue du Loess, 67034 Strasbourg Cedex, France

# Spatial correlation functions for non-ergodic stochastic processes of
macroscopic systems

J.P. Wittmer<EMAIL_ADDRESS>A.N. Semenov  J. Baschnagel

(Received: date / Revised version: date; )

###### Abstract

Focusing on non-ergodic macroscopic systems we reconsider the variances
$\delta{\cal O}^{2}$ of time averages ${\cal O}[\mathbf{x}]$ of time-series
$\mathbf{x}$. The total variance $\delta{\cal
O}^{2}_{\mathrm{tot}}=\delta{\cal O}^{2}_{\mathrm{int}}+\delta{\cal
O}^{2}_{\mathrm{ext}}$ (direct average over all time-series) is known to be
the sum of an internal variance $\delta{\cal O}^{2}_{\mathrm{int}}$
(fluctuations within the meta-basins) and an external variance $\delta{\cal
O}^{2}_{\mathrm{ext}}$ (fluctuations between meta-basins). It is shown that
whenever ${\cal O}[\mathbf{x}]$ can be expressed as a volume average of a
local field ${\cal O}_{{\bf r}}$ the three variances can be written as volume
averages of correlation functions $C_{\mathrm{tot}}({\bf r})$,
$C_{\mathrm{int}}({\bf r})$ and $C_{\mathrm{ext}}({\bf r})$ with
$C_{\mathrm{tot}}({\bf r})=C_{\mathrm{int}}({\bf r})+C_{\mathrm{ext}}({\bf
r})$. The dependences of the $\delta{\cal O}^{2}$ on the sampling time
$\Delta\tau$ and the system volume $V$ can thus be traced back to
$C_{\mathrm{int}}({\bf r})$ and $C_{\mathrm{ext}}({\bf r})$. Various relations
are illustrated using lattice spring models with spatially correlated spring
constants.

##  1 Introduction

###  1.1 Background

Let us consider a stochastic dynamical variable $x(\tau)$, like certain
density or stress fields averaged over the system volume $V$, characterizing a
large physical system as a function of (continuous) time $\tau$. Extending
recent work on stationary stochastic processes in non-ergodic macroscopic
systems lyuda19a ; spmP1 ; spmP2 ; spmP3  we investigate here quite generally
the variances $\delta{\cal O}^{2}(\Delta\tau,V)$ of observables ${\cal
O}[\mathbf{x}]$ of time-series $\mathbf{x}$. As further specified in Sec. 3, a
time-series $\mathbf{x}$ stands for an ensemble of discrete data entries
$x_{t}$ sampled over a “sampling time” $\Delta\tau$ and ${\cal O}[\mathbf{x}]$
for a time-averaged moment over the data entries $x_{t}$. While for ergodic
systems independently created configurations $c$ are able in principle given
enough time to explore the complete (generalized) phase space, for strictly
non-ergodic systems they are permanently trapped in meta-basins GoetzeBook ;
Heuer08 . The different time-series $k$ of the same independent configuration
$c$ are then correlated being all confined to the same basin even if separated
by arbitrarily long spacer (tempering) time intervals spmP2 . A time-series
$\mathbf{x}_{ck}$ must now be characterized by two indices $c$ and $k$ and it
becomes crucial in which order $c$\- and $k$-averages are taken. This implies
that the commonly used total variance spmP2

$\delta{\cal O}^{2}_{\mathrm{tot}}(\Delta\tau,V)=\delta{\cal
O}^{2}_{\mathrm{int}}(\Delta\tau,V)+\delta{\cal
O}^{2}_{\mathrm{ext}}(\Delta\tau,V)$ (1)

becomes the sum of two independent terms: an internal variance $\delta{\cal
O}^{2}_{\mathrm{int}}$, measuring the typical fluctuations within each meta-
basins, and an external variance $\delta{\cal O}^{2}_{\mathrm{ext}}$,
comparing the different meta-basins. Importantly, $\delta{\cal
O}_{\mathrm{int}}$ and $\delta{\cal O}_{\mathrm{ext}}$ depend differently on
$\Delta\tau$ and $V$. For $\Delta\tau$ larger than the typical relaxation time
$\tau_{\mathrm{b}}$ of the meta-basins, $\delta{\cal
O}_{\mathrm{int}}(\Delta\tau,V)$ decays as
$\sqrt{\tau_{\mathrm{b}}/\Delta\tau}$ while $\delta{\cal
O}_{\mathrm{ext}}(\Delta\tau,V)$ becomes a $\Delta\tau$-independent constant.
This large-$\Delta\tau$ limit

$\Delta_{\mathrm{ne}}(V)\equiv\lim_{\Delta\tau\to\infty}\delta{\cal
O}_{\mathrm{ext}}(\Delta\tau,V)$ (2)

is our definition of the “non-ergodicity parameter” spmP1 ; spmP2 ; spmP3
,111Other definitions of non-ergodicity parameters may be found in the
literature GoetzeBook . Our definition does not rely on the properties of a
specific model or a theoretical assumption. It can be made system-size
independent by multiplying it with the appropriate non-universal volume
dependence $V^{\gamma}$. an important order parameter vanishing for ergodic
stochastic processes but remaining positive definite for non-ergodic systems
spmP2 .

###  1.2 Motivations

For macroscopic systems without long-range spatial correlations it is not
difficult to predict the system-size scaling of $\Delta_{\mathrm{ne}}(V)$
spmP2 . Quite generally, this leads to a power law
$\Delta_{\mathrm{ne}}(V)\simeq 1/V^{\gamma}$ where the exponent $\gamma$
naturally depends on the considered observable ${\cal O}[\mathbf{x}]$.
Deviations from this exponent suggest long-range correlations. Such deviations
have, e.g., been observed for the non-ergodicity parameter
$\Delta_{\mathrm{ne}}(V)$ associated to the elastic shear shear modulus spmP1
; Procaccia16  obtained by means of the stress-fluctuation formalism Lutsko88
; Lutsko89 ; Barrat06 ; WXP13  or the (closely related) variance of the shear
stresses lyuda19a ; spmP1 ; spmP2 ; spmP3  in viscoelastic and/or glass-
forming colloidal systems. Unfortunately, it becomes numerically rapidly
demanding to precisely obtain $\Delta_{\mathrm{ne}}(V)$ for increasingly large
systems and, quite generally, it gets impossible to characterize the spatial
correlations just by measuring the $V$-dependence of macroscopic properties
such as $\Delta_{\mathrm{ne}}(V)$. It is thus crucial to directly measure the
correlations Lemaitre14 ; Lemaitre15  and to do this consistently with the
non-ergodicity of the systems.

###  1.3 New key results

We assume in the present work that the macroscopic observable ${\cal
O}[\mathbf{x}]$ can be written as a linear superposition ${\cal
O}[\mathbf{x}]=\mathbf{E}^{{\bf r}}{\cal O}_{{\bf r}}$ of an associated local
field ${\cal O}_{{\bf r}}$. (Using the notation introduced in Sec. 2.1,
$\mathbf{E}^{{\bf r}}$ denotes here a spatial average over microcells at a
position ${\bf r}$ of the system.) One main point of the present study is to
show that it is then both possible and useful to write the three different
variances as volume averages

$\displaystyle\delta{\cal O}^{2}_{\mathrm{tot}}(\Delta\tau,V)$
$\displaystyle=$ $\displaystyle\mathbf{E}^{{\bf r}}C_{\mathrm{tot}}({\bf
r},\Delta\tau,V)$ (3) $\displaystyle\delta{\cal
O}^{2}_{\mathrm{int}}(\Delta\tau,V)$ $\displaystyle=$
$\displaystyle\mathbf{E}^{{\bf r}}C_{\mathrm{int}}({\bf r},\Delta\tau,V)$ (4)
$\displaystyle\delta{\cal O}^{2}_{\mathrm{ext}}(\Delta\tau,V)$
$\displaystyle=$ $\displaystyle\mathbf{E}^{{\bf r}}C_{\mathrm{ext}}({\bf
r},\Delta\tau,V)$ (5)

over the corresponding spatial correlation functions $C_{\mathrm{tot}}$,
$C_{\mathrm{int}}$ and $C_{\mathrm{ext}}$ which are properly defined below in
Sec. 5.1 and Appendix A, Eqs. (68-70). Moreover, this can be done in such a
way that

$C_{\mathrm{tot}}({\bf r},\Delta\tau,V)=C_{\mathrm{int}}({\bf
r},\Delta\tau,V)+C_{\mathrm{ext}}({\bf r},\Delta\tau,V)$ (6)

holds in analogy of Eq. (1). This makes it possible to trace back the
$\Delta\tau$\- and $V$-dependences of the three different macroscopic
variances to the two correlation functions $C_{\mathrm{int}}$ and
$C_{\mathrm{ext}}$. Various relations will be illustrated by means of a simple
“Lattice Spring Model” (LSM) characterized by two quenched and spatially
correlated lattice fields.

###  1.4 Outline

We begin by addressing in Sec. 2 some technicalities such as useful
conventions (Sec. 2.1 and Sec. 2.2), the description of the LSM (Sec. 2.3 and
Sec. 2.6) and the determination and use of spatial correlation functions (Sec.
2.4 and Sec. 2.5). We construct then in Sec. 3 from the instantaneous
stochastic processes $x_{t}$ and fields $x_{t{\bf r}}$ (Sec. 3.1) the time-
averaged observables ${\cal O}[\mathbf{x}]$ and fields ${\cal O}_{{\bf r}}$
(Sec. 3.2) Summarizing Refs. spmP1 ; spmP2 ; spmP3  we remind in Sec. 4.1
various features of the three variances $\delta{\cal O}^{2}$. The
corresponding $\Delta\tau$\- and $V$-dependences are illustrated in,
respectively, Sec. 4.3 and Sec. 4.4 using numerical results obtained by means
of LSM simulations. We turn in Sec. 5 to the spatial correlation functions.
Examples from the LSM simulations are discussed in Sec. 5.2. We conclude the
paper in Sec. 6 where we shall finally hint on results of preliminary related
work investigating the internal and external spatial correlation functions of
time-averaged fields obtained from instantaneous stress fields in amorphous
colloidal glasses. The derivation of Eqs. (3-6) is presented in Appendix A.
The general scaling of the internal correlation function
$C_{\mathrm{int}}[v]({\bf r},\Delta\tau,V)$ of the important local variance
field ${\cal O}_{{\bf r}}=v_{{\bf r}}$ (as defined in Sec. 3.2) is discussed
in detail in Appendix B.

##  2 Conventions and technicalities

###  2.1 Notations

We use the same compact operator notations as in Refs. spmP2 ; spmP3 . The
arithmetic $l$-average operator

$\mathbf{E}^{l}{\cal
O}_{lmn\ldots}\equiv\frac{1}{N_{l}}\sum_{l=1}^{N_{l}}{\cal O}_{lmn\ldots}$ (7)

takes a property ${\cal O}_{lmn\ldots}$ depending possibly on several indices
$l,m,\ldots$ and projects out the specified index $l$, i.e. the $l$-average
${\cal O}_{mn\ldots}(N_{l})\equiv\mathbf{E}^{l}{\cal O}_{lmn\ldots}$ does not
depend on $l$, but as marked by the argument may depend on the upper bound
$N_{l}$. The $l$-variance operator $\mathbf{V}^{l}$ is defined by

$\displaystyle\mathbf{V}^{l}{\cal O}_{lmn\ldots}$ $\displaystyle\equiv$
$\displaystyle\mathbf{E}^{l}{\cal O}_{lmn\ldots}^{2}-\left(\mathbf{E}^{l}{\cal
O}_{lmn\ldots}\right)^{2}$ (8) $\displaystyle\equiv$
$\displaystyle\frac{1}{N_{l}}\sum_{l=1}^{N_{l}}\left({\cal
O}_{lmn\ldots}-{\cal O}_{mn\ldots}\right)^{2}.$

Note that the “empirical variance” $\delta{\cal
O}_{mn\ldots}^{2}(N_{l})\equiv\mathbf{V}^{l}{\cal O}_{lmn\ldots}$ vanishes

$\delta{\cal O}_{mn\ldots}^{2}(N_{l})\to 0\mbox{ for }N_{l}\to 1.$ (9)

In many cases ${\cal O}_{mn\ldots}(N_{l})$ and $\delta{\cal
O}_{mn\ldots}(N_{l})$ converge for large $N_{l}$ or become stationary for a
large $N_{l}$-window of the experimentally or numerically accessible
$N_{l}$-range. To simplify notations we often denote this limit by ${\cal
O}_{mn\ldots}$ and $\delta{\cal O}_{mn\ldots}$ without the argument $N_{l}$.
As discussed in Ref. spmP2 , we have defined the empirical variance as an
uncorrected (biased) sample variance without the usual Bessel correction
LandauBinderBook , i.e. we normalize in Eq. (8) with $N_{l}$ and not with
$N_{l}-1$. This implies

$\delta{\cal
O}_{mn\ldots}^{2}(N_{l})\simeq\left(1-\frac{1}{N_{l}}\right)\delta{\cal
O}_{mn\ldots}^{2}$ (10)

for variances obtained with finite $N_{l}$. This relation is used below to
extrapolate finite-$N_{l}$ observables to $N_{l}\to\infty$.

###  2.2 Periodic grid of microcells

Figure 1: Sketch of a two-dimensional ($d=2$) square lattice with
$n_{\mathrm{grid}}=L/a_{\mathrm{grid}}=2^{3}$. The filled circles indicate
microcells of the principal box, the open circles some periodic images. The
spatial position ${\bf r}$ of a microcell is either given by the $r_{x}$\- and
$r_{y}$-coordinates (in the principle box) or by the distance $r=|{\bf r}|$
from the origin (large circle) and the angle $\theta$.

We shall illustrate below various properties by means of (real and discrete)
fields $f_{{\bf r}}$ (${\bf r}$ labeling the microcell position) corresponding
to $N_{{\bf r}}$ microcells on regular grids in $d=2$ dimensions as sketched
in Fig. 1. For simplicity we assume square periodic lattices of linear
dimension $L=n_{\mathrm{grid}}a_{\mathrm{grid}}$, i.e. of ($d$-dimensional)
volume $V=L^{d}=N_{{\bf r}}\delta V$ with $\delta V=a_{\mathrm{grid}}^{d}$
being the microcell volume. To characterize spatial correlations (cf. Sec.
2.4) it is convenient numrec  to focus not on $f_{{\bf r}}$ but on its
discrete Fourier transform

$f_{{\bf q}}={\cal F}\\{f_{{\bf r}}\\}=\mathbf{E}^{{\bf r}}f_{{\bf
r}}\exp(i{\bf q}\cdot{\bf r})$ (11)

with $\mathbf{E}^{{\bf r}}$ being the average over all $N_{{\bf r}}$
microcells, using the notation Eq. (7), and ${\bf q}$ the discrete wavevectors
(being commensurate with the periodic grid).222The inverse Fourier transform
is $f_{{\bf r}}=\sum_{{\bf q}}f_{{\bf q}}\exp(-i{\bf q}\cdot{\bf r})$. Due to
our Fourier transform convention the sum rule $f_{{\bf q}=0}=\mathbf{E}^{{\bf
r}}f_{{\bf r}}$ holds. We denote by $f$ the field irrespective of its
representation and specific values while $f_{{\bf r}}$ refers to the
instantaneous field in real space and $f_{{\bf q}}$ to the corresponding
discrete field in reciprocal space. Fast Fourier Transforms (FFT) numrec  are
naturally used for the efficient transformation between real and reciprocal
space and it is thus convenient to set $n_{\mathrm{grid}}$ to be an integer
power of $2$. Moreover, we set $a_{\mathrm{grid}}=1$, i.e.
$L=n_{\mathrm{grid}}$ and $V=N_{{\bf r}}$.

