On the three-dimensional spatial correlations of curved dislocation systems

Coarse-grained descriptions of dislocation motion in crystalline metals inherently represent a loss of information regarding dislocation-dislocation interactions. In the present work, we consider a coarse-graining framework capable of re-capturing these interactions by means of the dislocation-dislocation correlation functions. The framework depends on a coarse-graining length to define slip-system-specific dislocation densities. Following a statistical definition of this coarse-graining process, we define a spatial correlation function which will allow relative positions of dislocation pairs, and thus the strength of their interactions at short range, to be recaptured into a mean field description of dislocation dynamics. Through a statistical homogeneity argument, we present a method of evaluating this correlation function from discrete dislocation dynamics simulations. Finally, results of this evaluation are shown in the form of self-correlation of dislocation densities on the same slip-system. These correlation functions are seen to depend weakly on plastic strain (and, in turn, the total dislocation density), but are seen to depend strongly on the coarse-graining length. Implications of these correlation functions in regard to continuum dislocation dynamics as well as future directions of investigation are also discussed.


Introduction
Dislocation-dislocation correlations represent an important link between the continuum and discrete descriptions of the dislocation dynamics. Many views on what this correlation represents, how to evaluate it, and what kinetically-relevant information it contains have been presented in recent years. The present work puts forward a clear and robust definition of the dislocationdislocation correlation functions and presents computations based upon simulations of discrete dislocation systems.
One may think of correlation functions as a certain error estimate on mean field representations of discrete systems (cf. self-consistent field theories, Hartree-type theories of electronic systems (Hartree, 1928)). Specifically in our case, the dislocation-dislocation correlation functions represent an error estimate on mean dislocation density field theories (El-Azab et al., 2018). Therefore, to even define a correlation, we must first have some idea of what we are referring to as our mean dislocation density field. Several descriptions have been proposed in recent years, both for the two-dimensional (2D) case (Groma, 1997, Groma et al., 1999, Valdenaire et al., 2016 of perfectly parallel edge dislocations and the 3D case of curved dislocations (Hochrainer, 2007, Sandfeld, 2010, Xia, 2016. For the purpose of this work, we will consider three-dimensional (3D) dislocation configurations in face-centered cubic (FCC) crystals by distinguishing each of the 12 slip systems [ ] as a separate (vector) density field The most recent of these studies (Valdenaire et al., 2016), has found that the spatial correlations in a dislocation system is dependent on the convolution length . We will follow this formalism as well, with our approach most closely following that of Valdenaire et al. (Valdenaire et al., 2016).
One of the major purposes behind this length scale dependent scheme is that it allows us to distinguish between dislocation structures which occur at two different scales. The local structure-at a length scale on the order of or below the convolution length -can be associated with the correlation, while the spatial variation of the mean-field density can be used to describe longer-length structures such as dislocation patterning. Such patterns have been observed in some of the mean-field theories already presented (Groma et al., 1999, , 2015. There is also evidence of these patterns in discrete dislocation dynamics simulations 1 (Deng et al., 2007).
Generally speaking, one of the major goals of continuum dislocation dynamics is to observe the formation of these patterns, as they are thought to play a significant role in the response of crystalline materials to monotonic and cyclic loading (Sauzay et al., 2011).
None of the above mean-field approaches are capable of capturing the true dynamics of a dislocation system. This is due to one unavoidable fact: dislocation interactions depend on the relative arrangement of dislocations, while mean-field theories all involve a systematic "forgetting" of this relative arrangement. This relative arrangement of the dislocations is represented by the dislocation correlation functions, and is precisely the information which our present formulation purports to recover.
The reason we wish to recover this relative arrangement information is that one of the most kinetically significant quantities pertaining to dislocation interactions is the elastic energy functional. This functional can under certain assumptions be expressed in terms of the mean field dislocation density and a certain integral of the correlation (Groma et al., 2015, Zaiser, 2015. As a result, there has been significant interest in calculating the form of the correlation functions. Two means have been explored to do so. The first follows statistical mechanical arguments to arrive at analytical forms of the dislocation-dislocation correlation function, while the other calculates the correlations brute-force from discrete simulations. Investigations along these two lines have elucidated some of the alterations which the correlation functions introduce into the dynamics. In short, the correlations produce additional stress terms (a friction and back stress) (Groma et al., 2003, Valdenaire et al., 2016, Zaiser, 2015, and alter the mobility of the mean-field density (Kooiman et al., 2015).
As mentioned, there have been attempts to analytically compute the geometrically necessary dislocation field induced in a homogenous dislocation field due to a dislocation pinned at the origin, controversially interpreted as a correlation. The analytical solutions obtained, however, still require a parameter which must be fit to discrete simulations (Groma et al., 2006, Limkumnerd et al., 2008, Zaiser, 2015 (For the clearest presentation of this parameter, see (Zaiser, 2015)). As a result, one goal of the present work is to present a formalism by which these correlation functions might be computed directly from discrete dislocation configurations.
The question then arises as to how one might compute these correlation functions from discrete data. There have been several attempts to accomplish this task. They all involve the simulation of a random, homogenous distribution of discrete (2D) edge dislocations which have been relaxed at zero stress. The resulting relative separation vectors of same-sign and different-sign dislocations are binned into a histogram, which is interpreted as the correlation function. The first investigations which used this method (Gulluoglu et al., 1988, Wang et al., 1997 were largely motivated by a characterization of the dislocation microstructure, and agree with later evaluations of the correlation which arose with interest to the dynamics (Groma et al., 2006, , 2003, Zaiser et al., 2001. However, the only attempt in 3-dimensions attempted to evaluate a radial distribution function of the scalar line density (Csikor et al., 2008), which does not contain kinetically useful information. Valdenaire et al. (Valdenaire et al., 2016), using their convolution length dependent mean-field theory, were able to ascertain a dependence of the correlation on the convolution length using this binning method. A dependence on convolution length should be anticipated, as the convolution length controls the partition of relative arrangement information between the correlation and the density field: as the convolution length decreases, the mean-field density represents a better picture of the relative arrangement of the dislocations, and less correction is needed from the correlation functions.
The present work represents an extension of Valdenaire's approach to 3D dislocation arrangements, while in the process it deepens the theoretical basis of the analysis. The work may be outlined as follows: in section 2, we define a measure theoretic picture of the dislocation densities, pair-distributions, and finally correlation functions; in section 3, we outline a means of evaluating the result from discrete simulations. In following sections, we apply this formalism to discrete dislocation configurations and present the correlation functions for dislocation pairs on like slip systems.

