Different types of spatial correlation functions for non-ergodic stochastic processes of macroscopic systems

Abstract

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

Graphical abstract

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Appendices

A Spatial correlations of periodic microcells

1.1 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 $K[y_{\mathbf{r}}](\mathbf{r})$ of a field $y_{\mathbf{r}}$ of volume average $y \equiv \mathbf {E}^{\mathbf{r}} y_{\mathbf{r}}$. Obviously,

$$\begin{aligned} K[y_{\mathbf{r}}-y](\mathbf{r}) = K[y_{\mathbf{r}}](\mathbf{r}) - y^2. \end{aligned}$$

(58)

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

$$\begin{aligned} \mathbf {E}^lK[y_{l\mathbf{r}}-y_l](\mathbf{r}) = \mathbf {E}^l K[y_{l\mathbf{r}}](\mathbf{r}) - \mathbf {E}^ly_l^2. \end{aligned}$$

(59)

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

$$\begin{aligned} \mathbf {E}^lK[y_{l\mathbf{r}}-\lambda y_{\mathbf{r}}](\mathbf{r}) = \mathbf {E}^lK[y_{l\mathbf{r}}](\mathbf{r}) - \lambda (2-\lambda ) K[y_{\mathbf{r}}](\mathbf{r})\qquad \end{aligned}$$

(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

$$\begin{aligned} \mathbf {V}^l y_l= & {} \mathbf {E}^{\mathbf{r}} C(\mathbf{r}) \text{ with } \end{aligned}$$

(61)

$$\begin{aligned} C(\mathbf{r})\equiv & {} \mathbf {E}^l K[y_{l\mathbf{r}}-y](\mathbf{r}) \equiv \mathbf {E}^l K[y_{l\mathbf{r}}](\mathbf{r}) - y^2 \end{aligned}$$

(62)

being the l-averaged correlation function in real space. By comparing Eq. (59) and Eq. (61), we may also write

$$\begin{aligned} C(\mathbf{r}) = \mathbf {E}^l K[y_{l\mathbf{r}}-y_l](\mathbf{r}) + \mathbf {V}^l y_l \end{aligned}$$

(63)

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

$$\begin{aligned} C(\mathbf{r}=\mathbf{0}) = \mathbf {V}^{l\mathbf{r}'} y_{l\mathbf{r}'} = \mathbf {E}^l \mathbf {V}^{\mathbf{r}'} y_{l\mathbf{r}'} + \mathbf {V}^l \mathbf {E}^{\mathbf{r}'} y_{l\mathbf{r}'} \end{aligned}$$

(64)

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

$$\begin{aligned} \mathbf {E}^l \mathbf {E}^{\mathbf{r}'} y_{\mathbf{r}'+\mathbf{r}} y_{\mathbf{r}'} = \mathbf {E}^l \mathbf {E}^{\mathbf{r}'} y_{\mathbf{r}'+\mathbf{r}} \times \mathbf {E}^l \mathbf {E}^{\mathbf{r}'} y_{\mathbf{r}'} = y^2 \end{aligned}$$

(65)

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

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

$$\begin{aligned} \mathbf {V}^l (y_l-z) = \mathbf {E}^{\mathbf{r}} \ \mathbf {E}^l K[(y_{l\mathbf{r}}-z_{\mathbf{r}})-(y-z)](\mathbf{r}). \end{aligned}$$

(66)

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

$$\begin{aligned} \mathbf {E}^{\mathbf{r}} \mathbf {E}^l K[\delta y_{l\mathbf{r}}](\mathbf{r}) = \mathbf {E}^{\mathbf{r}} \mathbf {E}^l K[\delta y_{l\mathbf{r}}-\delta z_{\mathbf{r}}](\mathbf{r}) \end{aligned}$$

(67)

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

1.2 A.2 Derivation of correlation functions

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

$$\begin{aligned} C_{\mathrm {tot}}(\mathbf{r})\equiv & {} \mathbf {E}^l K[\mathcal{O}_{l\mathbf{r}}-\mathcal{O}](\mathbf{r}) = \mathbf {E}^l K[\mathcal{O}_{l\mathbf{r}}](\mathbf{r}) - \mathcal{O}^2 \end{aligned}$$