LSM | description | $C[a]({\bf r})$ | $C[b]({\bf r})$ | $J$
---|---|---|---|---
A | uncorrelated sites | $a_{{\bf r}}\in{\cal U}[-0.1,0.1]$ | $b_{{\bf r}}\in{\cal U}[0.1,1.9]$ | 0
B | exponential decay | $C[a]({\bf r})\approx c_{0}\exp(-r/\xi)$ |  $c_{{\bf r}}=a_{{\bf r}}$,$b_{{\bf r}}=1+c_{{\bf r}}^{2}$ | 0, 0.1, 1
C | power law decay |  $C[a]({\bf r})\approx c_{0}/r^{\alpha}$ with $\alpha=1,2,3$ |  $c_{{\bf r}}=a_{{\bf r}}$,$b_{{\bf r}}=(1+0.1c_{{\bf r}})^{2}$ | 0
D | anisotropic decay | $C[a]({\bf q})=8\pi c_{0}(q_{x}q_{y}/q^{2})^{2}/N_{{\bf r}}$ |  $c_{{\bf r}}=a_{{\bf r}}$,$b_{{\bf r}}=(1+0.1c_{{\bf r}})^{2}$ | $-1,0,\ldots,10$
Table 1: LSM variants studied with the third column indicating the imposed
$C[a]({\bf r})$ and the forth $C[b]({\bf r})$. The uncorrelated random fields
$a_{{\bf r}}$ and $b_{{\bf r}}$ of LSM-A are taken from the given uniform
distributions ${\cal U}[\ldots]$. In all other cases $b_{{\bf r}}$ is computed
using the indicated relation from an auxiliary field $c_{{\bf r}}$. Naturally,
the inverse spring constant $b_{{\bf r}}=1/k_{{\bf r}}$ is always positive.
The coupling parameter $J$ (fifth column) for springs of neighboring grid
sites is switched off but for LSM-B and LSM-D. All correlations are isotropic
for LSM-B and LSM-C while they are anisotropic for LSM-D.

###  2.3 Lattice spring model

We present below MC simulations of a “Lattice Spring Model” (LSM) with
$x_{{\bf r}}$ being the linear length of the ideal springs and $a_{{\bf r}}$
and $b_{{\bf r}}=1/k_{{\bf r}}>0$ two quenched fields imposing, respectively,
the average length of a spring and its variance. In addition, neighboring
springs may be coupled by tuning a “coupling parameter” $J$. The energy
$E_{{\bf r}}$ of a microcell at ${\bf r}$ is thus given by

$E_{{\bf r}}=\frac{k_{{\bf r}}}{2}\left(x_{{\bf r}}-a_{{\bf
r}}\right)^{2}+\frac{J}{2}\sum_{{\bf r}^{\prime}}(x_{{\bf r}^{\prime}}-x_{{\bf
r}})^{2}$ (12)

where the sum over ${\bf r}^{\prime}$ runs over the $2d$ nearest-neighbors of
${\bf r}$ on the periodic grid. In the limit where the interactions between
springs are switched off ($J=0$) or are small, this implies the thermal
averages $\left<x_{{\bf r}}\right>=a_{{\bf r}}$ and $\left<\delta x_{{\bf
r}}^{2}\right>=b_{{\bf r}}T>0$ with $T$ being the temperature and setting
Boltzmann’s constant $k_{\mathrm{B}}$ to unity. We impose $T=1$ in all
presented simulations. A summary of the studied model variants is given in
Table 1. How spatially correlated fields $f=a$ and $f=b$ are generated is
explained in Sec. 2.6. Using these fields we perform Metropolis MC simulations
with local moves LandauBinderBook ; AllenTildesleyBook . Results are recorded
in time intervals $\delta\tau=10$ measured in MC steps.

###  2.4 Spatial correlation functions

In this work we shall impose or sample auto-correlation functions $C[f]({\bf
r})=\langle K[f_{{\bf r}}]({\bf r})\rangle-\left<\mathbf{E}^{{\bf r}}f_{{\bf
r}}\right>^{2}$ of various fields $f$.333We note $f$ for the functional
argument of the averaged correlation function $C$ and $f_{{\bf r}}$ for the
functional argument of the non-averaged correlation function $K$.
$\left<\ldots\right>$ stands here for some general average (to be specified
below), ${\bf r}$ for any site (microcell) of the principal simulation box and

$K[f_{{\bf r}}]({\bf r})\equiv\mathbf{E}^{{\bf r}^{\prime}}f_{{\bf r}+{\bf
r}^{\prime}}f_{{\bf r}^{\prime}}$ (13)

for the non-averaged correlation function of one given field $f_{{\bf r}}$.
All correlation functions are even and periodic (Fig. 1). Periodicity is most
readily implemented in reciprocal space using the Wiener-Khinchin theorem
(WKT) numrec

$K[f_{{\bf q}}]({\bf q})\equiv{\cal F}\\{K[f_{{\bf r}}]({\bf r})\\}=|f_{{\bf
q}}|^{2}=f_{{\bf q}}f_{-{\bf q}}$ (14)

for $f_{{\bf q}}={\cal F}\\{f_{{\bf r}}\\}$. The Fourier transformed auto-
correlation functions are thus real and positive for all wavevectors ${\bf
q}$.

All (averaged) correlation functions $C[f]({\bf r})$ or $C[f]({\bf q})$
considered here have in addition $x\leftrightarrow y$-symmetry, but are not
necessarily radial symmetric (isotropic) spmP3 . Instead of the
$d$-dimensional fields $C[f]({\bf r})$ we present below the weighted
projections

$C[f,p](r=|{\bf r}|)\equiv\left<C[f](r,\theta)\cos(2p\theta)\right>_{\theta}$
(15)

averaged over all lattice sites (angles $\theta$) at the same (or similar) $r$
with $p=0,1,2,\ldots$ Due to the $x\leftrightarrow y$-symmetry only even
$p=0,2,4,\ldots$ are allowed. We focus here on $p=0$ (“isotropic projection”)
and $p=2$ (“anisotropic projection”) spmP3 . If not stated otherwise $p=0$ is
assumed.

###  2.5 $V$-dependence of observables

As explained in the Introduction it is a general problem to explain or predict
the system-size dependence of an observable ${\cal P}(V)$ for asymptotically
large volumes $V$. The idea is to express ${\cal P}(V)^{2}=\mathbf{E}^{{\bf
r}}C[f]({\bf r})$ as an average of a suitable correlation function $C[f]({\bf
r})$ of a field $f$ which can be independently obtained numerically or
understood on theoretical grounds. Using the isotropic ($p=0$) projection
$C[f](r)$ of $C[f]({\bf r})$ we have

$V\ {\cal P}(V)^{2}\approx I(V)\equiv\int\ \mathrm{d}r\ r^{d-1}\ C[f](r)$ (16)

in $d$ dimensions. Let us write ${\cal P}(V)\sim 1/V^{\gamma}$ for large $V$
using the phenomenological exponent $\gamma$. Several cases are important. If
$C[f](r)$ vanished more rapidly then $1/r^{d}$, the integral $I(V)$ is
dominated by its lower bound and $V{\cal P}^{2}(V)$ becomes constant, hence,
$\gamma=1/2$ if the lower bound does not explicitly dependent on $V$. If on
the other hand $C[f](r)\simeq c_{0}>0$ for large $r$ with $c_{0}$ being a
constant, $I(V)\simeq c_{0}V$ for large $V$ and, hence, $\gamma=0$ if $c_{0}$
is $V$-independent. More generally, LSM-C (Table 1) illustrates power-law
correlations with

$C[f](r)\simeq c_{0}/r^{\alpha}\mbox{ for }1\ll r\ll L/2$ (17)

with $c_{0}$ being a $V$-independent constant. While (as already said)
$\gamma=1/2$ for $\alpha>d$, this implies $\gamma=\alpha/2d<1/2$ for long-
range correlations ($\alpha<d$), i.e. $\gamma\to 0$ for $\alpha\to 0$.
Finally, we note that the intermediate case with $\alpha=d$ yields the
logarithmic relation

${\cal P}(V)\simeq\sqrt{(c_{1}+c_{2}\ln(V))/V}\mbox{ for }V\to\infty$ (18)

with $c_{1}$ and $c_{2}>0$ being again $V$-independent constants.

###  2.6 Imposing $C[a]({\bf r})$ and $C[b]({\bf r})$

Figure 2: Isotropically averaged ($p=0$) correlation functions $C[f,p=0](r)$
for LSM-B with $c_{0}=1$, $\xi=16$ and $L=512$ characterizing the quenched
fields $f=a$ and $f=b$ for all $J$. Note that $C[a,0](r)=\exp(-r/\xi)$ as
imposed (bold solid line). For $f=b$ we compare the two closures $b_{{\bf
r}}=1+a_{{\bf r}}^{2}$ (squares) and $b_{{\bf r}}=(1+0.1a_{{\bf r}})^{2}$. The
dashed lines indicate the expected behavior for $L\gg\xi$.

Figure 3: Double-logarithmic representation of $C[f,p=0](r)$ for LSM-C and
$-C[f,p=2](r)$ for LSM-D for the two quenched fields $f=a$ and $f=b$ assuming
that $b_{{\bf r}}=(1+0.1a_{{\bf r}})^{2}$.

As indicated in Table 1 the frozen fields $f=a$ and $f=b$ of LSM-A, our
simplest LSM variant, are spatially decorrelated and uniformly distributed
random variables. In all other considered cases these fields are spatially
correlated as shown in Fig. 2 for LSM-B and in Fig. 3 for LSM-C and LSM-D. We
explain here how this is done. We remind that, quite generally, a spatially
correlated field with $C[f]({\bf r})=C_{\mathrm{imp}}({\bf r})$ is generated
by setting FractalConcepts ; spmP3

$f_{{\bf q}}=\sqrt{C_{\mathrm{imp}}({\bf q})N_{{\bf r}}}\ g_{{\bf q}}\mbox{
with }g_{{\bf q}}={\cal F}\\{g_{{\bf r}}\\}$ (19)

being the Fourier transform of a (decorrelated) random Gaussian field $g_{{\bf
r}}\in{\cal N}(0,1)$ of zero mean and unit variance. As a consequence

$\left<|f_{{\bf q}}|^{2}\right>=C_{\mathrm{imp}}({\bf q})N_{{\bf
r}}\left<g_{{\bf q}}g_{-{\bf q}}\right>=C_{\mathrm{imp}}({\bf q}),$ (20)

i.e. according to the WKT we have $C[f]({\bf r})=C_{\mathrm{imp}}({\bf r})$
upon inverse Fourier transform back to real space.444Using Parseval’s theorem
it is seen that $\langle g_{{\bf q}}g_{-{\bf q}}\rangle=1/N_{{\bf r}}$. It is
assumed here that (in addition of being even and periodic functions) the
imposed $C_{\mathrm{imp}}({\bf q})$ must be for all ${\bf q}$ both real and
positive in agreement with Eq. (14). If $C_{\mathrm{imp}}$ is known (stated)
in real space, it may be thus necessary to regularize the desired relation.
For instance, the power law $C_{\mathrm{imp}}({\bf r})=c_{0}/r^{\alpha}$ must
be changed to

$C_{\mathrm{imp}}({\bf r})=c_{0}(1+r^{2})^{-\alpha/2}$ (21)

to avoid the singularity at the origin FractalConcepts . This is done, e.g.,
with $c_{0}=1$ and $\alpha=2$ for $f=a$ of LSM-C as shown in Fig. 3. To avoid
such numerical problems $C_{\mathrm{imp}}({\bf q})$ is best directly imposed
in reciprocal space. This is the case for $f=a$ and $f=b$ of LSM-D where we
impose the anisotropic correlation function

$C_{\mathrm{imp}}({\bf q})=8\pi\frac{c_{0}}{N_{{\bf r}}}\
(q_{x}q_{y}/q^{2})^{2}$ (22)

($q_{x}$ and $q_{y}$ being the discrete components of the wavevector)
motivated by the theoretically predicted shear-stress correlations of
viscoelastic liquids and glasses Fuchs17 ; Fuchs18 ; Fuchs19 ; lyuda18 . As
may be seen for $f=a$ of LSM-D in Fig. 3 (squares) Eq. (22) thus leads in real
space to lyuda18

$C({\bf r})=C_{\mathrm{imp}}({\bf r})=-2c_{0}\cos(4\theta)/r^{2}\mbox{ for
}r>0,$ (23)

i.e. the isotropic average vanishes (not shown) and the anisotropic average
$C[f,p=2](r)=-c_{0}/r^{2}$ is long-ranged (bold solid line).

An additional problem arises for the correlated $f=b$ since all inverse spring
constants $b_{{\bf r}}$ must be positive. To impose this we generate first an
auxiliary field $c_{{\bf r}}$ using the above scheme. In all cases presented
here $c_{{\bf r}}=a_{{\bf r}}$. From this $b_{{\bf r}}$ is obtained by
setting, e.g., $b_{{\bf r}}=1+c_{{\bf r}}^{2}$. As seen in Fig. 2 this
“closure” leads for LSM-B to $C[b]({\bf r})\approx 2\exp(-2r/\xi)$ for
$L\gg\xi$ as can be also proved theoretically. Another possibility is to set

$b_{{\bf r}}=(1+\lambda c_{{\bf r}})^{2}\approx 1+2\lambda c_{{\bf r}}\mbox{
for }|\lambda|\ll 1.$ (24)

With $C[c]({\bf r})$ being the correlation function of the auxiliary variable
this implies to leading order

$C[b]({\bf r})\approx(2\lambda)^{2}C[c]({\bf r})\mbox{ for }|\lambda|\ll 1$
(25)

which is merely a shift in logarithmic coordinates. That this works well can
be seen (triangles) in Fig. 2 for LSM-B and in Fig. 3 for LSM-C and LSM-D.

##  3 Stochastic processes, observables and corresponding fields

###  3.1 Time-series and associated local fields

It is common to characterize a stochastic process $x(\tau)$ using ensembles
$\\{\mathbf{x}\\}$ of discrete time-series

$\mathbf{x}=\\{x_{t}=x(\tau=t\delta\tau),t=1,\ldots,N_{\mathrm{t}}\\}$ (26)

with $t$ being the discrete time, $\delta\tau$ the time interval between the
equidistant measurements and $\Delta\tau=N_{\mathrm{t}}\delta\tau$ the
available “sampling time”. We assume that the global stochastic process is a
$d$-dimensional volume average

$x_{t}=\mathbf{E}^{{\bf r}}x_{t{\bf r}}\approx\frac{1}{V}\int\mathrm{d}{\bf
r}\ x_{t{\bf r}}$ (27)

over a discrete field $x_{t{\bf r}}$ of same dimension. As a specific example
we consider the spatial average $x_{t}=\mathbf{E}^{{\bf r}}x_{t{\bf r}}$ of
the LSM spring lengths $x_{t{\bf r}}$ (cf. Sec. 2.3). It is useful to directly
measure $x_{t{\bf q}}={\cal F}\\{x_{t{\bf r}}\\}$ in reciprocal space. Since
we consider stochastic processes in non-ergodic systems $\mathbf{x}_{ck}$,
$x_{ckt}$, $x_{ckt{\bf r}}$ and $x_{ckt{\bf q}}$ are additionally
characterized by the index $c$ of the independent configuration and the index
$k$ of the time-series of a given $c$.