Measure Theoretic Definition of Correlations
To arrive at a definition of the correlation function, we first motivate the discussion with a definition of the energy of a discrete dislocation configuration. We then follow with a discussion of mesoscopic density fields (mean-fields) and arrive at a definition of the correlation function which reveals a clear path forward in evaluation.

Energy Functional of a Discrete Dislocation Configuration
Let us consider a dislocated FCC crystal. The dislocation configuration represents 12 1dimensional manifolds ℒ [ ] embedded in the crystal manifold ℳ, which we consider identical to where [ ] denotes the unit tangent vector of ℒ [ ] , and [ , ] denotes an energetic interaction kernel, a second rank tensor representing the energetic interaction of two differential segments and ′ on slip systems [ ] and [ ], respectively. The interaction kernel is of the form (Hirth et al., 1982, Zaiser, 2015: Now stated, we will decline to use this expression in further analysis. For the sake of brevity, the dependence of equation (1) on the slip systems will be put aside to be reinserted at a later point in the analysis.
We choose to represent our system with a spatial field describing the density of lines around a given point in space (time dependence is implicit throughout the formulation presented here). As a result, we must define our basic (discrete) system in terms of a singular dislocation density ( ): where we have used the vector-valued differential line element ≔ ( ).
This dislocation density defines two measures on ℳ: These measures represent the geometrically necessary dislocation content and total dislocation line length contained in Ω, respectively. These are singular measures with respect to the volume measure, as they are non-zero on sets of zero measure (subsets of ℒ).
The density above allows us to re-express the energy functional by the following integration: where , represent the vector components of ( ) and Einstein's summation convention has been used. In this form, it becomes apparent that the energy functional represents a sum of nine integrations of ℰ against nine measures . These measures, however, are distinct from the measures in equations (5). Rather, represent measures of the product space ℳ 2 . In the discrete case which we are considering, this product measure is simply expressed as the product of the discrete measures: = 3 3 ′ . However, we are interested in a statistical description of the dislocation configuration; in such a description, this product measure no longer has such a trivial form. In the following subsections, we will consider a definition of our statistical description, in the course of which it will be apparent why this product measure requires additional considerations. We will then return to examine equation (6) in light of this statistical description.