(68)

$$\begin{aligned} C_{\mathrm {ext}}(\mathbf{r})\equiv & {} \mathbf {E}^c K[\mathcal{O}_{c\mathbf{r}}-\mathcal{O}](\mathbf{r}) = \mathbf {E}^c K[\mathcal{O}_{c\mathbf{r}}](\mathbf{r}) - \mathcal{O}^2 \end{aligned}$$

(69)

$$\begin{aligned} C_{\mathrm {int}}(\mathbf{r})\equiv & {} \mathbf {E}^c \mathbf {E}^k K[\mathcal{O}_{ck\mathbf{r}}-\mathcal{O}_{c\mathbf{r}}](\mathbf{r}) \end{aligned}$$

(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 $\mathcal{O}$ 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 $\delta \mathcal{O}^2_{\mathrm {tot}}= \mathbf {E}^{\mathbf{r}} C_{\mathrm {tot}}(\mathbf{r})$ and Eq. (69) with $\delta \mathcal{O}^2_{\mathrm {ext}}= \mathbf {E}^{\mathbf{r}} C_{\mathrm {ext}}(\mathbf{r})$ is directly implied by Eqs. (61) and (61). To show that Eq. (70) is consistent with $\delta \mathcal{O}^2_{\mathrm {int}}= \mathbf {E}^{\mathbf{r}} C_{\mathrm {int}}(\mathbf{r})$ 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

$$\begin{aligned} C_{\mathrm {int}}(\mathbf{r}) = \mathbf {E}^c \left\{ \mathbf {E}^k K[\mathcal{O}_{ck\mathbf{r}}](\mathbf{r}) - K[\mathcal{O}_{c\mathbf{r}}](\mathbf{r})\right\} . \end{aligned}$$

(71)

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

$$\begin{aligned} \mathbf {E}^{\mathbf{r}} C_{\mathrm {int}}(\mathbf{r})= & {} \mathbf {E}^{\mathbf{r}} \left( C_{\mathrm {tot}}(\mathbf{r}) - C_{\mathrm {ext}}(\mathbf{r})\right) \nonumber \\= & {} \delta \mathcal{O}^2_{\mathrm {tot}}- \delta \mathcal{O}^2_{\mathrm {ext}}= \delta \mathcal{O}^2_{\mathrm {int}}\end{aligned}$$

(72)

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

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

$$\begin{aligned} C_{\mathrm {int}}(\mathbf{r},\lambda ) \equiv \mathbf {E}^c \mathbf {E}^k K[(\mathcal{O}_{ck\mathbf{r}}-\mathcal{O}_c)-\lambda (\mathcal{O}_{c\mathbf{r}}-\mathcal{O}_c)](\mathbf{r})\qquad \end{aligned}$$

(73)

which reduces to Eq. (70) for $\lambda =1$, since it is possible to shift $\mathcal{O}_{ck\mathbf{r}}-\mathcal{O}_c$ with the k-independent field $\lambda (\mathcal{O}_{c\mathbf{r}}-\mathcal{O}_c)$ 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

$$\begin{aligned} C_{\mathrm {tot}}(\mathbf{r}) = C_{\mathrm {int}}(\mathbf{r},\lambda ) + \lambda (2-\lambda ) C_{\mathrm {ext}}(\mathbf{r}) + (\lambda -1)^2 \delta \mathcal{O}^2_{\mathrm {ext}}.\nonumber \\ \end{aligned}$$

(74)

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

1.3 A.3 Important limits

We have omitted for clarity in the preceding subsection all possible dependences on $N_\mathrm {c}$, $N_\mathrm {k}$, $\varDelta \tau$ 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 $\varDelta \tau \gg \tau _{\mathrm {b}}$, i.e., both $\mathcal{O}_{c\mathbf{r}} = \mathbf {E}^k \mathcal{O}_{ck\mathbf{r}}$ and its average $\mathcal{O}= \mathbf {E}^c \mathbf {E}^{\mathbf{r}} \mathcal{O}_{c\mathbf{r}}$ are $\varDelta \tau$-independent to leading order. Due to Eq. (69) the same holds for the external correlation function, i.e.,

$$\begin{aligned} C_{\mathrm {ext}}(\mathbf{r},\varDelta \tau ,V) \simeq C_{\mathrm {ext}}(\mathbf{r},V) \text{ for } \varDelta \tau \gg \tau _{\mathrm {b}}. \end{aligned}$$