###  3.2 $t$-averaged observables and fields

Importantly, it is generally not possible to store all sets of time-series
$\mathbf{x}$ and associated fields but one normally only computes and stores
functionals (moments) ${\cal O}[\mathbf{x}]$ of each time-series, called here
“$t$-averages” or “observables”. The two observables we shall focus on are the
arithmetic mean

${\cal O}[\mathbf{x}]=m[\mathbf{x}]\equiv\mathbf{E}^{t}x_{t}$ (28)

and the empirical variance

${\cal O}[\mathbf{x}]=v[\mathbf{x}]\equiv\beta V\ \mathbf{V}^{t}x_{t}$ (29)

with $\beta=1/T$ being the inverse temperature ($k_{\mathrm{B}}=1$). The
prefactor $\beta V$, introduced for consistency with previous work spmP1 ;
spmP2 ; spmP3 , is natural for stochastic processes $x_{t}$ corresponding to
intensive thermodynamic variables Lebowitz67 .555In this case $v[\mathbf{x}]$
has the dimension of a (free) energy density just as the stress (pressure) of
the system. We often write below compactly ${\cal O}_{ck}={\cal
O}[\mathbf{x}_{ck}]$.

As already pointed out in the Introduction, we assume that, as the stochastic
process $x_{t}$, also the observables ${\cal O}$ may be written as linear
volume averages ${\cal O}[\mathbf{x}]=\mathbf{E}^{{\bf r}}{\cal O}_{{\bf r}}$
of local contributions ${\cal O}_{{\bf r}}$. For ${\cal
O}[\mathbf{x}]=m[\mathbf{x}]$ these local contributions are given by ${\cal
O}_{{\bf r}}=m_{{\bf r}}\equiv\mathbf{E}^{t}x_{t{\bf r}}$. Importantly, it is
also possible to write the $t$-averaged variance as
$v[\mathbf{x}]=\mathbf{E}^{{\bf r}}v_{{\bf r}}$ defining the “local variance”

$\displaystyle v_{{\bf r}}$ $\displaystyle\equiv$ $\displaystyle\beta V\
\mathbf{E}^{t}(x_{t{\bf r}}-x_{{\bf r}})(x_{t}-x)$ (30) $\displaystyle=$
$\displaystyle\beta V\ \left(\mathbf{E}^{t}x_{t{\bf r}}x_{t}-x_{{\bf
r}}x\right)$

with $x_{{\bf r}}=\mathbf{E}^{t}x_{t{\bf r}}$ and $x=\mathbf{E}^{{\bf
r}}x_{{\bf r}}$. Strictly speaking, $v_{{\bf r}}$ is a “co-variance”
correlating the local field to the total average. Such local variances appear
in the stress-fluctuation formulae for local elastic moduli Lutsko88 ;
Lutsko89 ; Barrat06 .666The covariance $v_{{\bf r}}$ must be distinguished
from the purely local variance $\tilde{v}_{{\bf r}}=\beta
V\mathbf{E}^{t}(x_{t{\bf r}}-x_{{\bf r}})^{2}$. For numerical reasons it is
convenient to compute the local fields $m_{{\bf r}}$ and $v_{{\bf r}}$ in
Fourier space from $x_{t{\bf q}}$ using $m_{{\bf q}}=\mathbf{E}^{t}x_{t{\bf
q}}$ and $v_{{\bf q}}=\beta V\mathbf{E}^{t}(x_{t{\bf q}}-x_{{\bf
q}})(x_{t}-x)$ with $x=x_{{\bf q}={\bf 0}}$.

We remark finally that for the LSM versions with no or weak interactions
between neighboring sites we have quite generally777To show the second
relation it is used that the $x_{{\bf r}}-a_{{\bf r}}$ are decorrelated for
$J\to 0$ albeit their first and second moments may be correlated. $v_{{\bf
r}}\to b_{{\bf r}}$ holds for all $V$ and $\beta$ due to prefactor $\beta V$
in the definition of $v_{{\bf r}}$.

$m_{{\bf r}}\to a_{{\bf r}},v_{{\bf r}}\to b_{{\bf r}}\mbox{ for }J\to 0\mbox{
and }N_{\mathrm{t}}\to\infty.$ (31)

In other words, since we know $C[a]({\bf r})$ and $C[b]({\bf r})$ by
construction, Eq. (31) determines (for the specified limits) the spatial
correlations for the local fields $m_{{\bf r}}$ and $v_{{\bf r}}$.

##  4 Global properties

###  4.1 Reminder of recent work

Summarizing recent work spmP2 ; spmP3  we discuss now several general
properties of expectation values and variances of observables ${\cal
O}_{ck}\equiv{\cal O}[\mathbf{x}_{ck}]$ in non-ergodic systems. We focus first
on the dependences on the number $N_{\mathrm{c}}$ of independent
configurations $c$ and the number $N_{\mathrm{k}}$ of time-series $k$ for each
$c$ and discuss then the dependences on sampling time $\Delta\tau$ and volume
$V$.

The first point to be made is that the total average ${\cal
O}(N_{\mathrm{c}},N_{\mathrm{k}})$ of the ${\cal O}_{ck}$ can be obtained
equivalently by

${\cal O}(N_{\mathrm{c}},N_{\mathrm{k}})=\mathbf{E}^{c}\mathbf{E}^{k}{\cal
O}_{ck}=\mathbf{E}^{k}\mathbf{E}^{c}{\cal O}_{ck}=\mathbf{E}^{l}{\cal O}_{l},$
(32)

i.e. $c$\- and $k$-averages commute and for such “simple averages” spmP2  the
two indices $c$ and $k$ can be “lumped” together in one index $l$ with
$N_{l}=N_{\mathrm{c}}N_{\mathrm{k}}$ as indicated by the last sum. The order
of averaging matters, however, for the variance of ${\cal O}_{ck}$ for which
three different definitions are relevant:

$\displaystyle\delta{\cal
O}^{2}_{\mathrm{tot}}(N_{\mathrm{c}},N_{\mathrm{k}})$ $\displaystyle\equiv$
$\displaystyle\mathbf{V}^{l}{\cal O}_{l},$ (33) $\displaystyle\delta{\cal
O}^{2}_{\mathrm{int}}(N_{\mathrm{c}},N_{\mathrm{k}})$ $\displaystyle\equiv$
$\displaystyle\mathbf{E}^{c}\mathbf{V}^{k}{\cal O}_{ck}\mbox{ and }$ (34)
$\displaystyle\delta{\cal
O}^{2}_{\mathrm{ext}}(N_{\mathrm{c}},N_{\mathrm{k}})$ $\displaystyle\equiv$
$\displaystyle\mathbf{V}^{c}\mathbf{E}^{k}{\cal O}_{ck}.$ (35)

As shown in Ref. spmP2 , with these definitions Eq. (1) exactly holds. The
“total variance” $\delta{\cal
O}^{2}_{\mathrm{tot}}(N_{\mathrm{c}},N_{\mathrm{k}})$ is the standard commonly
computed variance lyuda19a ; spmP1 ; Procaccia16 . We emphasize that
$\delta{\cal O}^{2}_{\mathrm{tot}}(N_{\mathrm{c}},N_{\mathrm{k}})$ is again a
“simple average”, i.e. all time-series $\mathbf{x}_{ck}$ are lumped together
(index $l$) as for the average ${\cal O}(N_{\mathrm{c}},N_{\mathrm{k}})$, Eq.
(32), while the order of the $c$\- and $k$-averaging matters for the “internal
variance” $\delta{\cal O}^{2}_{\mathrm{int}}(N_{\mathrm{c}},N_{\mathrm{k}})$
and the “external variance” $\delta{\cal
O}^{2}_{\mathrm{ext}}(N_{\mathrm{c}},N_{\mathrm{k}})$.

Let us assume next that $N_{\mathrm{c}}$ becomes arbitrarily large.
Importantly, the large-$N_{\mathrm{c}}$ limits ${\cal O}$ and $\delta{\cal
O}_{\mathrm{tot}}$ of ${\cal O}(N_{\mathrm{c}},N_{\mathrm{k}})$ and
$\delta{\cal O}_{\mathrm{tot}}(N_{\mathrm{c}},N_{\mathrm{k}})$ do neither
depend on $N_{\mathrm{c}}$ nor on $N_{\mathrm{k}}$ and may, especially, also
be computed by using only one time-series for each configuration
($N_{\mathrm{k}}=1$). At variance with this, internal and external variances
still depend on $N_{\mathrm{k}}$, i.e. $\delta{\cal
O}_{\mathrm{int}}(N_{\mathrm{c}},N_{\mathrm{k}})\to\delta{\cal
O}_{\mathrm{int}}(N_{\mathrm{k}})$ and $\delta{\cal
O}_{\mathrm{ext}}(N_{\mathrm{c}},N_{\mathrm{k}})\to\delta{\cal
O}_{\mathrm{ext}}(N_{\mathrm{k}})$ in general for $N_{\mathrm{c}}\to\infty$.
Note that

$\delta{\cal O}_{\mathrm{int}}(N_{\mathrm{k}})\to 0,\delta{\cal
O}_{\mathrm{ext}}(N_{\mathrm{k}})\to\delta{\cal O}_{\mathrm{tot}}\mbox{ for
}N_{\mathrm{k}}\to 1.$ (36)

For large spacer times $\tau_{\mathrm{spacer}}\gg\tau_{\mathrm{b}}$ between
time-series the $N_{\mathrm{k}}$-dependence is given using Eq. (10) by spmP2

$\displaystyle\delta{\cal O}^{2}_{\mathrm{int}}(N_{\mathrm{k}})$
$\displaystyle\simeq$ $\displaystyle\left(1-\frac{1}{N_{\mathrm{k}}}\right)\
\delta{\cal O}^{2}_{\mathrm{int}}$ (37) $\displaystyle\delta{\cal
O}^{2}_{\mathrm{ext}}(N_{\mathrm{k}})$ $\displaystyle\simeq$
$\displaystyle\delta{\cal O}^{2}_{\mathrm{ext}}+\frac{1}{N_{\mathrm{k}}}\
\delta{\cal O}^{2}_{\mathrm{int}}$ (38)

where $\delta{\cal O}_{\mathrm{int}}$ and $\delta{\cal O}_{\mathrm{ext}}$
without the argument $N_{\mathrm{k}}$ stand for the limit
$N_{\mathrm{k}}\to\infty$. Using these relations it is possible (with a bit of
care as discussed in Sec. 4.2) to extrapolate internal and external variances
measured at finite $N_{\mathrm{k}}$ to the respective large-$N_{\mathrm{k}}$
limits. We focus below on properties corresponding to the
large-$N_{\mathrm{c}}$ and large-$N_{\mathrm{k}}$ limits.

The above properties may depend additionally on the sampling time $\Delta\tau$
and the volume $V$. The first dependence is relevant for all considered
properties below and around the basin relaxation time $\tau_{\mathrm{b}}$. In
this work we shall mainly focus on the opposite large-$\Delta\tau$ limit
($\Delta\tau\gg\tau_{\mathrm{b}}$). In this limit the typical $k$-averaged
${\cal O}_{ck}$ become $\Delta\tau$-independent. Hence, ${\cal
O}(\Delta\tau,V)\to{\cal O}(V)$ and $\delta{\cal
O}_{\mathrm{ext}}(\Delta\tau,V)\to\Delta_{\mathrm{ne}}(V)$ with
$\Delta_{\mathrm{ne}}(V)$ being the “non-ergodicity parameter” defined in the
Introduction, Eq. (2).888Following Ref. spmP2  one simple possibility to
characterize $\tau_{\mathrm{b}}$ is to set ${\cal
O}(\Delta\tau=\tau_{\mathrm{b}},V)=f{\cal O}(V)$ using a fixed fraction $f$
close to unity. We use $f=0.95$. At variance with this $\delta{\cal
O}_{\mathrm{int}}(\Delta\tau,V)$ remains $\Delta\tau$-dependent decaying as

$\delta{\cal
O}_{\mathrm{int}}(\Delta\tau,V)\propto\sqrt{\tau_{\mathrm{b}}/\Delta\tau}\mbox{
for }\Delta\tau\gg\tau_{\mathrm{b}}$ (39)

since we average over $\Delta\tau/\tau_{\mathrm{b}}$ independent subintervals
spmP2 . Let us define the “non-ergodicity time”
$\tau_{\mathrm{ne}}(V)\gg\tau_{\mathrm{b}}$ by $\delta{\cal
O}_{\mathrm{int}}(\tau_{\mathrm{ne}},V)=\Delta_{\mathrm{ne}}(V)$. $\delta{\cal
O}_{\mathrm{tot}}(\Delta\tau,V)$ is dominated by the internal fluctuations,
Eq. (39), for $\Delta\tau\ll\tau_{\mathrm{ne}}(V)$ while

$\delta{\cal O}_{\mathrm{tot}}(\Delta\tau,V)\to\delta{\cal
O}_{\mathrm{ext}}(\Delta\tau,V)\approx\Delta_{\mathrm{ne}}(V)$ (40)

in the large-$\Delta\tau$ limit ($\Delta\tau\gg\tau_{\mathrm{ne}}(V)$). If
only the standard total variance is probed the non-ergodicity of the system
may remain unnoticed for $\Delta\tau\ll\tau_{\mathrm{ne}}(V)$. As further
emphasized below, it is then necessary to systematically check the
$\Delta\tau$-dependence of $\delta{\cal O}_{\mathrm{tot}}(\Delta\tau,V)$ and
to carefully extrapolate to $\Delta\tau\to\infty$ spmP2 ; spmP3 . The volume
dependence will be addressed in more detail in Sec. 4.4. As a consequence
$\tau_{\mathrm{ne}}(V)$ is found to strongly increase with $V$ since
$\Delta_{\mathrm{ne}}(V)/\delta{\cal O}_{\mathrm{int}}(\Delta\tau,V)$ quite
generally decreases with the system size. Assuming the latter ratio to decay
as $1/V^{\gamma}$ this implies $\tau_{\mathrm{ne}}(V)\propto V^{2\gamma}$. The
determination of $\Delta_{\mathrm{ne}}(V)$ by means Eq. (40) thus becomes
increasingly difficult.

###  4.2 Focus and examples

We focus now on ${\cal O}[\mathbf{x}]=v[\mathbf{x}]$ and the corresponding
expectation value $v(\Delta\tau,V)$, Eq. (32), and the three associated
standard deviations $\delta v_{\mathrm{tot}}(\Delta\tau,V)$, $\delta
v_{\mathrm{int}}(\Delta\tau,V)$ and $\delta v_{\mathrm{ext}}(\Delta\tau,V)$
determined according to Eqs. (33-35). We begin by discussing
$\Delta\tau$-effects (Sec. 4.3) and turn then to the $V$-dependence of these
properties (Sec. 4.4). We illustrate various points made above by means of MC
simulations of the LSM introduced in Sec. 2. For all cases we have $T=1$,
$\delta\tau=10$, $N_{\mathrm{c}}=200$ and at least $N_{\mathrm{k}}=100$. Using
Eq. (37) and Eq. (38) we extrapolate to $N_{\mathrm{k}}\to\infty$. $\delta
v_{\mathrm{int}}(\Delta\tau,V,N_{\mathrm{k}})$ can readily be extrapolated to
$\delta v_{\mathrm{int}}(\Delta\tau,V)$ even using small $N_{\mathrm{k}}<10$
as discussed in Ref. spmP2 . At variance with this the extrapolation from
$\delta v_{\mathrm{ext}}(\Delta\tau,V,N_{\mathrm{k}})$ to $\delta
v_{\mathrm{ext}}(\Delta\tau,V)$ turns out to be inaccurate if the correction
term

$\frac{1}{N_{\mathrm{k}}}\delta
v^{2}_{\mathrm{int}}(\Delta\tau,V)\propto\frac{\tau_{\mathrm{b}}}{N_{\mathrm{k}}\Delta\tau}\mbox{
for }\Delta\tau\gg\tau_{\mathrm{b}}$ (41)

in Eq. (38) is not small compared to $\delta
v^{2}_{\mathrm{ext}}(\Delta\tau,V,N_{\mathrm{k}})$. This matters especially
for $\Delta\tau\ll 10^{3}$ and $J>0.1$ when the stochastic process becomes
slow, increasing thus $\tau_{\mathrm{b}}(J)$. Occasionally, we have thus been
forced to use $N_{\mathrm{k}}=1000$.999A spacer time interval
$\tau_{\mathrm{spacer}}\approx\Delta\tau$ is used between each measured time
series of length $\Delta\tau$. It may have been more efficient to use instead
$\tau_{\mathrm{spacer}}\approx\max(\Delta\tau,\tau_{\mathrm{b}}(J))$ to make
the only asymptotically exact Eq. (38) applicable for smaller
$N_{\mathrm{k}}$.