The Dislocation Ensemble and the Mesoscopic Dislocation Density Field
The fundamental problem of statistical mechanics is one of coarse-graining: purposely throwing away some amount of information about a dynamical system. This raises some obvious questions: how much information should we throw away, and are the dynamics still recoverable from the information that is retained in our model? The answers are somewhat related: we would like to throw away as much information as we possibly can while still being able to recover the dynamics.
The recoverability of the dynamics hinges upon the energy being recoverable from the coarse description of the system (Öttinger, 2005), which is why we have motivated this discussion with the energy functional of a discrete dislocation system (equations 1 and 2).
In more rigorous terms, this "throwing away" of information translates to the action of a projection operator, the so-called ensemble average. The completely defined system resides in some space of microstates Γ, but we wish to project this onto a lower dimensional space Τ, the coarse-grained space or space of macrostates. We do so through a many-to-one function: This map creates equivalence classes in our space of microstates corresponding to the level sets of Ψ: We then define on each equivalence class Γ a unit measure, a map from a -algebra on Γ to the unit interval: such that (Γ ) is equal unity (cf. regular conditional probabilities (Durrett, 2019)). We then define the projection operation which we will refer to as the ensemble average as a projection from functions of microstate variables ∈ Γ to functions of macrostate variables ∈ Τ: We realize that the set theoretic notation used above may not be accessible to the average reader and as such it has been explained in appendix 1, along with an attempt to build a physical intuition into the definitions presented based on Gibbs' canonical ensemble.
In the case of a dislocation configuration, a completely determined description is a state such as we have already discussed: the collection of the twelve line objects corresponding to the twelve species of dislocation line {ℒ [ ] } =1 12 . Also as discussed, this is equivalent to the singular densities [ ] . As these represent (piecewise) continuous space curves, this represents a vast amount of information. The space of microstates, then, is the set of all space curves, with a few minor constraints that are difficult to formulate in the sense of this space.
There is an equivalent operation, one where we choose a particular function 0 ( ), for which we define its ensemble average ⟨ 0 ⟩( ). This induces the in-between steps, although not uniquely by any means. We may say that there exists some map Ψ and a measure on its level sets such that: This is the operation we would like to perform on the density fields . In this sense, we now look at the field itself as a map from the space of lines to the space of distributions on ℳ, and we can ensemble average this over many configurations of lines.
Motivated by the discussion above, let us consider the ensemble of states which are somehow "near" a parent state ℒ 0 , with singular density ( ). We define the ensemble (combination of map Ψ and measures ) as any ensemble which returns the following projection operation: where ( ) is a function with compact support Ω which is characterized by some length parameter . We will denote this average as ( ), and refer to it as the mesoscopic dislocation density vector associated with the dislocation configuration ℒ 0 . This definition of the ensemble average density follows a similar line of reasoning as Valdenaire et al. (Valdenaire et al., 2016) and the aforementioned loose suggestion by Groma (Groma et al., 1999). Thus we have entirely avoided defining: 1) which microstates are considered in our ensemble (up to a small caveat in footnote 2), 2) a map from the microstate space to the projected space, and 3) measures on the level sets of this map. This relation in equation (12) is the definition of our ensemble. With this ensemble in hand, we refocus our attention to the issues of product measures and energy.

Product Measures and Correlation
We now wish to return to the problem of the recoverability of the energy (and thereby the dynamics) from the mesoscopic dislocation density. This requires an examination of ⟨ ⟩. By linearity of the integral, this is simply: Now we are interested in the measure produced by prod = ⟨ ( ) ( ′ )⟩ 3 ′ 3 . We would like, however, to integrate against the "naïve" product measure naïve = ( ) ( ′ ) 3 ′ 3 .
To do this, we must first examine why these two are not equal.
Consider for each microstate ℒ, the singular density field is rather expressed ( ℒ ) = + ( ℒ ) . By definition, ⟨( ℒ ) ⟩ = 0. However, in the projection of its product, we obtain: We may not assume that the second term on the right-hand side is necessarily equal to zero.
However, this is not to say that we are no longer able to integrate against the naïve product measure; we may do so in the following way: where ( , ′ ) here denotes the Radon-Nikodym derivative 2 . For our case in which both prod and naïve have density functions, the Radon-Nikodym derivative is simply expressible as: This definition does not involve any form of summation. Some may object to the use of vector index notation in the above equation, but we have considered each component of ( ) ( ′ ) and ( ) ( ′ ) as separate scalar quantities throughout our discussion of measures. This equation is no exception to that rule. The above equation is best understood in the sense that ⟨ ( ) ( ′)⟩ = ( , ) ( , ′ ) ( ) ( ′), with ( , ) being a scalar correlation transforming the tensor product ( ) ( ′) of the mean field densities to the ensemble average of the singular density product, which is also a tensor. We also note that by the linearity of the ensemble average: where we have introduced a field which we will refer to as a "proto-correlation density" Thus, we have arrived at a form of the correlation which informs us how to evaluate it from discrete data. Given some way of evaluating this ensemble average, we must merely examine the average product of proto-correlation densities at two points.
The energy of the system (integration of the interaction kernel against the product measure), then now be expressed: Before we move on, however, we reintroduce in a straightforward manner the multi-slip aspect of the dislocation configuration (previously dropped from equation (1)) in the following two equations:

Evaluation Scheme
In this section we present a scheme by which we may evaluate the expression for the correlation function seen in equation (21). This is a two-step process. The first step involves a discretization scheme in which we mollify the singular densities present in ̃[ ] ( ). The second step is to define a certain statistical homogeneity assumption that will allow us to empirically measure the

Regularization Scheme
In order to evaluate any expression containing ̃[ ] ( ) from simulation data, we must mollify the singular character of the discrete density [ ] ( ). In order to perform this, we perform a double convolution with some weight function 0 , suppressing for the moment the slip system notation: The prime or lack thereof denotes whether the convolution is over or ′ . The weight function is arbitrary, so long as it has unit integral and is of small, compact support characterized by some length 0 . If the convolution length in the mean field calculation ( ) is significantly longer than 0 , then we may treat the mean field density as constant over the support of 0 , simplifying our expression of ̃ * 0 : Note that we have incorporated the weight function convolution into a compact notation.

Empirical Measurement
Given some random variable and independent measurements of that variable , we may be confident that by the law of large numbers the empirical mean approaches the ensemble average: We also know that by the central limit theorem, the following normalized sum converges in distribution to a standard normal random variable: It follows by continuity of the inverse square root that: As a result, we may quantify our uncertainty by noting that 68% of measurements will fall in the range: where ≔ √ 2 ̅̅̅̅ / we define as the standard error of the empirical measurement.
To begin the analogy to the question of correlations, consider first the case where only one slip system is present in the crystal. In this case, the averages which we would like to consider are of the form: Turning to the simulated crystal, we may choose a sample grid to be a finite, countable collection of position vectors: We then have a measurement of ̃( )̃ * ( ′ ) at all points in × , 2 in total. However, since the average in equation (29) is taken over a very poorly defined ensemble, some discretion is necessary in terms of which points we choose to include in our empirical average.
To discriminate between measurements, we consider two factors. We consider points equivalent if they are kinetically or kinematically equivalent up to the value of the microscopic dislocation density. We call points kinetically equivalent if the interaction energy would be equivalent, and we call points kinematically equivalent if the transport relations of the discrete density would be equivalent at the two points.
We will now translate these requirements into partitions of our measurement space × .
Consider first the kinetic equivalence classes. Examining the dependency of the interaction kernel ℰ (equation (2)) allows us to note that two pairs of points are kinetically equivalent if they share the same separation vector − ′ . This allows us to partition × into the equivalence classes: Furthermore, we note that the transport behavior of a location with zero density field is significantly different from locations where there is a density field present. Thus we consider the set: Considering the points which are kinetically and kinematically equivalent results in a partition of the measurement space into sets of interest: and irrelevant sets ( × ) ∖ (̃×̃).
Treating these sets of interest as equivalent measurements of ̃ * ( )̃ * ( + ), we may now apply the law of large numbers and central limit theorem to the correlation expression in equation (22): where we have represented by the cardinality of . We note two things regarding the standard error functions. First, this standard error does not represent the standard error in the calculation of ( , ) ( ) but rather in the double convolution 0 * ( , ) ( ) * 0 ′ . Secondly, the standard error varies with the components and as well as being a spatially varying field.

Multi-slip Considerations
Further discrimination among our sample points becomes necessary when we consider systems with twelve slip system proto-correlations ̃[ ] ( ) (we momentarily suppress the convolution notation in favor of slip system dependence). We first consider an altered form of ̃ which is unique to each slip-system: the dislocation density transport equations must be suspended at these points. For this reason, we do not consider these points equivalent to points where only one density field is present.
To be precise in this omission, consider the sets [ ] and [ , ] (glissile and junction): Thus, there are three new types of pairs in our sample partition: glissile-glissile pairs, junctionjunction pairs, and glissile-junction pairs. We will in the present work only consider the glissileglissile pairs: To summarize, in a multi-slip dislocation system, there are several classes of correlation functions, calculated as in equation (34) with varying types of pair sets considered. Most broadly there are the glissile correlations and the junction correlations, the distinction of which we have treated immediately above. Secondly, there are what we will refer to as self-correlations and crosscorrelations, considering like-slip-system densities and unlike-slip system densities, respectively.
That is, their pair sets are of the form of equation (39) with = and ≠ .

Calculations
As a preliminary consideration, we have calculated a small subset of the correlation functions from a set of discrete dislocation dynamics simulations. Specifically, we consider the glissile selfcorrelations only.