(75)

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

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

$$\begin{aligned} \lim _{\varDelta \tau \rightarrow \infty } C_{\mathrm {int}}(\mathbf{r},\varDelta \tau ,V) = 0. \end{aligned}$$

(76)

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

$$\begin{aligned} C_{\mathrm {int}}(\mathbf{r}=\mathbf{r}_2-\mathbf{r}_1,\varDelta \tau ,V) = \mathbf {E}^c \mathbf {E}^{\mathbf{r}_1} \left( \mathbf {E}^{t_1} \mathbf {E}^{t_2} \mathbf {E}^k \ldots \right) \end{aligned}$$

(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 $N_\mathrm {t}\propto \varDelta \tau$, this shows that quite generally the internal correlation function must decay to leading order for all $\mathbf{r}$ as

$$\begin{aligned} C_{\mathrm {int}}(\mathbf{r},\varDelta \tau ,V) \propto \frac{1}{\varDelta \tau } \text{ for } \varDelta \tau \gg \tau _{\mathrm {b}}\end{aligned}$$

(78)

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

B Scaling of $C_{\mathrm {int}}[v](\mathbf{r},\varDelta \tau ,V)$

1.1 B.1 Predictions for LSM-A

Table 2 k-averages $\mathbf {E}^k A_1A_2$, $\mathbf {E}^k A_1B_2$, $\mathbf {E}^k A_2B_1$ and $\mathbf {E}^k B_1B_2$ for LSM-A. The different relevant cases for $\mathbf{r}_1$, $\mathbf{r}_2$, $\mathbf{r}_3$, $\mathbf{r}_4$ are indicated in the first column. Note that $r = |\mathbf{r}|$ with $\mathbf{r}=\mathbf{r}_2-\mathbf{r}_1$. The first two cases indicate contributions for $r=0$, the last two cases contributions for $r>0$. $\mathbf {E}^k A_1B_2$ and $\mathbf {E}^k A_2B_1$ are identical by symmetry. The leading contributions of order $\mathcal{O}(\varDelta \tau ^{-1})$ are due to $\mathbf {E}^k A_1A_2$ (second column). The second case ($\mathbf{r}_1=\mathbf{r}_2 \ne \mathbf{r}_3=\mathbf{r}_4$) yields contributions proportional to the system size

As noted in Appendix A.1, all correlation functions $C[f](\mathbf{r})$ 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 $\mathbf{r}$ are uncorrelated. Here we draw attention to the fact that although the primary instantaneous field $x_{t\mathbf{r}}$ may be uncorrelated this may not be the case for the field $\mathcal{O}_{\mathbf{r}}$ associated with the time-averaged functional $\mathcal{O}[\mathbf {x}]$ of the time series $\mathbf {x}$.²⁰ As we shall see, this matters specifically for the covariance field, Eq. (30),

$$\begin{aligned} v_{ck\mathbf{r}}= & {} V \left\{ \mathbf {E}^t x_{ckt\mathbf{r}}x_{ckt} - x_{ck\mathbf{r}} x_{ck} \right\} \nonumber \\= & {} \mathbf {S}^{\mathbf{r}'} \left\{ \mathbf {E}^t x_{ckt\mathbf{r}} x_{ckt\mathbf{r}'} - \mathbf {E}^t \mathbf {E}^{t'} x_{ckt\mathbf{r}} x_{ckt'\mathbf{r}'}\right\} \end{aligned}$$

(79)

restated for convenience omitting the irrelevant prefactor $\beta$ and introducing the sum $\mathbf {S}^{\mathbf{r}'} \equiv V\mathbf {E}^{\mathbf{r}'}$. In the second step we have made explicit the crucial contributions with $\mathbf{r}'\ne \mathbf{r}$ 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 $x_{ckt\mathbf{r}}$ of different microcells $\mathbf{r}$ are uncorrelated. Ultimately, we want to expand $v_{ck\mathbf{r}}$ for large $\varDelta \tau$. Reminding Eq. (31), it is thus useful that the $x_{ckt\mathbf{r}}$ in Eq. (79) can be replaced by $\delta x_{ckt\mathbf{r}} \equiv x_{ckt\mathbf{r}}-a_{c\mathbf{r}}$ without changing $v_{ck\mathbf{r}}$. Moreover, let us assume that the data is sampled with large time increments $\delta \tau \gg \tau _{\mathrm {b}}$, i.e., different times $t'$ and $t''$ can be considered to be uncorrelated. This implies that