###  4.3 Sampling time dependence

Figure 4: Sampling time dependence of $v(\Delta\tau)$, $\delta
v_{\mathrm{int}}(\Delta\tau)$, $\delta v_{\mathrm{ext}}(\Delta\tau)$ and
$\delta v_{\mathrm{tot}}(\Delta\tau)$ for LSM-B with $\xi=1$, $J=0$ and
$L=16$. It is seen that $\delta v_{\mathrm{int}}(\Delta\tau)\propto
1/\sqrt{\Delta\tau}$ (bold solid line) while all other properties become
constant for large $\Delta\tau$. Note that $\delta
v_{\mathrm{ext}}(\Delta\tau)\approx\Delta_{\mathrm{ne}}$ much faster than
$\delta v_{\mathrm{tot}}(\Delta\tau)$.

As a generic example we present in Fig. 4 data obtained for LSM-B
(exponentially decaying $a$\- and $b$-fields) using a correlation length
$\xi=1$ and a small simulation box with $L=16$.101010We often suppress in this
subsection the possible additional $V$-dependences, i.e. we write, e.g.,
$v(\Delta\tau)$ instead of $v(\Delta\tau,V)$. All interactions between
neighboring springs $x_{{\bf r}}$ are switched off ($J=0$). As can be seen,
$v(\Delta\tau)$, $\delta v_{\mathrm{int}}(\Delta\tau)$ and $\delta
v_{\mathrm{ext}}(\Delta\tau)$ reach rapidly for
$\Delta\tau\gg\tau_{\mathrm{b}}\approx 100$ the asymptotic behavior expected
from the general discussion in Sec. 4.1, i.e. $v(\Delta\tau)$ and $\delta
v_{\mathrm{ext}}(\Delta\tau)$ approach the respective constants $v$ and
$\delta v_{\mathrm{ext}}=\Delta_{\mathrm{ne}}$ while $\delta
v_{\mathrm{int}}(\Delta\tau)\propto 1/\sqrt{\Delta\tau}$ (bold solid line).
Importantly, due to Eq. (1)

$\delta v_{\mathrm{tot}}(\Delta\tau,V)=\sqrt{\delta
v^{2}_{\mathrm{int}}(\Delta\tau,V)+\delta v^{2}_{\mathrm{ext}}(\Delta\tau,V)}$
(42)

approaches $\Delta_{\mathrm{ne}}$ much later than $\delta
v_{\mathrm{ext}}(\Delta\tau)$, i.e. only for
$\Delta\tau\gg\tau_{\mathrm{ne}}\approx 5600\gg\tau_{\mathrm{b}}$. For this
reason it is problematic to determine $\Delta_{\mathrm{ne}}$ only by measuring
$\delta v_{\mathrm{tot}}(\Delta\tau)$ for one sampling time.111111If $\delta
v_{\mathrm{tot}}(\Delta\tau)$ is known for a broad range of $\Delta\tau$ one
may plot $\delta v_{\mathrm{tot}}(\Delta\tau)$ as a function of
$1/\sqrt{\Delta\tau}$ in linear coordinates. $\Delta_{\mathrm{ne}}$ may then
be obtained for $N_{\mathrm{k}}=1$ from the intercept of the vertical axis of
a linear data fit. This procedure allows to avoid the determination of $\delta
v_{\mathrm{ext}}(\Delta\tau,N_{\mathrm{k}})$.

Figure 5: $v(\Delta\tau)$ (upper data with open symbols) and $\delta
v_{\mathrm{int}}(\Delta\tau)$ (symbols filled with pattern) for LSM-B with
$\xi=1$ and $J=1$ for a broad range of $L$ demonstrating the $V$-independence
of both properties. Increasing $J$ increases $\tau_{\mathrm{b}}$, decreases
$v$ (horizontal dashed line) and increases the power-law amplitude for $\delta
v_{\mathrm{int}}(\Delta\tau)\simeq 1/\sqrt{\Delta\tau}$ for
$\Delta\tau\gg\tau_{\mathrm{b}}$ (bold solid line).

Deviations from this generic asymptotic behavior are visible for small
$\Delta\tau$ around and below the basin relaxation time $\tau_{\mathrm{b}}$.
This can be seen from Fig. 4 but more clearly from Fig. 5 where we present
$v(\Delta\tau)$ and $\delta v_{\mathrm{int}}(\Delta\tau)$ for LSM-B with $J=1$
and for a broad range of system-sizes. Remarkably, $\delta
v_{\mathrm{int}}(\Delta\tau)$ reveals non-monotonic behavior with a maximum
below the basin relaxation time $\tau_{\mathrm{b}}(J=1)\approx 10^{3}$. Being
generally due to relaxation processes within each meta-basin this
small-$\Delta\tau$ regime is more relevant for more realistic models as
discussed elsewhere lyuda19a ; spmP1 ; spmP2 . For the present work it is only
important to stress that the general $\Delta\tau$-dependence of
$v(\Delta\tau)$ and $\delta v_{\mathrm{int}}(\Delta\tau)$ can be traced back
to the “mean-square displacement” (MSD) $h(\tau)$ of the stochastic process.
This is defined by

$h(\tau=t\delta\tau)\equiv h_{t=|i-j|}\equiv\frac{\beta
V}{2}\left<(x_{i}-x_{j})^{2}\right>$ (43)

averaged over all time entries $i$ and $j$ of a long trajectory with
$t=|i-j|$. For stationary processes spmP1

$v(\Delta\tau)=\frac{2}{N_{\mathrm{t}}^{2}}\sum_{t=1}^{N_{\mathrm{t}}-1}(N_{\mathrm{t}}-t)\
h_{t}$ (44)

must hold.121212In in statistical mechanics Eq. (44) is closely related to the
equivalence of the Green-Kubo and the Einstein relations for transport
coefficients HansenBook ; spmP1 ; AllenTildesleyBook . The sampling time
dependence of $\delta v_{\mathrm{int}}(\Delta\tau)$ can be understood and
described assuming a stationary Gaussian stochastic process lyuda19a ; spmP1 ;
spmP2 . This implies that

$\displaystyle\delta v^{2}_{\mathrm{int}}(\Delta\tau)$ $\displaystyle=$
$\displaystyle\delta
v^{2}_{\mathrm{G}}[h]\equiv\frac{1}{2N_{\mathrm{t}}^{4}}\sum_{i,j,k,l=1}^{N_{\mathrm{t}}}\
g_{ijkl}^{2}\mbox{ with }$ (45) $\displaystyle g_{ijkl}$ $\displaystyle\equiv$
$\displaystyle(h_{i-j}+h_{k-l})-(h_{i-l}+h_{j-k}).$

Numerical more convenient alternative representations are given elsewhere
lyuda19a ; spmP1 . By analyzing the functional $\delta v_{\mathrm{G}}[h]$ it
is seen lyuda19a ; spmP1  that while $\delta
v_{\mathrm{int}}(\Delta\tau)\propto 1/\sqrt{\Delta\tau}$ for (to leading
order) $h(t)\approx const$ for $t\approx\Delta\tau$, it may become large with
$\delta v_{\mathrm{int}}(\Delta\tau)\approx v(\Delta\tau)$ for sampling times
$\Delta\tau$ corresponding to a strong change of
$h(t\approx\Delta\tau)$.131313No general relation such as Eq. (45) for $\delta
v_{\mathrm{int}}(\Delta\tau)$ is known at present for $\delta
v_{\mathrm{ext}}(\Delta\tau)$. We emphasize finally that since $h(\tau)$,
$v(\Delta\tau)$ and $\delta v_{\mathrm{int}}(\Delta\tau)$ are connected
through Eq. (44) and Eq. (45) all three quantities must have the same system-
size dependence and this for all times. That $v(\Delta\tau)$ and $\delta
v_{\mathrm{int}}(\Delta\tau)$ in Fig. 5 are both $V$-independent is one
consequence.

###  4.4 Volume dependence

We turn now to system-size effects. Let us focus first on the limit
$\Delta\tau\gg\tau_{\mathrm{ne}}(V)$ where $\delta
v_{\mathrm{int}}(\Delta\tau)$ becomes negligible and $\delta
v_{\mathrm{tot}}(\Delta\tau)\approx\delta
v_{\mathrm{ext}}(\Delta\tau)\approx\Delta_{\mathrm{ne}}$. Examples for $v(V)$
and $\Delta_{\mathrm{ne}}(V)$ are given for $J=0$ in Fig. 6 and Fig. 7 and for
$\Delta_{\mathrm{ne}}(V)$ comparing different $J$ for LSM-D in Fig. 8.

Figure 6: $V$-dependence for $v$ and $\Delta_{\mathrm{ne}}$ (filled symbols)
for LSM-A, LSM-B with $\xi=1$ and $\xi=2$, LSM-C with $\alpha=3$ and LSM-D.
$v$ is always $V$-independent (horizontal dashed line) and
$\Delta_{\mathrm{ne}}\propto 1/V^{\gamma}$ with $\gamma=1/2$ (bold solid line)
for the given examples with short-range correlations.
$\Delta_{\mathrm{ne}}(V)$ for LSM-D (filled squares) is finite, but much
smaller than all other cases.

Figure 7: Volume dependence of $v$ and $\Delta_{\mathrm{ne}}$ for LSM-C and
three different values of $\alpha$. We observe short-range behavior with
$\gamma=1/2$ for $\alpha>d$, long range behavior with $\gamma=\alpha/2d$ for
$\alpha<d$ and, as expected from Eq. (18), logarithmic decay (thin solid line)
for the intermediate case with $\alpha=d=2$. The latter case is well fitted by
an exponent $\gamma=0.4$ (bold dashed line).

Figure 8: $\Delta_{\mathrm{ne}}(V,J)$ for LSM-D for different $J$. Main panel:
$\gamma=1/2$ holds for all $J$ (lines). Inset: Power-law amplitude
$a(J)\equiv\Delta_{\mathrm{ne}}(V,J)V^{1/2}$ vs. coupling constant $J$.
Apparently, the spring coupling introduces isotropic ($p=0$) correlations of
the $x_{t{\bf r}}$\- and, hence, $v_{{\bf r}}$-fields which remain, however,
short-ranged.

The first point to be made is that the variance $v$ is always $V$-independent
(as already seen in Fig. 5) due to the prefactor $\beta V$ introduced in Eq.
(29). In fact this scaling is expected to hold for all stochastic processes
describing intensive system properties if the $c$-trajectories are at thermal
equilibrium in their respective basins spmP2 . (Note that each stochastic
process is ergodic in its basin.) Using the standard fluctuation-dissipation
relation for the fluctuation of intensive thermodynamic variables Lebowitz67 ;
WXP13 ; AllenTildesleyBook  this implies that $v_{c}$ does not depend
explicitly on $V$ and, hence, neither does $v=\mathbf{E}^{c}v_{c}$.141414It is
well known that $v_{c}$ depends on whether the average intensive variable of
the basin is imposed or its conjugated extensive variable Lebowitz67 . This
argument even holds for systems with long-range correlations if standard
thermostatistics can be used for each basin. This can be seen from the
variances $v$ of LSM-C (power-law correlations) for exponents $\alpha<d$ as
shown in Fig. 7 for $\alpha=1$.

Interestingly, the same thermodynamic reasoning cannot be made for
$\Delta_{\mathrm{ne}}$. However, it can be readily demonstrated that quite
generally $\Delta_{\mathrm{ne}}\propto 1/V^{\gamma}$ with $\gamma=1/2$ for
systems without spatial correlations spmP2 . This is the case for LSM-A with
$J=0$ on which we may focus without loss of generality. According to Eq. (31)
we have $b_{c{\bf r}}=v_{c{\bf r}}$ and thus $v_{c}=\mathbf{E}^{{\bf
r}}v_{c{\bf r}}$ is given by the spatial average $b_{c}=\mathbf{E}^{{\bf
r}}b_{c{\bf r}}$. This implies in turn that
$v\equiv\mathbf{E}^{c}v_{c}=\mathbf{E}^{c}b_{c}\equiv b$.151515Without
invoking here thermostatistics this argument demonstrates that $v$ must be
$V$-independent whenever the spatial correlations are short-ranged. To get the
variance of the variance $\mathbf{V}^{c}v_{c}$ one uses again that the
variance of the sum of stochastic independent variables is the sum of the
variances of those variables

$\Delta_{\mathrm{ne}}^{2}=\mathbf{V}^{c}v_{c}=\mathbf{V}^{c}\left(\frac{1}{N_{{\bf
r}}}\sum_{{\bf r}}b_{c{\bf r}}\right)=\frac{1}{N_{{\bf
r}}}\times\underline{\mathbf{E}^{{\bf r}}\mathbf{V}^{c}b_{c{\bf r}}}$ (46)

and the fact that the underlined term does not depend on the number of grid
sites $N_{{\bf r}}\propto V$ for large systems. Hence, $\gamma=1/2$.
Naturally, this does not only hold for systems with strictly decorrelated
fields but also if short-range correlations are present (which may be
renormalized away) as confirmed by the various additional examples with short-
range correlations161616While “short-range” is often reserved for ultimately
exponentially decaying correlation functions, it is used here also for
correlations decaying sufficiently fast such that the volume average does not
depend on the upper integration boundary $L$. presented in Fig. 6, Fig. 7 and
Fig. 8. (As shown in the latter plot for LSM-D, the coupling parameter $J$ has
apparently only a weak quantitative effect on the range of the effective
spatial correlations.) The above argument breaks down, however, if long-range
correlations are present as for the power-law exponent $\alpha=1$ of LSM-C
shown in Fig. 7. The observed power law with $\gamma=\alpha/2d$ is, of course,
expected from Sec. 2.5 as we shall corroborate below in Sec. 5.

Figure 9: $V$-dependence of $\delta v_{\mathrm{tot}}(\Delta\tau)$ and $\delta
v_{\mathrm{ext}}(\Delta\tau)$ for LSM-A for different sampling times
$\Delta\tau$ as indicated. The bold green line indicates the known
$\Delta_{\mathrm{ne}}\propto 1/V^{\gamma}$ with $\gamma=1/2$. While $\delta
v_{\mathrm{ext}}(\Delta\tau,V)\approx\Delta_{\mathrm{ne}}(V)$ for all $L$ and
$\Delta\tau\gg\tau_{\mathrm{b}}\approx 10$, a much slower convergence to this
limit is seen for $\delta v_{\mathrm{tot}}(\Delta\tau,V)$ due to the
$V$-independent contribution $\delta v_{\mathrm{int}}(\Delta\tau)$ to $\delta
v_{\mathrm{tot}}(\Delta\tau)$.