Dislocation Dynamics Simulations
Discrete dislocation dynamics simulations of copper were carried out using microMegas (Devincre et al., 2011). 45 simulations were performed, all beginning from initial configurations of dipolar loops in a periodic box of dimensions 4.4 × 4.90 × 5.8 μm. 15 random seeds were used to create the initial configurations, and simulations were run in a strain-controlled mode to 0.3% plastic strain. Parameters used to create the initial configurations can be found in Table 1, while Simulation parameters for the dislocation dynamics simulations can be found in directions respectively. This was done to suppress any dependence which may have arisen in the correlations due to the loading direction, as such a dependence has been seen to occur in 2D dislocation dynamics simulations (Valdenaire et al., 2016).
Following these simulations, instantaneous dislocation configurations were extracted at 0.075%, 0.15%, 0.225%, and 0.3% plastic strain. These extractions will allow us to examine how the correlation is affected as the simulation progresses and the total dislocation density rises.

Calculation of Density Fields
The scheme for post-processing the dislocation configuration data to obtain correlation functions follows the line of reasoning resulting in equation (34). The crystal was discretized into an array of sample points 720 points long in the longest direction; this amounts to an 8.1 nm distance between points. The discrete-level convolution length 0 was chosen to be twice this distance (16.2 nm) as this is the largest discretization distance in the simulation (the distance between discrete slip planes). Subsequent convolution length are multiples of the sample distance ranging from 24.3 nm to 162 nm.
All convolutions were performed using a cloud-in-cell weight function: originally exposited in (Birdsall et al., 1969), and used previously in discrete-to-continuum treatments of dislocations in (Bertin, 2019) on account of its analytical solution for line integrals.
For each simulation output, we now have in hand the dislocation density and GND density at all points in the crystal at 10 different levels of coarseness. Since the correlation calculation in equation (33) only involves the proto-correlation, and the support of the proto-correlation is the support of the "discrete" density, we only evaluate the higher convolutions on the support of ̃ * .
Only glissile segments were used in the calculation of these densities: sessile junction segments (having Burgers vectors which are sums of the basic FCC Burgers vectors) were ignored.

Computational Results
The main goal of the present work is to present a formulation which allows these dislocation correlation functions to be calculated. As proof of the validity of this formulation we will show results from a small (but important) class of correlations. While the free indices on ( , ) [ , ] ( ) imply dependence on 3 vector components and 12 slip systems, we considered the correlations for which = . We will refer to such correlations as self-correlations. Since the two dislocation densities lie in the same slip system, we refer to all separation distances and density components in terms of a slip-system coordinate system consisting of the Burgers vector direction ̂, the slip plane normal ̂, and the binormal vector ̂≔̂×̂. Together these form a right-handed coordinate system . For all present intents and purposes, the density vector is a planar quantity, having only screw (̂) and edge (̂) components. The self-correlations between the screw-screw, screw-edge, and edge-edge components are discussed. The edge-screw component will not be discussed, as it is symmetric by parity to the screw-edge component.
Among these results, we first present in Fig. 2 the dependence of the self-correlations on the convolution length . These are all shown in the Δ = 0 plane (the slip plane itself). We quickly note that all the features of the correlation function seem to be relative to the convolution length.
Past some minimal convolution length (≥65 nm), we see qualitatively similar spatial variation up to some spatial rescaling due to the convolution length. We suggest that the obscurity of some of the small features near the origin is not qualitatively different behavior, but rather arises due to the convolution on the order of 8.1 nm discussed in equation (29). For this reason, we choose the largest convolution length (162 nm) for the subsequent presentation of self-correlations, as it presents the clearest picture of the correlation, being less obscured by this effect.