$$\begin{aligned} \mathbf {E}^k \delta x_{ckt'\mathbf{r}'}^{p'} \delta x_{ckt''\mathbf{r}''}^{p''} = \mathbf {E}^k \delta x_{ckt'\mathbf{r}'}^{p'} \mathbf {E}^k \delta x_{ckt''\mathbf{r}''}^{p''} \end{aligned}$$

(80)

whenever $t'\ne t''$ or $\mathbf{r}'\ne \mathbf{r}''$ ($p'$ and $p''$ being integers) and, moreover, the following useful relations hold:

$$\begin{aligned} \mathbf {E}^{t}\mathbf {E}^{t'} \delta _{tt'}= & {} 1/N_\mathrm {t}, \end{aligned}$$

(81)

$$\begin{aligned} \mathbf {E}^k \delta x_{ckt\mathbf{r}}= & {} 0, \end{aligned}$$

(82)

$$\begin{aligned} \mathbf {E}^k \delta x_{ckt\mathbf{r}} \delta x_{ckt'\mathbf{r}'}= & {} b_{c\mathbf{r}} \ \delta _{\mathbf{r}\mathbf{r}'} \delta _{tt'}, \end{aligned}$$

(83)

$$\begin{aligned} \mathbf {E}^k \delta x_{ckt\mathbf{r}}^2 \delta x_{ckt\mathbf{r}'}^2= & {} 3 b_{c\mathbf{r}}^2 \ \delta _{\mathbf{r}\mathbf{r}'} + b_{c\mathbf{r}}b_{c\mathbf{r}'}(1-\delta _{\mathbf{r}\mathbf{r}'}). \end{aligned}$$

(84)

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

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

(85)

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

To compute the internal correlation function, we write

$$\begin{aligned} C_{\mathrm {int}}[v](\mathbf{r})= & {} \mathbf {E}^c C_c(\mathbf{r}) \text{ with } \end{aligned}$$

(86)

$$\begin{aligned} C_c(\mathbf{r})= & {} \mathbf {E}^k K[(v_{ck\mathbf{r}}-b_{c\mathbf{r}})-(v_{c\mathbf{r}}-b_{c\mathbf{r}})](\mathbf{r}) \end{aligned}$$

(87)

$$\begin{aligned}= & {} \mathbf {E}^k K[v_{ck\mathbf{r}}-b_{c\mathbf{r}}](\mathbf{r}) - K[v_{c\mathbf{r}}-b_{c\mathbf{r}}](\mathbf{r}) \end{aligned}$$

(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 $\mathcal{O}(\varDelta \tau ^{-2})$. We can focus on the first term which we rewrite as volume average $\mathbf {E}^{\mathbf{r}_1} {\tilde{C}}_c(\mathbf{r}_1,\mathbf{r}_2=\mathbf{r}_1+\mathbf{r})$ with

$$\begin{aligned} {\tilde{C}}_{c\mathbf{r}_1\mathbf{r}_2}(\mathbf{r}_1,\mathbf{r}_2)\equiv & {} \mathbf {E}^k (v_{ck\mathbf{r}_1}-b_{c\mathbf{r}_1}) (v_{ck\mathbf{r}_2}-b_{c\mathbf{r}_2}) \end{aligned}$$

(89)

$$\begin{aligned}= & {} \mathbf {E}^k (A_1 - B_1) (A_2 -B_2) \end{aligned}$$

(90)

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

$$\begin{aligned} A_1= & {} \mathbf {S}^{\mathbf{r}_3} \mathbf {E}^{t_1} \left( \delta x_{ckt_1\mathbf{r}_1} \delta x_{ckt_1\mathbf{r}_3} - b_{c\mathbf{r}_1} \delta _{\mathbf{r}_1\mathbf{r}_3}\right) , \end{aligned}$$