Let us instead end this paragraph with some comments on the
$\Delta\tau$-dependence of the system-size effects. We remind that
$v(\Delta\tau)$, $\delta v_{\mathrm{int}}(\Delta\tau)$ and $h(t)$ are related
via Eq. (44) and Eq. (45). In view of the observed $V$-independence of $v$ it
is thus not surprising that $v(\Delta\tau)$ and $\delta
v_{\mathrm{int}}(\Delta\tau)$ are found to be $V$-independent for all
$\Delta\tau$ as shown in Fig. 5. Since $\delta
v_{\mathrm{int}}(\Delta\tau)\propto V^{0}/\sqrt{\Delta\tau}$ for
$\Delta\tau\gg\tau_{\mathrm{b}}$ the non-ergodicity crossover time
$\tau_{\mathrm{ne}}(V)\simeq V^{2\gamma}$ rapidly increases with $V$. This
implies that the regime with
$\tau_{\mathrm{b}}\ll\Delta\tau\ll\tau_{\mathrm{ne}}(V)$ where $\delta
v_{\mathrm{tot}}(\Delta\tau)\approx\delta v_{\mathrm{int}}(\Delta\tau)$
strongly increases with $V$. If computed at constant $\Delta\tau$ as in most
computational studies Procaccia16 , $\delta v_{\mathrm{tot}}(\Delta\tau,V)$ as
a function of $V$ must thus become $V$-independent for large $V$. This
behavior can clearly be seen from the data presented in Fig. 9 for LSM-A
(filled symbols).171717We use here $N_{\mathrm{k}}=1000$ for $\Delta\tau\leq
10^{4}$ to obtain for $\delta v_{\mathrm{ext}}(\Delta\tau,N_{\mathrm{k}})$ a
sufficiently accurate $N_{\mathrm{k}}$-extrapolation $\delta
v_{\mathrm{ext}}(\Delta\tau)$ for small sampling times. The crossing over from
$\delta v_{\mathrm{tot}}(\Delta\tau,V)\approx\Delta_{\mathrm{ne}}(V)\propto
1/V^{\gamma}$ for small $V$ (bold solid line) to $\delta
v_{\mathrm{tot}}(\Delta\tau,V)\approx\delta
v_{\mathrm{int}}(\Delta\tau)\propto V^{0}$ for large $V$ makes it likely that
in turn a too small apparent exponent $\gamma$ may be fitted. Should $\delta
v_{\mathrm{ext}}(\Delta\tau,V)$ not be available one needs at least to compare
$\delta v_{\mathrm{tot}}(\Delta\tau,V)$ for several $\Delta\tau$. Only the
$V$-regime where the highest $\Delta\tau$-data do indeed collapse can be used
for a fit of the exponent $\gamma$. Without such a crosscheck all fits
claiming an exponent $\gamma<1/2$ and, hence, (according to the preceding
paragraph) long-range spatial correlations are questionable.

##  5 Associated spatial correlation functions

###  5.1 General relations for non-ergodic systems

As demonstrated in detail in Appendix A the integrals, Eqs. (3-5), are solved
by

$\displaystyle C_{\mathrm{tot}}({\bf q})$ $\displaystyle=$
$\displaystyle\mathbf{E}^{l}K[{\cal O}_{l{\bf q}}]({\bf q})-{\cal
O}^{2}\delta_{{\bf q}{\bf 0}}$ (47) $\displaystyle C_{\mathrm{int}}({\bf q})$
$\displaystyle=$ $\displaystyle\mathbf{E}^{c}\mathbf{E}^{k}K[{\cal O}_{ck{\bf
q}}-{\cal O}_{c{\bf q}}]({\bf q})$ (48) $\displaystyle C_{\mathrm{ext}}({\bf
q})$ $\displaystyle=$ $\displaystyle\mathbf{E}^{c}K[{\cal O}_{c{\bf q}}]({\bf
q})-{\cal O}^{2}\delta_{{\bf q}{\bf 0}}$ (49)

where for numerical convenience we have stated all correlation functions in
reciprocal space (with $\delta_{{\bf q}{\bf 0}}$ denoting Kronecker’s symbol
for the zero-wavevector contribution). The “simple average” $C_{\mathrm{tot}}$
corresponds to the standard commonly measured correlation function. The
internal correlation function $C_{\mathrm{int}}$ characterizes the
correlations of the difference ${\cal O}_{ck{\bf q}}-{\cal O}_{c{\bf q}}$ with
respect to the $k$-average ${\cal O}_{c{\bf q}}=\mathbf{E}^{k}{\cal O}_{ck{\bf
q}}$. Moreover, as shown by Eq. (71) the “total” correlation function
$C_{\mathrm{tot}}$ is the sum of an “internal” contribution $C_{\mathrm{int}}$
and an “external” contribution $C_{\mathrm{ext}}$

$C_{\mathrm{tot}}({\bf q})=C_{\mathrm{int}}({\bf q})+C_{\mathrm{ext}}({\bf
q})$ (50)

in agreement with Eq. (6) stated in the Introduction.

Just as $\delta{\cal O}_{\mathrm{tot}}$, $\delta{\cal O}_{\mathrm{int}}$ and
$\delta{\cal O}_{\mathrm{ext}}$ the correlation functions $C_{\mathrm{tot}}$,
$C_{\mathrm{int}}$ and $C_{\mathrm{ext}}$ depend in general on
$N_{\mathrm{c}}$ and $N_{\mathrm{k}}$. As above in Sec. 4.1 we assume that
$N_{\mathrm{c}}$ is arbitrarily large. This implies that

$\lim_{N_{\mathrm{c}}\to\infty}C_{\mathrm{tot}}({\bf
q},N_{\mathrm{c}},N_{\mathrm{k}})=C_{\mathrm{tot}}({\bf q})$ (51)

and similarly in real space, i.e. not only the $N_{\mathrm{c}}$\- but also the
$N_{\mathrm{k}}$-dependence drops out since the total correlation function is
a simple $l$-average. Consistently with Eq. (37) we have for the internal
correlation function

$C_{\mathrm{int}}({\bf
q},N_{\mathrm{k}})\simeq\left(1-\frac{1}{N_{\mathrm{k}}}\right)C_{\mathrm{int}}({\bf
q})$ (52)

for $N_{\mathrm{c}}\to\infty$. Using Eq. (51), Eq. (52) together with Eq. (6)
it is then seen that

$C_{\mathrm{ext}}({\bf q},N_{\mathrm{k}})\simeq C_{\mathrm{ext}}({\bf
q})+\frac{1}{N_{\mathrm{k}}}C_{\mathrm{int}}({\bf q}).$ (53)

These two relations should be used to extrapolate for the asymptotic
$C_{\mathrm{int}}({\bf q})$ and $C_{\mathrm{ext}}({\bf q})$ using the
$C_{\mathrm{int}}({\bf q},N_{\mathrm{k}})$ and $C_{\mathrm{ext}}({\bf
q},N_{\mathrm{k}})$ measured at finite $N_{\mathrm{k}}$. While the
$N_{\mathrm{k}}$-dependent correction term is less crucial for
$C_{\mathrm{int}}({\bf q},N_{\mathrm{k}})$, it is important, as above for
$\delta{\cal O}_{\mathrm{ext}}(N_{\mathrm{k}})$, to take advantage of Eq.
(53), especially if $C_{\mathrm{int}}({\bf q})$ is large. We focus below on
the $N_{\mathrm{k}}$-extrapolated correlation functions in real space.

The correlation functions may a priori also depend explicitly on the sampling
time $\Delta\tau$ and the system volume $V$. One trivial reason for a
$V$-dependency is that the linear box length $L$ sets a cut-off. Fortunately,
this only matters for large distances $r\approx L/2$ (and for the
corresponding small wavevectors), i.e. this effect becomes irrelevant for
large $L$ as one verifies by systematically increasing the box size. Since
$\delta{\cal O}_{\mathrm{ext}}(\Delta\tau,V)$ becomes $\Delta\tau$-independent
and $\delta{\cal O}_{\mathrm{int}}(\Delta\tau,V)\propto 1/\sqrt{\Delta\tau}$
for $\Delta\tau\gg\tau_{\mathrm{b}}$ this naturally suggests

$\left.\begin{array}[]{ll}C_{\mathrm{ext}}({\bf
r},\Delta\tau,V)\simeq&C_{\mathrm{ext}}({\bf r},V)\\\ C_{\mathrm{int}}({\bf
r},\Delta\tau,V)\propto&1/\Delta\tau\end{array}\right\\}\mbox{ for
}\Delta\tau\gg\tau_{\mathrm{b}}$ (54)

as discussed theoretically in more detail in Appendix A.3 and Appendix B. We
shall verify numerically in the next subsection whether this holds for our
model systems.

###  5.2 Examples for lattice spring models

Figure 10: $-C[f,p=2](r)$ for LSM-D for $J=0$ and
$\Delta\tau\gg\tau_{\mathrm{ne}}\gg\tau_{\mathrm{b}}$ confirming that
$C[a]({\bf r})\approx C_{\mathrm{ext}}[m]({\bf r})$ and $C[b]({\bf r})\approx
C_{\mathrm{ext}}[v]({\bf r})$ as expected from Eq. (31).

We present now various (projected) correlation functions $C[f,p](r)$ from our
LSM simulations. We begin in Fig. 10 with data from LSM-D obtained for $J=0$
and a large sampling time $\Delta\tau=10^{6}$. We remind that LSM-D is defined
by Eq. (23) for the $a$-field and by Eq. (24) for the $b$-field. All indicated
correlation functions are obtained by anisotropic projection ($p=2$). Since
all spring interactions are switched off ($J=0)$ and since
$\Delta\tau\gg\tau_{\mathrm{b}}$ we have $m_{{\bf r}}\approx a_{{\bf r}}$ and
$v_{{\bf r}}\approx b_{{\bf r}}$. As expected from Eq. (23) and Eq. (25), Fig.
10 confirms

$\displaystyle-C_{\mathrm{ext}}[m,p](r)$ $\displaystyle\approx$
$\displaystyle-C[a,p](r)\approx 1/r^{2}$ (55) $\displaystyle-
C_{\mathrm{ext}}[v,p](r)$ $\displaystyle\approx$
$\displaystyle-C[b,p](r)\approx(2\lambda)^{2}/r^{2}$ (56)

with $p=2$ and $\lambda=0.1$. Similar results have been found for all model
cases with $|J|\ll 1$ and $\Delta\tau\gg\tau_{\mathrm{b}}$. Since, moreover,
$\Delta\tau\gg\tau_{\mathrm{ne}}$ for the presented data, the internal
correlation functions $C_{\mathrm{int}}[f]({\bf r},\Delta\tau)$ are negligible
small and $C_{\mathrm{tot}}[f]({\bf r},\Delta\tau)\approx
C_{\mathrm{ext}}[f]({\bf r})$ holds (not shown).

Figure 11: Isotropic projection $C[f,p=0](r)$ for LSM-B with $\xi=8$, $J=0.1$
and $\Delta\tau=10^{5}$. Open symbols refer to $L=256$, small filled symbols
to $L=128$. As expected $C_{\mathrm{ext}}[m](r)$ and $C_{\mathrm{ext}}[v](r)$
decrease exponentially just as $C[a](r)$ and $C[b](r)$.
$C_{\mathrm{int}}[v](r)$ decays exponentially (dashed line) for
$r\ll\xi_{\star}$ ($\xi_{\star}$ being a crossover length) but becomes
constant for large $r$.

All correlation functions presented below in this section are isotropically
projected. “$p=0$” is often suppressed for clarity. We consider now finite
spring interactions and smaller sampling times. As an example we show in Fig.
11 correlation functions obtained for LSM-B with $\xi=8$, $J=0.1$ and
$\Delta\tau=10^{5}$. Data for two system sizes are compared.
$C_{\mathrm{ext}}[m](r)$ and $C_{\mathrm{ext}}[v](r)$ are $V$-independent for
all $r\ll L/2$. The small, finite $J$ only has a minor effect on the
prefactors: As for $J=0$ we observe $C_{\mathrm{ext}}[m](r)\approx
C[a](r)\approx\exp(-r/\xi)$ and $C_{\mathrm{ext}}[v](r)\approx
C[b](r)\approx\exp(-2r/\xi)$. The observed short-range correlations are
consistent with $\gamma=1/2$ (cf. Fig. 6).

Figure 12: $y=C_{\mathrm{int}}[v,0](r)/\delta
v^{2}_{\mathrm{int}}(\Delta\tau)$ versus $r$ for LSM-B ($\xi=1$, $L=256$) with
$J=0$ (open symbols), $J=0.1$ (filled symbols) and $J=1$ (symbols with
pattern). Data from a broad $\Delta\tau$-range scale nicely for each $J$. Note
the huge peak at $r=0$ (arrow) for $J=0$ and the exponential decays (dashed
lines) with $y(r)\sim V\exp(-r/\xi_{\mathrm{ind}})$ for $J>0$ and
$r\ll\xi_{\star}(J)$. For large $r$ all data approach $y=1/2$ (bold horizontal
line).

We turn next to the scaling of the internal correlation function
$C_{\mathrm{int}}[v](r)$. Focusing on LSM-B this is presented in Fig. 11 and
Fig. 12. As we shall see, all our numerical data are consistent with the
general scaling

$C_{\mathrm{int}}[v](r,\Delta\tau,V)=[V(1-\alpha)c(r)+\alpha]\ \delta
v^{2}_{\mathrm{int}}(\Delta\tau)$ (57)

with $\alpha=1/2$ and $c(r)$ being a $\Delta\tau$-independent function,
depending somewhat on the model (especially on the coupling parameter $J$),
vanishing for large distances $r$ and being normalized as $V\mathbf{E}^{{\bf
r}}c({\bf r})=1$. In fact, this scaling is natural for a large class of models
as further discussed in Appendix B.

Let us focus first on the $\Delta\tau$-dependence of the internal correlation
function. We present in Fig. 12 the rescaled correlation function
$y=C_{\mathrm{int}}[v](r)/\delta v^{2}_{\mathrm{int}}(\Delta\tau)$ as a
function of $r$ for LSM-B with $J=0$, $J=0.1$ and $J=1$. A perfect data
collapse is observed for each $J$ confirming thus Eq. (57). Since $\delta
v^{2}_{\mathrm{int}}(\Delta\tau)\propto 1/\Delta\tau$ for the presented
sampling times, we could have also used as vertical axis
$C_{\mathrm{int}}[v](r,\Delta\tau)\times\Delta\tau$ to scale the data.
Importantly, the dimensionless scaling variable $y$ is more general allowing
the scaling for all $\Delta\tau$, i.e. also for
$\Delta\tau\ll\tau_{\mathrm{b}}$.