Fig. 2
Dependence of the correlation tensor on the convolution length. All correlations shown are calculated from the 45 configurations at 0.30% strain, and the correlations are relative to a "discrete density" which is calculated with a 16 nm convolution length. The rows show the edge-edge, screw-edge, and screw-screw components of the correlation function and the columns demonstrate progressively longer correlation lengths. All figures show the correlation on the same color scale and relative to the same spatial dimensions. For reference, a white dotted circle is shown with a radius equal to the convolution length.
The second relation we would like to demonstrate is the dependence of the self-correlations on the plastic strain, shown in Fig. 3. Self-correlations were calculated separately from dislocation configurations at each strain step using 81 and 162 nm convolution lengths. In the course of the simulation (from 0.075% to 0.3%), the total dislocation density roughly doubles. However, we notice very little qualitative difference between these correlation functions as they run to higher strains.
Next we show the out-of-plane behavior of these self-correlation functions calculated at 0.3% strain with a convolution length of 162 nm. In Fig. 4, the first two columns show the Δ = 0 and Fig. 3 Dependence of the correlation tensor on the plastic strain (a surrogate for the dislocation density). This dependence (or rather lack thereof) is shown for two convolution lengths, 81 and 162 nm. White dotted circles have radius equal to the convolution length. Each component of the correlation tensor is shown for both convolution lengths. Plastic strain increases with descending row, with strain steps at every 0.075% Δ = 0 planes respectively. In these plots we notice marked anti-correlation ( < 1) for the offplane separation vectors. In the subsequent columns, slices parallel to the slip plane are shown for normal distances up to two times the convolution length, which equates to 324 nm in this case. We notice that this anti-correlation begins at distances as small as one tenth of the convolution length, and that the self-correlations converge to an uncorrelated state ( = 1) at distances greater than the convolution length.
To see this radial convergence to an uncorrelated state, we consider two types of radial correlation functions. The first is integrated over circles in the slip plane with radius , while the second is integrated over the spherical surface with radius : Columns 3-7 show slices parallel to the slip plane at the indicated values of Δ n . Taken together, they show that the correlation function is largely relevant only on the slip plane itself, seeing slight anti-correlations for small outof-plane separation vectors, and rapidly decaying to an uncorrelated state as Δ n > (the convolution length). Again, the outline of a sphere with radius equal to the convolution length is shown for reference on slices which pass through said sphere.
These are plotted in Fig. 5. The in-plane radial correlation function is not seen to converge to an uncorrelated state ( − 1 = 0). However, the spherical radial correlation function does converge to zero within 5 convolution lengths of the origin.
Lastly, we turn our attention to the convergence behavior of the self-correlation calculations themselves. Using all the data available, we first calculate the self-correlation ( , ) [ , ] . We then use this in equation (34) and calculate the standard error using only a restricted dataset corresponding to a partial number of configurations. Each such calculation returns a spatial field of standard error. To easily understand the spatial variation of this field, this standard error is averaged over discs of radius /2, , 3 /2, and 2 in planes with normal distance 0, /2, , and 2 . These averages are shown in Fig. 6 for the screw-screw standard error. Each plot shows this convergence for regions at a constant normal distance, while the regions of different radii are shown in series on each plot. The other components (screw-edge and edge-edge) are similar as the greatest influence is due to the multiplicity , which is roughly equivalent for the three Fig. 5 Radial correlation functions. Shown with ±1 standard error region. The correlations shown were calculated with a convolution length of 64 nm to examine large separation distances. The in-plane standard errors are too small to display on these axes. The in-plane radial correlation does not converge to an uncorrelated state ( − 1 = 0) in the spatial region examined. However, the spherical radial correlation function decays quickly to zero, with the error bars including zero by = 5 .
components. Of note is an order of magnitude difference upon increasing radial distance from the cylindrical axis, as well as an order of magnitude difference upon leaving the slip plane with a weaker dependence between higher off-plane distances. This is due to a slight sampling bias which favors data with in-slip-plane separation vectors. The magnitude of even the highest standard error, however, is small compared to the range of the correlation functions (~10 0 ). For this reason we are confident in the convergence of our self-correlation calculation to a meaningful average.

Discussion
There is an important feature of the correlation function which warrants further discussion. This is the tensor nature of the correlation. In the most basic sense, the correlation is simply a transformation between two tensorial quantities: Fig. 6 Convergence behavior of the screw-screw self-correlation functions in various regions of separation space. The standard error (see equation 35) was calculated with only some of the dislocation configurations included in the calculation. The resulting standard error was then averaged over discs parallel to the slip plane, with the outof-plane distance shown at the top of the column. The convergence of the standard error (plotted logarithmically) with respect to the number of configurations included in the calculation is shown for each disc and out-of-plane distance. The spatial regions (and correlations) are shown in the bottom row for reference. Of note are two trends: standard error tends to increase at distances away from the cylindrical axis, and also increases by an order of magnitude upon leaving the slip plane.
In the above equation and in all treatment throughout this work, the correlation ( , ) represents a scalar transformation between one component of two tensor fields. However, it can be expressed as a fourth order tensor in the following way: where , with being the Kronecker delta. Under transformation of the underlying spaces by the same coordinate transformation , expression (44) transforms as follows: We thus see that ( − ′ ) transforms as a fourth rank two-point tensor (second rank in each leg). However, due to the two Kronecker deltas in the definition of , it has the same number of non-zero components as a second rank tensor. For simplicity, the non-zero components were referred to by their second rank equivalents throughout the work; ( , ) was treated as having screw-screw, screw-edge, and edge-edge components, respectively.
Having presented a number of dependencies of the self-correlation functions, we would like to discuss some of the implications of the findings with respect to incorporation into continuum dislocation dynamics schemes. This will be followed by a discussion of some open questions which were not settled by the present investigation.