(91)

$$\begin{aligned} A_2= & {} \mathbf {S}^{\mathbf{r}_4} \mathbf {E}^{t_2} \left( \delta x_{ckt_2\mathbf{r}_2} \delta x_{ckt_2\mathbf{r}_4} - b_{c\mathbf{r}_2} \delta _{\mathbf{r}_2\mathbf{r}_4}\right) , \end{aligned}$$

(92)

$$\begin{aligned} B_1= & {} \mathbf {S}^{\mathbf{r}_3} \mathbf {E}^{t_1} \mathbf {E}^{t_3} \delta x_{ckt_1\mathbf{r}_1} \delta x_{ckt_3\mathbf{r}_3} \ \text{ and } \end{aligned}$$

(93)

$$\begin{aligned} B_2= & {} \mathbf {S}^{\mathbf{r}_4} \mathbf {E}^{t_2} \mathbf {E}^{t_4} \delta x_{ckt_2\mathbf{r}_2} \delta x_{ckt_4\mathbf{r}_4}. \end{aligned}$$

(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 $\mathbf{r}_1$, $\mathbf{r}_2$, $\mathbf{r}_3$ and $\mathbf{r}_4$. Most contributions are of order $\mathcal{O}(\varDelta \tau ^{-2})$ and only three contributions of $\mathbf {E}^kA_1A_2$ (second column) do matter. A central result is that due to the last case ($\mathbf{r}_1=\mathbf{r}_4 \ne \mathbf{r}_2=\mathbf{r}_3$), the internal correlation function must remain finite for $r > 0$. Note also that all terms for the second case ($\mathbf{r}_1=\mathbf{r}_2 \ne \mathbf{r}_3=\mathbf{r}_4$) increase linearly with V due to the sum $\mathbf {S}^{\mathbf{r}_3}\mathbf {S}^{\mathbf{r}_4} \delta _{\mathbf{r}_3\mathbf{r}_4} (1-\delta _{\mathbf{r}_1\mathbf{r}_3})$. In fact, using $b_c=\mathbf {E}^{\mathbf{r}_3}b_{c\mathbf{r}_3}$, the indicated term for $\mathbf {E}^kA_1A_2$ can be rewritten as

$$\begin{aligned} b_{c\mathbf{r}_1} (\mathbf {S}^{\mathbf{r}_3\ne \mathbf{r}_1}b_{c\mathbf{r}_3})/N_\mathrm {t}= b_{c\mathbf{r}_1} b_c V/N_\mathrm {t}- b_{c\mathbf{r}_1}^2/N_\mathrm {t}. \end{aligned}$$

(95)

We finally average over $\mathbf{r}_1$ and c using that the $b_{c\mathbf{r}'}$ are decorrelated for different $\mathbf{r}'$. Summarizing all terms, we obtain to leading order

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

(96)

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

1.2 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 $\delta v^2_{\mathrm {int}}(\varDelta \tau )$. It is in this context of relevance that the above result Eq. (96) can be recast as

$$\begin{aligned} C_{\mathrm {int}}[v](\mathbf{r}) \simeq \left\{ V (1-\alpha ) c(\mathbf{r})+\alpha \right\} \delta v^2_{\mathrm {int}}(\varDelta \tau ) \end{aligned}$$

(97)

with $\mathbf {S}^{\mathbf{r}} c(\mathbf{r}) = 1$, $c(\mathbf{r}) \rightarrow 0$ for large r (with $r \ne 0$ for LSM-A) and $\alpha =1/2$. Note that $\mathbf {E}^{\mathbf{r}} C_{\mathrm {int}}[v](\mathbf{r}) = \delta v^2_{\mathrm {int}}(\varDelta \tau )$ holds for all coefficients $\alpha$. As discussed in Sect. 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 $x_{t\mathbf{r}}$-fluctuations are often nearly Gaussian. [For the LSM variants the joint distributions of the $x_{t\mathbf{r}}$ are in fact exactly Gaussian since the total energy is quadratic in $x_{t\mathbf{r}}$, Eq. (12).] This allows for a theoretical treatment of $C_{\mathrm {int}}[v](\mathbf{r})$ based on the cumulant formalism (“Wick’s theorem”) similar to the calculation which leads to Eq. (45) for the global variance $\delta v^2_{\mathrm {int}}(\varDelta \tau )$ [2, 21]. 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.