Turning to the $r$-dependence we note that Eq. (57) implies that
$C_{\mathrm{int}}[v](r,\Delta\tau,V)$ should level off to a plateau with
$\alpha\delta v^{2}_{\mathrm{int}}(\Delta\tau)>0$ for sufficiently large
$r\gg\xi_{\star}$.181818According to Eq. (57) and assuming $c(r)$ to be
continuous, the crossover length $\xi_{\star}$ may be defined by
$c(r=\xi_{\star})\approx 1/V$. As emphasized by the bold horizontal lines in
Fig. 11 and Fig. 12 this is indeed the case. Moreover, the latter figure
confirms $\alpha=1/2$, i.e. quite generally we have
$C_{\mathrm{int}}[v](r)\to\delta v^{2}_{\mathrm{int}}(\Delta\tau)/2$ for large
$r$. That $C_{\mathrm{int}}[v](r)$ becomes a finite constant for large $r$,
albeit the instantaneous $x_{t{\bf r}}$-field is decorrelated, has to do with
the definition of the $v_{{\bf r}}$-field, Eq. (30), as further explained in
Appendix B.1. As can be seen from the latter calculation the function $c(r)$
for LSM-A with $J=0$ has a jump singularity at $r=0$, Eq. (96). That this is
also the case for all other models with $J=0$ can be seen for LSM-B in Fig. 12
(arrow). This becomes different if the interaction between the springs is
switched on ($J>0$). As seen in Fig. 11 and Fig. 12 we then observe for
$r\ll\xi_{\star}$ a continuous exponential decay
$C_{\mathrm{int}}[v](r)\propto c(r)\propto\exp(-r/\xi_{\mathrm{ind}})$ with a
finite induced correlation length $\xi_{\mathrm{ind}}$ weakly increasing with
$J$.

Moreover, as can be also seen in Fig. 11, the internal correlations increase
in the first $r$-regime with $V$. Confirming the $V$-dependence indicated in
Eq. (57), a systematic comparison of a broad range of $L$ reveals that
$C_{\mathrm{int}}[v](r)\propto V$ for small $r$ while it is strictly
$V$-independent for large $r$ (not shown). Due to both contributions the
volume average $\mathbf{E}^{{\bf r}}C_{\mathrm{int}}[v]({\bf r})$ is thus
$V$-independent consistently with the $V$-independence of $\delta
v^{2}_{\mathrm{int}}(\Delta\tau)$ demonstrated above (cf. Fig. 5 and Sec.
4.4). 191919Data collapse for different $V$ and $J>0$ can be achieved (not
shown) by obtaining first $c(r)=(2y-1)/V$ and by plotting then $c(r)/c(0)$ as
a function of $x=r/\xi_{\mathrm{ind}}(J,V)$.

Figure 13: Isotropic projections $C_{\mathrm{tot}}[v](r,\Delta\tau)$ and
$C_{\mathrm{ext}}[v](r,\Delta\tau)$ for LSM-C with $\alpha=2$, $J=0$ and
$L=256$. Also indicated is the imposed asymptotic limit $C[b](r)\approx
0.04/r^{2}$ (bold solid line) for large $\Delta\tau$ and $V$ for both
correlation functions. While $C_{\mathrm{ext}}[v](r,\Delta\tau)\approx
C[b](r)$ for $\Delta\tau\gg\tau_{\mathrm{b}}$,
$C_{\mathrm{tot}}[v](r,\Delta\tau)$ is seen to converge much more slowly to
the large-$\Delta\tau$ limit.

The scaling of $C_{\mathrm{ext}}[v](r,\Delta\tau)$ and
$C_{\mathrm{tot}}[v](r,\Delta\tau)$ with $\Delta\tau$ is illustrated in Fig.
13. We present here data obtained for LSM-C with $\alpha=2$, $J=0$ and
$L=256$. Importantly, both $C_{\mathrm{tot}}[v](r,\Delta\tau)$ and
$C_{\mathrm{ext}}[v](r,\Delta\tau)$ must approach for sufficiently large
$\Delta\tau$ the (known) asymptotic limit $C[b](r)\approx 0.04/r^{2}$ (bold
solid line) imposed by construction. $N_{\mathrm{k}}=1000$ is used for
$\Delta\tau\leq 10^{3}$ to improve the precision of the
$N_{\mathrm{k}}$-extrapolation for the external correlation function. While
$C_{\mathrm{ext}}[v](r,\Delta\tau)$ becomes rapidly $\Delta\tau$-independent
($\Delta\tau\gg\tau_{\mathrm{b}}$) several orders of magnitude larger sampling
times are needed for $C_{\mathrm{tot}}[v](r,\Delta\tau)$. This is caused by
the internal contribution $C_{\mathrm{int}}[v](r,\Delta\tau)\propto
1/\Delta\tau$ to the total correlation function. This is also responsible for
the leveling-off of $C_{\mathrm{tot}}[v](r,\Delta\tau)$ for large $r\gg
r_{\mathrm{ne}}(\Delta\tau)$ with $r_{\mathrm{ne}}(\Delta\tau)$ being a
crossover distance defined by
$C_{\mathrm{int}}[v](r_{\mathrm{ne}},\Delta\tau)=C_{\mathrm{ext}}[v](r_{\mathrm{ne}})$.
For the same reasons that $\delta v_{\mathrm{tot}}(\Delta\tau,V)$ is
problematic for the determination of the system-size exponent $\gamma$, only
computing $C_{\mathrm{tot}}[v](r,\Delta\tau)$ for one $\Delta\tau$ may
incorrectly suggest a weak (possibly long-ranged) decay of the correlations.
Only the $r$-regime where the data sets for the largest available $\Delta\tau$
clearly collapse can be used. This would be in the presented case less than an
order of magnitude. Similar behavior has been found for all LSM versions.

Figure 14: Isotropic projection $C_{\mathrm{ext}}[v,0](r,\Delta\tau)$ for
LSM-D for different $J$. A $\delta(r)$-peak is observed for $J=0$ while
$C_{\mathrm{ext}}[v,0](r,\Delta\tau)\propto\exp(-r/\xi_{\mathrm{ind}}(J))$ for
$J>0$.

As a last example we come back to LSM-D and present the isotropic ($p=0$)
external correlations $C_{\mathrm{ext}}[v,0](r)$ for different $J$ obtained
for $\Delta\tau\gg\tau_{\mathrm{b}}(J)$. This is shown in Fig. 14 using half-
logarithmic coordinates. As expected from the imposed (quenched) anisotropic
$a$\- and $b$-fields of the LSM-D (Table 1) a $\delta(r)$-peak at the origin
is observed if all spring interactions are switched off ($J=0$). At variance
with this the external correlation functions decay exponentially for $J>0$.
Apparently, the corresponding induced correlation length
$\xi_{\mathrm{ind}}(J)$ increases with $J$ but remains finite. The range of
the correlations thus increases but stays short-ranged in agreement with the
exponent $\gamma=1/2$ seen in Fig. 8.

##  6 Conclusion

###  6.1 General points made

Extending recent work on stochastic processes in non-ergodic macroscopic
systems spmP1 ; spmP2 ; spmP3  we have investigated the different types of
spatial correlation functions $C({\bf r})$ related to the macroscopic
variances $\delta{\cal O}^{2}$ of observables ${\cal O}[\mathbf{x}]$ of time-
series $\mathbf{x}$. As reminded in Sec. 4 the standard total variance
$\delta{\cal O}^{2}_{\mathrm{tot}}(\Delta\tau)$ is the sum of an internal
variance $\delta{\cal O}^{2}_{\mathrm{int}}(\Delta\tau)$ and an external
variance $\delta{\cal O}^{2}_{\mathrm{ext}}(\Delta\tau)$, cf. Eq. (1). While
$\delta{\cal O}_{\mathrm{int}}(\Delta\tau)\simeq 1/\sqrt{\Delta\tau}$ for
$\Delta\tau\gg\tau_{\mathrm{b}}$, $\delta{\cal
O}_{\mathrm{ext}}(\Delta\tau,V)\simeq\Delta_{\mathrm{ne}}(V)$ becomes
constant. One motivation of this work (cf. Sec. 1.2) was to understand the
$V$-dependence of the non-ergodicity parameter $\Delta_{\mathrm{ne}}$ in
systems with (possibly long-ranged) spatial correlations. The generally
important novel key results of this study were presented in Sec. 5 and
Appendix A. As shown there, assuming ${\cal O}[\mathbf{x}]$ to be a spatial
average of a local field ${\cal O}_{{\bf r}}$, the three global variances can
be written as volume averages of the three spatial correlation functions
$C_{\mathrm{tot}}({\bf r})$, $C_{\mathrm{int}}({\bf r})$ and
$C_{\mathrm{ext}}({\bf r})$ and, moreover, $C_{\mathrm{tot}}({\bf
r})=C_{\mathrm{int}}({\bf r})+C_{\mathrm{ext}}({\bf r})$ holds. The
$\Delta\tau$\- and $V$-dependences of the global variances can thus be traced
back to the internal and external correlation functions $C_{\mathrm{int}}({\bf
r})$ and $C_{\mathrm{ext}}({\bf r})$.

###  6.2 Specific fields considered

Focusing on the arithmetic mean ${\cal
O}[\mathbf{x}]=m[\mathbf{x}]=\mathbf{E}^{t}x_{t}$ and especially on the
empirical variance ${\cal O}[\mathbf{x}]=v[\mathbf{x}]=\mathbf{V}^{t}x_{t}$ of
time-series (cf. Sec. 3) we illustrated the various general theoretical
relations by means of MC simulations of variants of a simple lattice spring
model (LSM) in two dimensions characterized by two quenched and spatial
correlated fields (Table 1). We have especially investigated $\delta
v_{\mathrm{int}}(\Delta\tau,V)$ and $\delta v_{\mathrm{ext}}(\Delta\tau,V)$
and the corresponding correlation functions $C_{\mathrm{int}}[v]({\bf
r},\Delta\tau,V)$ and $C_{\mathrm{ext}}[v]({\bf r},\Delta\tau,V)$. As
discussed in Sec. 5.2 and Appendix B, under general assumptions the internal
correlation function is given by Eq. (57), i.e. it decreases inversely with
$\Delta\tau$ for $\Delta\tau\gg\tau_{\mathrm{b}}$ and becomes constant,
$C_{\mathrm{int}}[v]({\bf r},\Delta\tau,V)\simeq\delta
v^{2}_{\mathrm{int}}(\Delta\tau)/2$, for large $r$ (cf. Fig. 12) albeit the
primary instantaneous field $x_{t{\bf r}}$ is decorrelated. The external
correlation function becomes $\Delta\tau$-independent for
$\Delta\tau\gg\tau_{\mathrm{b}}$ (cf. Fig. 13). The last statement requires a
proper $N_{\mathrm{k}}$-extrapolation by means of Eq. (53) or that the data
are sampled using sufficiently large $\Delta\tau$ and $N_{\mathrm{k}}$, i.e.
the correction term $C_{\mathrm{int}}[v]({\bf r},\Delta\tau,V)/N_{\mathrm{k}}$
in Eq. (53) must be negligible. Importantly, $C_{\mathrm{int}}[v]({\bf
r},\Delta\tau)\gg C_{\mathrm{ext}}[v]({\bf r})$ for small $\Delta\tau$, large
$V$ and large $r$. In these limits and due to large crossover effects
$C_{\mathrm{tot}}[v]({\bf r},\Delta\tau)$ may deviate from its
large-$\Delta\tau$ limit $C_{\mathrm{ext}}[v]({\bf r})$. Importantly, this may
lead to an overestimation of the range of correlations as shown in Fig. 13.

###  6.3 Outlook

Some of the presented relations and technical features will be used in a
presentation currently under preparation focusing on shear stresses
$x_{t}=\sigma_{t}$ and associated shear-stress fields $x_{t{\bf
r}}=\sigma_{t{\bf r}}$ Lemaitre14 ; Lemaitre15 . Again focusing on
$\Delta\tau\gg\tau_{\mathrm{b}}$ we investigate $C_{\mathrm{int}}[f]({\bf r})$
and $C_{\mathrm{ext}}[f]({\bf r})$ for $f=m$ and $f=v$ for a broad range of
particle numbers $n\approx V$. As central results it will be shown that
$C_{\mathrm{ext}}[f]({\bf r})$ is long-ranged for both $f=m$ and $f=v$
decreasing as a power law $1/r^{\alpha}$ with $\alpha\approx d$. While the
scaling of $C_{\mathrm{ext}}[m]({\bf r})$ is expected from previous
simulations Lemaitre14 ; Lemaitre15  and recent theoretical work Fuchs17 ;
Fuchs18 ; Fuchs19 ; lyuda18  the long-range decay of $C_{\mathrm{ext}}[v]({\bf
r})$ is non-trivial and, as we shall discuss, indicates that the corresponding
local elastic field is also long-ranged.

## Author contribution statement

JPW designed and wrote the project benefiting from contributions of all
authors.

## Acknowledgments

We acknowledge computational resources from the HPC cluster of the University
of Strasbourg.

## Data availability statement

It was not possible to store all the $N_{\mathrm{c}}\times
N_{\mathrm{k}}\times N_{\mathrm{t}}\times L^{2}$ primary fields $x_{ckt{\bf
q}}$ which were immediately deleted after having been analyzed. Tables of the
global averages have been kept, however, for a broad range of $\Delta\tau$ and
$V$ and different LSM variants. These data sets are available from the
corresponding author on reasonable request.

##  Appendix A Spatial correlations of periodic microcells

###  A.1 Some useful general relations

Let us begin by stating several useful general relations for spatial
correlation functions of $d$-dimensional, real, discrete and periodic fields.
As defined by Eq. (13) or Eq. (14) we consider the instantaneous correlation
function $K[y_{{\bf r}}]({\bf r})$ of a field $y_{{\bf r}}$ of volume average
$y\equiv\mathbf{E}^{{\bf r}}y_{{\bf r}}$. Obviously,

$K[y_{{\bf r}}-y]({\bf r})=K[y_{{\bf r}}]({\bf r})-y^{2}.$ (58)

Let us assume that the field $y_{l{\bf r}}$ additionally depends on an index
$l$. We use below the averages $y_{l}=\mathbf{E}^{{\bf r}}y_{l{\bf r}}$,
$y_{{\bf r}}=\mathbf{E}^{l}y_{l{\bf r}}$ and
$y=\mathbf{E}^{l}y_{l}=\mathbf{E}^{{\bf r}}y_{{\bf r}}$. Rewriting Eq. (58)
and summing over $l$ gives

$\mathbf{E}^{l}K[y_{l{\bf r}}-y_{l}]({\bf r})=\mathbf{E}^{l}K[y_{l{\bf
r}}]({\bf r})-\mathbf{E}^{l}y_{l}^{2}.$ (59)

Also it is seen by expansion using Eq. (13) that

$\mathbf{E}^{l}K[y_{l{\bf r}}-\lambda y_{{\bf r}}]({\bf
r})=\mathbf{E}^{l}K[y_{l{\bf r}}]({\bf r})-\lambda(2-\lambda)K[y_{{\bf
r}}]({\bf r})$ (60)

for any real constant $\lambda$.