Incorporation in Continuum Dislocation Dynamics
It was noted that the most significant non-spatial dependence of the self-correlation functions is on the convolution length . This implies that any density-based continuum treatment of dislocations is dependent on the length-scale used to describe the system. The convolution length was used to force a distance of slow variation in the dislocation density field, and as a result has a close analog in the mesh size used to describe the dislocation density field in a spatially discretized continuum model. The findings imply that the correlation functions should be scaled with respect to that mesh size. The constant variation with respect to the mesh rather than the unscaled space will also allow for simpler integration in finite-element schemes, as they scale identically to the underlying shape functions.
Since it was observed that the self-correlation was stable with respect to the plastic strain (and as a result, total dislocation density), it is only necessary to supply a single form of the selfcorrelation fields at the beginning of a continuum simulation. It was initially a concern that these correlation functions would vary over the course of a simulation, greatly increasing the complexity as some sort of parallel simulation would have been needed to model the evolution of the correlation functions themselves. However, this does not seem to be the case. To incorporate these self-correlation functions, they must only be specified as a sort of initial condition to the simulation.
These two general considerations aside, we would like to speculate on how these selfcorrelations might be systematically incorporated into a continuum model. The most significant influence would involve a revision of the Peach-Koehler interactions of the dislocation densities.
Treating this interaction as the conjugate configurational force to the dislocation density allows us to express this force as follows: If we assume that ( , ) / is vanishingly small, we can neglect this second term. We assert that we can neglect this term, as part of the impetus of our formulation was to remove any relation between the local density field and the correlation. As a result we are left with a simple integration: where we have incorporated the correlation as simply an alteration to the spatial dependence of the interaction kernel: The effects due to the correlated regions ( − 1 ≠ 0) would introduce terms interpreted elsewhere as back and friction stresses. However, this energy kernel alteration circumvents the local density approximation (Zaiser, 2015) which underpins such back and friction stresses; such a local density assumption would be a poor approximation in our formulation given the significant variation of the correlation functions up to and past the convolution length.

Future Work
Some questions regarding the self-correlation function remain open, and we would like to discuss them here. The present work did not establish upper bounds to any of the relations demonstrated.
It was not shown whether the simple, linear scaling of the self-correlation with convolution length breaks down at large convolution lengths. It is also unknown whether the strain-independence of the self-correlation functions continues to hold at larger strains.
The dependences on convolution length and plastic strain may break down for the same reasons. The first possibility is that as the convolution length approaches the mean dislocation spacing 0 −1/2 (~1.5 μm in the simulations presented), the interactions being captured in the correlation functions are qualitatively different. Whereas at small convolution lengths the correlation functions capture line effects due to single dislocations, approaching the mean dislocation spacing will capture multi-dislocation effects such as dislocation patterning. We predict that upon convolution over lengths greater than this spacing, the correlations would become stable with respect to convolution length. As the strain increases, the length at which this transition might occur naturally decreases with the mean spacing. The second possible reason these relations could change at higher strains would be the introduction of lattice rotations. This would certainly affect the off-plane correlations as cross-slip becomes more common and slip planes are activated closer together. We can, however, assert that the relations demonstrated hold in the low-strain, low-convolution length regime which we have examined here. Discrete dislocation dynamics simulations might be run to slightly higher strain, but computation time needed to run a statistically significant number of such simulations would increase. Correlations at finite strains might necessitate other methods of investigation besides the discrete methods presented here.
A great deal is left to learn even in the present regimes of strain and convolution length by applying the formulation presented. For example, we have only considered the self-correlation of dislocation densities on the same slip system. The cross-correlations, on the other hand, contain information regarding the relative arrangement of different dislocation 'species.' This is especially important information which could inform corrections to dislocation reaction rates, which would in turn affect the strain hardening behavior of the crystal.
Additionally, the influence of the crystallography could introduce not only spatial symmetry breaking but also slip-system symmetry breaking. Efforts were made to suppress this dependence by averaging over the three strain directions as well as the twelve slip systems. However, a more nuanced investigation might reveal anisotropic effects due to the loading. On a similar note, it is unclear whether these correlation functions would change significantly for relaxed dislocation systems, as the configurations considered were all instantaneous snapshots of a dynamic simulation.