We remind that
$\mathbf{V}^{l}y_{l}=\mathbf{E}^{l}y_{l}^{2}-y^{2}=\mathbf{E}^{l}(y_{l}-y)^{2}$.
Using again Eq. (58) and the periodicity of the grid the variance
$\mathbf{V}^{l}y_{l}$ may be written as the volume average

$\displaystyle\mathbf{V}^{l}y_{l}$ $\displaystyle=$
$\displaystyle\mathbf{E}^{{\bf r}}C({\bf r})\mbox{ with }$ (61) $\displaystyle
C({\bf r})$ $\displaystyle\equiv$ $\displaystyle\mathbf{E}^{l}K[y_{l{\bf
r}}-y]({\bf r})\equiv\mathbf{E}^{l}K[y_{l{\bf r}}]({\bf r})-y^{2}$ (62)

being the $l$-averaged correlation function in real space. By comparing Eq.
(59) and Eq. (62) we may also write

$C({\bf r})=\mathbf{E}^{l}K[y_{l{\bf r}}-y_{l}]({\bf r})+\mathbf{V}^{l}y_{l}$
(63)

which using Eq. (61) implies $\mathbf{E}^{{\bf r}}\mathbf{E}^{l}K[y_{l{\bf
r}}-y_{l}]({\bf r})=0$. It is useful to state two important limits for $C({\bf
r})$: (i) At the origin we have

$C({\bf r}={\bf 0})=\mathbf{V}^{l{\bf r}^{\prime}}y_{l{\bf
r}^{\prime}}=\mathbf{E}^{l}\mathbf{V}^{{\bf r}^{\prime}}y_{l{\bf
r}^{\prime}}+\mathbf{V}^{l}\mathbf{E}^{{\bf r}^{\prime}}y_{l{\bf r}^{\prime}}$
(64)

and (ii) $C({\bf r})$ exactly vanishes if and only if

$\mathbf{E}^{l}\mathbf{E}^{{\bf r}^{\prime}}y_{{\bf r}^{\prime}+{\bf
r}}y_{{\bf r}^{\prime}}=\mathbf{E}^{l}\mathbf{E}^{{\bf r}^{\prime}}y_{{\bf
r}^{\prime}+{\bf r}}\times\mathbf{E}^{l}\mathbf{E}^{{\bf r}^{\prime}}y_{{\bf
r}^{\prime}}=y^{2}$ (65)

as it happens for most (albeit not all) fields for sufficiently large $r=|{\bf
r}|$. See Appendix B.1 for an exception relevant for the present study.

Moreover, with $z=\mathbf{E}^{{\bf r}}z_{{\bf r}}$ being an $l$-independent
quantity it follows from Eq. (61) that

$\mathbf{V}^{l}(y_{l}-z)=\mathbf{E}^{{\bf r}}\ \mathbf{E}^{l}K[(y_{l{\bf
r}}-z_{{\bf r}})-(y-z)]({\bf r}).$ (66)

Since $\mathbf{V}^{l}y_{l}=\mathbf{V}^{l}(y_{l}-z)$ this implies quite
generally that

$\mathbf{E}^{{\bf r}}\mathbf{E}^{l}K[\delta y_{l{\bf r}}]({\bf
r})=\mathbf{E}^{{\bf r}}\mathbf{E}^{l}K[\delta y_{l{\bf r}}-\delta z_{{\bf
r}}]({\bf r})$ (67)

with $\delta y_{l{\bf r}}=y_{l{\bf r}}-y$ and $\delta z_{{\bf r}}=z_{{\bf
r}}-z$, i.e. the correlation function of a field $y_{l{\bf r}}$ can be shifted
by an $l$-independent field $z_{{\bf r}}$ without changing the $l$-averaged
volume average.

###  A.2 Derivation of correlation functions

Using these general relations it is readily seen that the correlation
functions defined as

$\displaystyle C_{\mathrm{tot}}({\bf r})$ $\displaystyle\equiv$
$\displaystyle\mathbf{E}^{l}K[{\cal O}_{l{\bf r}}-{\cal O}]({\bf
r})=\mathbf{E}^{l}K[{\cal O}_{l{\bf r}}]({\bf r})-{\cal O}^{2}$ (68)
$\displaystyle C_{\mathrm{ext}}({\bf r})$ $\displaystyle\equiv$
$\displaystyle\mathbf{E}^{c}K[{\cal O}_{c{\bf r}}-{\cal O}]({\bf
r})=\mathbf{E}^{c}K[{\cal O}_{c{\bf r}}]({\bf r})-{\cal O}^{2}$ (69)
$\displaystyle C_{\mathrm{int}}({\bf r})$ $\displaystyle\equiv$
$\displaystyle\mathbf{E}^{c}\mathbf{E}^{k}K[{\cal O}_{ck{\bf r}}-{\cal
O}_{c{\bf r}}]({\bf r})$ (70)

are consistent with Eqs. (3-6). The index $l$ runs again over all independent
configurations $c$ and all time-series $k$ for each $c$ and the expectation
value ${\cal O}$ is defined in Eq. (32). The corresponding equations in
reciprocal space are given in Sec. 5.1, Eqs. (47-49). That Eq. (68) is
consistent with $\delta{\cal O}^{2}_{\mathrm{tot}}=\mathbf{E}^{{\bf
r}}C_{\mathrm{tot}}({\bf r})$ and Eq. (69) with $\delta{\cal
O}^{2}_{\mathrm{ext}}=\mathbf{E}^{{\bf r}}C_{\mathrm{ext}}({\bf r})$ is
directly implied by Eq. (61) and Eq. (62). To show that Eq. (70) is consistent
with $\delta{\cal O}^{2}_{\mathrm{int}}=\mathbf{E}^{{\bf
r}}C_{\mathrm{int}}({\bf r})$ and that all three correlation functions sum up
according to Eq. (6) let us first note that due to Eq. (60) for $\lambda=1$
the internal correlation function may be rewritten as

$C_{\mathrm{int}}({\bf r})=\mathbf{E}^{c}\left\\{\mathbf{E}^{k}K[{\cal
O}_{ck{\bf r}}]({\bf r})-K[{\cal O}_{c{\bf r}}]({\bf r})\right\\}.$ (71)

Using Eq. (68) and Eq. (69) this implies $C_{\mathrm{int}}({\bf
r})=C_{\mathrm{tot}}({\bf r})-C_{\mathrm{ext}}({\bf r})$ in agreement with the
key relation Eq. (6) stated in the Introduction. In turn we thus have

$\displaystyle\mathbf{E}^{{\bf r}}C_{\mathrm{int}}({\bf r})$ $\displaystyle=$
$\displaystyle\mathbf{E}^{{\bf r}}\left(C_{\mathrm{tot}}({\bf
r})-C_{\mathrm{ext}}({\bf r})\right)$ (72) $\displaystyle=$
$\displaystyle\delta{\cal O}^{2}_{\mathrm{tot}}-\delta{\cal
O}^{2}_{\mathrm{ext}}=\delta{\cal O}^{2}_{\mathrm{int}}$

where we have used Eq. (1) in the last step.

Please note that due to Eq. (67) $\delta{\cal
O}^{2}_{\mathrm{int}}=\mathbf{E}^{{\bf r}}C_{\mathrm{int}}({\bf r})$ would
also be solved by the more general internal correlation function

$C_{\mathrm{int}}({\bf r},\lambda)\equiv\mathbf{E}^{c}\mathbf{E}^{k}K[({\cal
O}_{ck{\bf r}}-{\cal O}_{c})-\lambda({\cal O}_{c{\bf r}}-{\cal O}_{c})]({\bf
r})$ (73)

which reduces to Eq. (70) for $\lambda=1$, since it is possible to shift
${\cal O}_{ck{\bf r}}-{\cal O}_{c}$ with the $k$-independent field
$\lambda({\cal O}_{c{\bf r}}-{\cal O}_{c})$ without changing the $k$-averaged
volume average. The trouble with such alternative definitions is that Eq. (6)
does not hold anymore in general, e.g., it can be shown that Eq. (73) leads to

$C_{\mathrm{tot}}({\bf r})=C_{\mathrm{int}}({\bf
r},\lambda)+\lambda(2-\lambda)C_{\mathrm{ext}}({\bf
r})+(\lambda-1)^{2}\delta{\cal O}^{2}_{\mathrm{ext}}.$ (74)

Due to the last term and since $\delta{\cal O}^{2}_{\mathrm{ext}}>0$ for non-
ergodic systems all three correlation functions may in principle only vanish
for the same ${\bf r}$ for $\lambda=1$ and for exactly this limit Eq. (74)
reduces to Eq. (6). We therefore set $\lambda=1$.

###  A.3 Important limits

We have omitted for clarity in the preceding subsection all possible
dependences on $N_{\mathrm{c}}$, $N_{\mathrm{k}}$, $\Delta\tau$ and $V$.
However, it is assumed below that $N_{\mathrm{c}}$ and $N_{\mathrm{k}}$ are
arbitrarily large, i.e. all properties are $N_{\mathrm{c}}$\- and
$N_{\mathrm{k}}$-independent. Moreover, we focus on the limit
$\Delta\tau\gg\tau_{\mathrm{b}}$, i.e. both ${\cal O}_{c{\bf
r}}=\mathbf{E}^{k}{\cal O}_{ck{\bf r}}$ and its average ${\cal
O}=\mathbf{E}^{c}\mathbf{E}^{{\bf r}}{\cal O}_{c{\bf r}}$ are
$\Delta\tau$-independent to leading order. Due to Eq. (69) the same holds for
the external correlation function, i.e.

$C_{\mathrm{ext}}({\bf r},\Delta\tau,V)\simeq C_{\mathrm{ext}}({\bf
r},V)\mbox{ for }\Delta\tau\gg\tau_{\mathrm{b}}.$ (75)

The indicated $V$-dependence drops out if ${\cal O}_{c{\bf r}}$ is
$V$-independent as in all the models of this work.

The internal correlation function, Eq. (70), characterizes the correlations of
the difference ${\cal O}_{ck{\bf r}}(\Delta\tau)-{\cal O}_{c{\bf r}}$. While
${\cal O}_{ck{\bf r}}(\Delta\tau)$ depends in general not only on $k$ but also
on $\Delta\tau$, both dependences drop out for $\Delta\tau\to\infty$. Hence,
${\cal O}_{ck{\bf r}}(\Delta\tau)\to{\cal O}_{c{\bf r}}$ and in turn

$\lim_{\Delta\tau\to\infty}C_{\mathrm{int}}({\bf r},\Delta\tau,V)=0.$ (76)

To obtain the internal correlation function for finite
$\Delta\tau\gg\tau_{\mathrm{b}}$ it should be remembered that ${\cal
O}_{ck{\bf r}}(\Delta\tau)$ is a time-averaged moment over
$N_{\mathrm{t}}=\Delta\tau/\delta\tau$ data entries from one stored time-
series. The internal correlation function can thus be written as an average

$C_{\mathrm{int}}({\bf r}={\bf r}_{2}-{\bf
r}_{1},\Delta\tau,V)=\mathbf{E}^{c}\mathbf{E}^{{\bf
r}_{1}}\left(\mathbf{E}^{t_{1}}\mathbf{E}^{t_{2}}\mathbf{E}^{k}\ldots\right)$
(77)

over entries measured at discrete times $t_{1}$ and $t_{2}$. A specific
example is worked out in Appendix B.1. If one assumes for simplicity that
$\delta\tau\gg\tau_{\mathrm{b}}$ only contributions with $t_{1}=t_{2}$ can
contribute. Using also that the time-average $\mathbf{E}^{t}$ is normalized by
$N_{\mathrm{t}}\propto\Delta\tau$, this shows that quite generally the
internal correlation function must decay to leading order for all ${\bf r}$ as

$C_{\mathrm{int}}({\bf r},\Delta\tau,V)\propto\frac{1}{\Delta\tau}\mbox{ for
}\Delta\tau\gg\tau_{\mathrm{b}}$ (78)

as expected from $\delta{\cal
O}^{2}_{\mathrm{int}}(\Delta\tau)=\mathbf{E}^{{\bf r}}C_{\mathrm{int}}({\bf
r})\propto 1/\Delta\tau$.

##  Appendix B Scaling of $C_{\mathrm{int}}[v]({\bf r},\Delta\tau,V)$

###  B.1 Predictions for LSM-A

case | $\mathbf{E}^{k}A_{1}A_{2}$ | $\mathbf{E}^{k}A_{1}B_{2}$ | $\mathbf{E}^{k}A_{2}B_{1}$ | $\mathbf{E}^{k}B_{1}B_{2}$
---|---|---|---|---
1\. ${\bf r}_{1}={\bf r}_{2}={\bf r}_{3}={\bf r}_{4}$ | $2b_{c{\bf r}_{1}}^{2}/N_{\mathrm{t}}$ | $3b_{c{\bf r}_{1}}^{2}/N_{\mathrm{t}}^{2}$ | $3b_{c{\bf r}_{1}}^{2}/N_{\mathrm{t}}^{2}$ | $b_{c{\bf r}_{1}}^{2}/N_{\mathrm{t}}^{3}$
2\. ${\bf r}_{1}={\bf r}_{2}\neq{\bf r}_{3}={\bf r}_{4}$ | $b_{c{\bf r}_{1}}(\mathbf{S}^{{\bf r}_{3}\neq{\bf r}_{1}}b_{c{\bf r}_{3}})/N_{\mathrm{t}}$ | $b_{c{\bf r}_{1}}^{2}(V-1)/N_{\mathrm{t}}^{2}$ | $b_{c{\bf r}_{1}}^{2}(V-1)/N_{\mathrm{t}}^{2}$ | $b_{c{\bf r}_{1}}^{2}(V-1)/N_{\mathrm{t}}^{2}$
3\. ${\bf r}_{1}={\bf r}_{3}\neq{\bf r}_{2}={\bf r}_{4}$ | 0 | 0 | 0 | $b_{c{\bf r}_{1}}b_{c{\bf r}_{2}}/N_{\mathrm{t}}^{2}$
4\. ${\bf r}_{1}={\bf r}_{4}\neq{\bf r}_{2}={\bf r}_{3}$ | $b_{c{\bf r}_{1}}b_{c{\bf r}_{2}}/N_{\mathrm{t}}$ | $b_{c{\bf r}_{1}}b_{c{\bf r}_{2}}/N_{\mathrm{t}}^{2}$ | $b_{c{\bf r}_{1}}b_{c{\bf r}_{2}}/N_{\mathrm{t}}^{2}$ | $b_{c{\bf r}_{1}}b_{c{\bf r}_{2}}/N_{\mathrm{t}}^{2}$
Table 2: $k$-averages $\mathbf{E}^{k}A_{1}A_{2}$, $\mathbf{E}^{k}A_{1}B_{2}$,
$\mathbf{E}^{k}A_{2}B_{1}$ and $\mathbf{E}^{k}B_{1}B_{2}$ for LSM-A. The
different relevant cases for ${\bf r}_{1}$, ${\bf r}_{2}$, ${\bf r}_{3}$,
${\bf r}_{4}$ are indicated in the first column. Note that $r=|{\bf r}|$ with
${\bf r}={\bf r}_{2}-{\bf r}_{1}$. The first two cases indicate contributions
for $r=0$, the last two cases contributions for $r>0$.
$\mathbf{E}^{k}A_{1}B_{2}$ and $\mathbf{E}^{k}A_{2}B_{1}$ are identical by
symmetry. The leading contributions of order ${\cal O}(\Delta\tau^{-1})$ are
due to $\mathbf{E}^{k}A_{1}A_{2}$ (second column). The second case (${\bf
r}_{1}={\bf r}_{2}\neq{\bf r}_{3}={\bf r}_{4}$) yields contributions
proportional to the system size.

As noted in Appendix A.1, all correlation functions $C[f]({\bf r})$ discussed
in the present work must vanish if Eq. (65) holds, i.e. if two typical points
of the field $f$ at a respective distance ${\bf r}$ are uncorrelated. Here we
draw attention to the fact that although the primary instantaneous field
$x_{t{\bf r}}$ may be uncorrelated this may not be the case for the field
${\cal O}_{{\bf r}}$ associated to the time-averaged functional ${\cal
O}[\mathbf{x}]$ of the time-series $\mathbf{x}$.202020As seen using Eq. (83)
this is, however, the case for $m_{{\bf r}}=\mathbf{E}^{t}x_{t{\bf r}}$ for
decorrelated primary instantaneous fields $x_{t{\bf r}}$. As we shall see,
this matters specifically for the covariance field, Eq. (30),

$\displaystyle v_{ck{\bf r}}$ $\displaystyle=$ $\displaystyle
V\left\\{\mathbf{E}^{t}x_{ckt{\bf r}}x_{ckt}-x_{ck{\bf r}}x_{ck}\right\\}$
(79) $\displaystyle=$ $\displaystyle\mathbf{S}^{{\bf
r}^{\prime}}\left\\{\mathbf{E}^{t}x_{ckt{\bf r}}x_{ckt{\bf
r}^{\prime}}-\mathbf{E}^{t}\mathbf{E}^{t^{\prime}}x_{ckt{\bf
r}}x_{ckt^{\prime}{\bf r}^{\prime}}\right\\}$

restated for convenience omitting the irrelevant prefactor $\beta$ and
introducing the sum $\mathbf{S}^{{\bf r}^{\prime}}\equiv V\mathbf{E}^{{\bf
r}^{\prime}}$. In the second step we have made explicit the crucial
contributions with ${\bf r}^{\prime}\neq{\bf r}$ to the averages $x_{ckt}$ and
$x_{ck}$.

Without restricting the generality of the argument let us focus on the model
LSM-A with $J=0$, i.e. the $x_{ckt{\bf r}}$ of different microcells ${\bf r}$
are uncorrelated. Ultimately, we want to expand $v_{ck{\bf r}}$ for large
$\Delta\tau$. Reminding Eq. (31) it is thus useful that the $x_{ckt{\bf r}}$
in Eq. (79) can be replaced by $\delta x_{ckt{\bf r}}\equiv x_{ckt{\bf
r}}-a_{c{\bf r}}$ without changing $v_{ck{\bf r}}$. Moreover, let us assume
that the data is sampled with large time increments
$\delta\tau\gg\tau_{\mathrm{b}}$, i.e. different times $t^{\prime}$ and
$t^{\prime\prime}$ can be considered to be uncorrelated. This implies that

$\mathbf{E}^{k}\delta x_{ckt^{\prime}{\bf r}^{\prime}}^{p^{\prime}}\delta
x_{ckt^{\prime\prime}{\bf
r}^{\prime\prime}}^{p^{\prime\prime}}=\mathbf{E}^{k}\delta x_{ckt^{\prime}{\bf
r}^{\prime}}^{p^{\prime}}\mathbf{E}^{k}\delta x_{ckt^{\prime\prime}{\bf
r}^{\prime\prime}}^{p^{\prime\prime}}$ (80)

whenever $t^{\prime}\neq t^{\prime\prime}$ or ${\bf r}^{\prime}\neq{\bf
r}^{\prime\prime}$ ($p^{\prime}$ and $p^{\prime\prime}$ being integers) and,
moreover, the following useful relations hold:

$\displaystyle\mathbf{E}^{t}\mathbf{E}^{t^{\prime}}\delta_{tt^{\prime}}$
$\displaystyle=$ $\displaystyle 1/N_{\mathrm{t}},$ (81)
$\displaystyle\mathbf{E}^{k}\delta x_{ckt{\bf r}}$ $\displaystyle=$
$\displaystyle 0,$ (82) $\displaystyle\mathbf{E}^{k}\delta x_{ckt{\bf
r}}\delta x_{ckt^{\prime}{\bf r}^{\prime}}$ $\displaystyle=$ $\displaystyle
b_{c{\bf r}}\ \delta_{{\bf r}{\bf r}^{\prime}}\delta_{tt^{\prime}},$ (83)
$\displaystyle\mathbf{E}^{k}\delta x_{ckt{\bf r}}^{2}\delta x_{ckt{\bf
r}^{\prime}}^{2}$ $\displaystyle=$ $\displaystyle 3b_{c{\bf r}}^{2}\
\delta_{{\bf r}{\bf r}^{\prime}}+b_{c{\bf r}}b_{c{\bf
r}^{\prime}}(1-\delta_{{\bf r}{\bf r}^{\prime}}).$ (84)

We have used above that the lattice springs are Gaussian variables, Eq. (12),
i.e. for each independent spring of length $x_{c{\bf r}}$ (dropping the
indices $k$ and $t$) we have $\left<\delta x_{c{\bf r}}\right>=0$,
$\left<\delta x_{c{\bf r}}^{2}\right>=b_{c{\bf r}}$ and $\left<\delta x_{c{\bf
r}}^{4}\right>=3b_{c{\bf r}}^{2}$ with $\left<\ldots\right>$ denoting the
thermal average within each basin $c$. Using Eqs. (81-83) we obtain, e.g., the
$k$-average

$v_{c{\bf r}}=\mathbf{E}^{k}v_{ck{\bf r}}=b_{c{\bf
r}}\left(1-1/N_{\mathrm{t}}\right)$ (85)

and, hence, the total average $v=\mathbf{E}^{c}\mathbf{E}^{{\bf r}}v_{c{\bf
r}}=b(1-1/N_{\mathrm{t}})$ with $b=\mathbf{E}^{c}\mathbf{E}^{{\bf r}}b_{c{\bf
r}}$.

To compute the internal correlation function we write

$\displaystyle C_{\mathrm{int}}[v]({\bf r})$ $\displaystyle=$
$\displaystyle\mathbf{E}^{c}C_{c}({\bf r})\mbox{ with }$ (86) $\displaystyle
C_{c}({\bf r})$ $\displaystyle=$ $\displaystyle\mathbf{E}^{k}K[(v_{ck{\bf
r}}-b_{c{\bf r}})-(v_{c{\bf r}}-b_{c{\bf r}})]({\bf r})$ (87) $\displaystyle=$
$\displaystyle\mathbf{E}^{k}K[v_{ck{\bf r}}-b_{c{\bf r}}]({\bf r})-K[v_{c{\bf
r}}-b_{c{\bf r}}]({\bf r})$ (88)

using Eq. (71) in the last step. Due to Eq. (85) it is clear that the second
term in Eq. (88) is of order ${\cal O}(\Delta\tau^{-2})$ We can focus on the
first term which we rewrite as volume average $\mathbf{E}^{{\bf
r}_{1}}\tilde{C}_{c}({\bf r}_{1},{\bf r}_{2}={\bf r}_{1}+{\bf r})$ with

$\displaystyle\tilde{C}_{c{\bf r}_{1}{\bf r}_{2}}({\bf r}_{1},{\bf r}_{2})$
$\displaystyle\equiv$ $\displaystyle\mathbf{E}^{k}(v_{ck{\bf r}_{1}}-b_{c{\bf
r}_{1}})(v_{ck{\bf r}_{2}}-b_{c{\bf r}_{2}})$ (89) $\displaystyle=$
$\displaystyle\mathbf{E}^{k}(A_{1}-B_{1})(A_{2}-B_{2})$ (90)

where Eq. (79) is used for the definition of the terms

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle\mathbf{S}^{{\bf
r}_{3}}\mathbf{E}^{t_{1}}\left(\delta x_{ckt_{1}{\bf r}_{1}}\delta
x_{ckt_{1}{\bf r}_{3}}-b_{c{\bf r}_{1}}\delta_{{\bf r}_{1}{\bf
r}_{3}}\right),$ (91) $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\mathbf{S}^{{\bf r}_{4}}\mathbf{E}^{t_{2}}\left(\delta
x_{ckt_{2}{\bf r}_{2}}\delta x_{ckt_{2}{\bf r}_{4}}-b_{c{\bf
r}_{2}}\delta_{{\bf r}_{2}{\bf r}_{4}}\right),$ (92) $\displaystyle B_{1}$
$\displaystyle=$ $\displaystyle\mathbf{S}^{{\bf
r}_{3}}\mathbf{E}^{t_{1}}\mathbf{E}^{t_{3}}\delta x_{ckt_{1}{\bf r}_{1}}\delta
x_{ckt_{3}{\bf r}_{3}}\ \mbox{and}$ (93) $\displaystyle B_{2}$
$\displaystyle=$ $\displaystyle\mathbf{S}^{{\bf
r}_{4}}\mathbf{E}^{t_{2}}\mathbf{E}^{t_{4}}\delta x_{ckt_{2}{\bf r}_{2}}\delta
x_{ckt_{4}{\bf r}_{4}}.$ (94)

We expand the different contributions and $k$-average using the identities
Eqs. (80-84). As summarized in Table 2 it is helpful to distinguish four cases
for ${\bf r}_{1}$, ${\bf r}_{2}$, ${\bf r}_{3}$ and ${\bf r}_{4}$. Most
contributions are of order ${\cal O}(\Delta\tau^{-2})$ and only three
contributions of $\mathbf{E}^{k}A_{1}A_{2}$ (second column) do matter. A
central result is that due to the last case (${\bf r}_{1}={\bf r}_{4}\neq{\bf
r}_{2}={\bf r}_{3}$), the internal correlation function must remain finite for
$r>0$. Note also that all terms for the second case (${\bf r}_{1}={\bf
r}_{2}\neq{\bf r}_{3}={\bf r}_{4}$) increase linearly with $V$ due to the sum
$\mathbf{S}^{{\bf r}_{3}}\mathbf{S}^{{\bf r}_{4}}\delta_{{\bf r}_{3}{\bf
r}_{4}}(1-\delta_{{\bf r}_{1}{\bf r}_{3}})$. In fact, using
$b_{c}=\mathbf{E}^{{\bf r}_{3}}b_{c{\bf r}_{3}}$, the indicated term for
$\mathbf{E}^{k}A_{1}A_{2}$ can be rewritten as

$b_{c{\bf r}_{1}}(\mathbf{S}^{{\bf r}_{3}\neq{\bf r}_{1}}b_{c{\bf
r}_{3}})/N_{\mathrm{t}}=b_{c{\bf r}_{1}}b_{c}V/N_{\mathrm{t}}-b_{c{\bf
r}_{1}}^{2}/N_{\mathrm{t}}.$ (95)

We finally average over ${\bf r}_{1}$ and $c$ using that the $b_{c{\bf
r}^{\prime}}$ are decorrelated for different ${\bf r}^{\prime}$. Summarizing
all terms we obtain to leading order

$C_{\mathrm{int}}[v]({\bf
r})\simeq\frac{b^{2}}{N_{\mathrm{t}}}\times\left\\{\begin{array}[]{ll}V+1&\mbox{
for }r=0,\\\ 1&\mbox{ for }r>0.\end{array}\right.$ (96)

As a consequence $\delta v^{2}_{\mathrm{int}}(\Delta\tau,V)=\mathbf{E}^{{\bf
r}}C_{\mathrm{int}}[v]({\bf r})=2b^{2}/N_{\mathrm{t}}\propto
V^{0}/\Delta\tau$, as expected.

###  B.2 Scaling for general Gaussian fields

It is clear that the above result can be generalized to other models with
short-range correlations and general $\delta\tau$ including
$\delta\tau\ll\tau_{\mathrm{b}}$. This merely requires a renormalization of
space and time. Especially this suggests to replace $1/N_{\mathrm{t}}$ by
$\delta v^{2}_{\mathrm{int}}(\Delta\tau)$. It is in this context of relevance
that the above result Eq. (96) can be recast as

$C_{\mathrm{int}}[v]({\bf r})\simeq\left\\{V(1-\alpha)c({\bf
r})+\alpha\right\\}\delta v^{2}_{\mathrm{int}}(\Delta\tau)$ (97)

with $\mathbf{S}^{{\bf r}}c({\bf r})=1$, $c({\bf r})\to 0$ for large $r$ (with
$r\neq 0$ for LSM-A) and $\alpha=1/2$. Note that $\mathbf{E}^{{\bf
r}}C_{\mathrm{int}}[v]({\bf r})=\delta v^{2}_{\mathrm{int}}(\Delta\tau)$ holds
for all coefficients $\alpha$. As discussed in Sec. 5.2, the numerical results
of all our LSM variants are consistent with this generalization of the direct
calculation for the simple LSM-A model. (As far as we can tell this even holds
reasonably for systems with long-range correlations.) There is in fact a
general reason for expecting Eq. (97) to hold for many models: For a given $c$
the $x_{t{\bf r}}$-fluctuations are often nearly Gaussian. [For the LSM
variants the joint distributions of the $x_{t{\bf r}}$ are in fact exactly
Gaussian since the total energy is quadratic in $x_{t{\bf r}}$, Eq. (12).]
This allows for a theoretical treatment of $C_{\mathrm{int}}[v]({\bf r})$
based on the cumulant formalism (“Wick’s theorem”) similar to the calculation
which leads to Eq. (45) for the global variance $\delta
v^{2}_{\mathrm{int}}(\Delta\tau)$ lyuda18 ; spmP1 . It is thus possible to
show that Eq. (97) must hold for general fluctuating Gaussian fields with
$\alpha=1/2$. This calculation is beyond of the scope of the present work.

## References

  * (1) L. Klochko, J. Baschnagel, J.P. Wittmer, A.N. Semenov, J. Chem. Phys. 151, 054504 (2019)
  * (2) G. George, L. Klochko, A. Semenov, J. Baschnagel, J.P. Wittmer, EPJE 44, 13 (2021)
  * (3) G. George, L. Klochko, A.N. Semenov, J. Baschnagel, J.P. Wittmer, EPJE 44, 54 (2021)
  * (4) G. George, L. Klochko, A.N. Semenov, J. Baschnagel, J.P. Wittmer, EPJE 44, 125 (2021)
  * (5) W. Götze, _Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory_ (Oxford University Press, Oxford, 2009)
  * (6) A. Heuer, J.Phys.: Condens. Matter 20, 373101 (2008)
  * (7) I. Procaccia, C. Rainone, C.A.B.Z. Shor, M. Singh, Phys. Rev. E 93, 063003 (2016)
  * (8) J.F. Lutsko, J. Appl. Phys 64, 1152 (1988)
  * (9) J.F. Lutsko, J. Appl. Phys 65, 2991 (1989)
  * (10) J.L. Barrat, _Microscopic Elasticity of Complex Systems_ , in _Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology - Vol. 2_ , edited by M. Ferrario, G. Ciccotti, K. Binder (Springer, Berlin and Heidelberg, 2006), Vol. 704, pp. 287—307
  * (11) J.P. Wittmer, H. Xu, P. Polińska, F. Weysser, J. Baschnagel, J. Chem. Phys. 138, 12A533 (2013)
  * (12) A. Lemaître, Phys. Rev. Lett. 113, 245702 (2014)
  * (13) A. Lemaître, J. Chem. Phys. 143, 164515 (2015)
  * (14) D.P. Landau, K. Binder, _A Guide to Monte Carlo Simulations in Statistical Physics_ (Cambridge University Press, Cambridge, 2000)
  * (15) W. Press, S. Teukolsky, W. Vetterling, B. Flannery, _Numerical Recipes in FORTRAN: the art of scientific computing_ (Cambridge University Press, Cambridge, 1992)
  * (16) M.P. Allen, D.J. Tildesley, _Computer Simulation of Liquids, 2nd Edition_ (Oxford University Press, Oxford, 2017)
  * (17) A.L. Barbabási, H. Stanley, _Fractal Concepts in Surface Growth_ (Cambridge University Press, Cambridge, 1995)
  * (18) M. Maier, A. Zippelius, M. Fuchs, Phys. Rev. Lett. 119, 265701 (2017)
  * (19) M. Maier, A. Zippelius, M. Fuchs, J. Chem. Phys. 149, 084502 (2018)
  * (20) F. Vogel, A. Zippelius, M. Fuchs, Europhys. Lett. 125, 68003 (2019)
  * (21) L. Klochko, J. Baschnagel, J.P. Wittmer, A.N. Semenov, Soft Matter 14, 6835 (2018)
  * (22) J.L. Lebowitz, J.K. Percus, L. Verlet, Phys. Rev. 153, 250 (1967)
  * (23) J.P. Hansen, I.R. McDonald, _Theory of simple liquids_ (Academic Press, New York, 2006), 3nd edition