Conclusions
In this work, we have outlined some of the difficulties surrounding a discussion of a dislocation ensemble. However, we defined an ensemble average operation based on a spatial convolution of a singular dislocation density. By treating spatial observations of the proto-correlation density products as independent observations of the underlying random variable, we were able to compute the ensemble average which we associate with the correlation function.
This method was used to evaluate the three independent components of self-correlation function (correlation between density components on the same slip system). These three independent components of the self-correlation functions were found to be strongly planar functions, with most of the interesting behavior being found at separation vectors falling in the slip plane. Moreover, the most significant factor affecting the form of the correlation function was found to be the convolution length: for lengths between 65 and 162 nm, the self-correlations are similar up to a rescaling of space proportional to the convolution length. No change in the correlation function was observed upon increase in plastic strain.
The implications which these findings have on continuum dislocation dynamics were discussed. It is the belief of the present authors that these correlation functions will provide an important correction to continuum dislocation dynamics models, introducing an altered form of the stress field and dislocation reaction rates. There are many features of these correlation functions which were beyond the scope of this particular document, but the results shown serve to demonstrate the validity of this approach to the calculation of correlation functions. It is our hope that this formulation will enable future studies of dislocation interactions in continuum dislocation models.

Appendix 1: Physical Intuition of Set-theoretic Definition of Ensembles
To aid in understanding the process of defining an ensemble, we will treat the Gibbs' canonical ensemble in the set theoretic terms which have been presented. While this is an equilibrium system as opposed to our (highly) nonequilibrium system of interest, the intuition of spaces and level sets should be helpful nonetheless. Also, although these considerations are for systems of point particles, an understanding of the coarse-graining process will be helpful for understanding conceptually the dislocation ensemble.

Spaces
Consider a system of particles, each with positions ∈ ℝ 3 and momenta ∈ ℝ 3 . In this case, the microstate of the system is a 6 -tuple: ≔ ( 1 , … , , 1 , … , ) and Γ represents the space of all such microstates, commonly referred to as the phase space (Nolte, 2010): Γ ≔ {( 1 , … , , 1 , … , ): ∈ ℝ 3 and ∈ ℝ 3 for all 1 ≤ ≤ } = ℝ 6 . (A2) Now we consider the quintessential coarse graining operation which involves level sets of the energy function. In this case, we consider as our coarse-graining function Ψ = ( 1 , … ). Since this returns a scalar, our coarse-grained space Τ is simply ℝ. For clarity, let us examine the form of this map: The level sets of this map are referred to as macrostates Γ : Notice that each of these macrostates represents a (hyper)sphere in the momentum portion of the space while extending as a cylinder in the position portion. These are all the equivalent configurations (microstates) which the particle system can occupy while retaining the same kinetic energy. There is a considerable amount of confusion about what such a macrostate map would look like in the case of dislocations, as the space is significantly more complex than the extremely tractable ℝ 6 .

Probabilities and Measures
Now we wish to assign probabilities to subsets of Γ (and Γ ). This requires two definitions: σalgebras and measures. Broadly speaking, the former represents a "sufficiently large" collection of subsets of a given space, while the latter represents a map from these subsets to the real numbers.
A σ-algebra on a set is a collection of subsets of which: 1) contains , 2) is closed under complement, and 3) is closed under countably infinite unions (Durrett, 2019). For an example of a σ-algebra, consider the Borel sets ℬ on ℝ: the smallest σ-algebra containing the open sets. We may note that these are all sets which are easily imagined: they can be formed by unions of small intervals or of individual points. The definition of such a collection might seem obscure, but it is mostly useful to define measures.
A measure is a set function: it assigns non-negative real numbers to sets. The space of sets which is its domain is a σ-algebra, as it allows the statement of a measure's defining (and most useful) property; for non-overlapping sets ( ∩ = ∅ for all ≠ ), the following decomposition must hold: Intuitively, this implies that a set will retain its measure regardless of how many pieces we cut it into. This definition also carries two useful consequences: ⊆ implies that ( ) ≤ ( ); (∅) = 0.
Two examples will be useful here. Firstly, consider the standard volume measure on the real numbers, referred to as the Lebesgue measure . This begins by simply assigning a map from all intervals to non-negative numbers: Defining this probability allows us to take averages as integration against this measure in the following way. The ensemble average of any quantity is now a function of the map : Now, all of the necessary machinery for the set-theoretic definition of ensembles and coarsegraining has been shown for the canonical ensemble. For the present consideration of dislocations, we assume the existence of this machinery so as to define ensemble averages similar to that of equation (A12). Intuition from the form of this machinery is then used to identify a certain quantity (the proto-correlation density ̃( )) which can represent an independent measurement of the quantity being averaged.

Availability of Data and Materials
The collection of discrete dislocation dynamics simulations as well as the code used to generate the self-correlation functions can be accessed by contacting the corresponding author.

Competing Interests
The authors declare no competing interests.

Funding
This work is supported by the US Department of Energy, Office of Science, Division of Materials Sciences and Engineering, through award number DE-SC0017718 at Purdue University.