On the equivalence of the two foundational formulations for atomistic flux in inhomogeneous transport processes

Abstract

Although there are numerous formulae for atomic-level fluxes, they are expressed either in terms of a singlet density, resulting from Irving and Kirkwood's statistical mechanics formulation of hydrodynamical equations, or a pair density, proposed in kinetic theories of transport processes. Flux formulae using singlet density have been further developed and widely implemented in molecular dynamics (MD) simulations by either replacing the Dirac delta with a volumetric averaging function or performing a surface average of the flux operators. Pair density-based flux formulae have also been further developed by using spatial-averaging kernels; these formulae, however, have rarely been implemented or used in modern MD. In this work, distributional calculus is used to reformulate the fluxes in momentum and energy transport processes. The formulation results demonstrate that these two types of existing flux formulae are mathematically equivalent when expressed with the Dirac delta. The lasting confusion regarding these two different types of flux formulae from two different formalisms is thus resolved.

1. Introduction

Transport phenomena are ubiquitous in physical and chemical processes. The non-equilibrium behaviour of transport processes is generally described by conservation equations, with flux being the measure of transport. Predicting transport flux based on the motion and interaction of molecules is becoming increasingly important as it can provide insights into the underlying physics of the processes and phenomena. This is especially true when the proliferation of molecular dynamics (MD) simulation has made it increasingly possible to simulate and quantify complex transport processes and properties of materials.

Historically, statistical mechanics has provided the major theoretical tool to formulate the transport equations based on the microscopic dynamics of particles. There is extensive literature on statistical mechanics formulations of stress and heat flux [1–25] and their applications to molecular dynamics simulation [21,26–29]; which are used commonplace in the very popular LAMMPs codebase [30,31]. One fundamental equation for such a formulation is the definition of local density. Such a definition defines the link between a continuum field variable, i.e. a local density, in the three-dimensional physical space and the dynamical variable in the 6N phase space, where N is the number of particles. Although there are numerous flux formulae resulting from such formulations, there are only two types of definitions in terms of the mathematical structures or molecular distribution functions: the singlet density and the pair or doublet density, as summarized by Kirkwood & Buff [7]. Singlet density formulations express local densities and fluxes in terms of one Dirac delta while pair density formulations describe the flux with a density involving pairs of molecules using two Dirac deltas.

The earliest description of fluxes in terms of molecular variables is the pair density definition of fluxes. This type of formulation can be traced back to the work by R. H. Fowler in 1936 [4], and is formally defined in Born and Green's volumes on the formulation of the kinetic theory of transport processes in 1946 [1,2] and 1947 [2], with the applications to liquids and their dynamics properties. The formulation of an explicit theory of interfacial tension in fluids using the pair density formalism is developed by Buff and Kirkwood in 1949. This is then followed by the work of MacLellan [12], Harasima [18], and Noll [32], the last of which is widely adopted by researchers in solid mechanics. In particular, Noll's pair density formulation inspired many theoretical works on fluxes using both ensemble averaging [33,34] and general space-averaging functions [5,35,36]. In addition, pair density fluxes are employed in the formulation of non-local continuum simulation methods such as peridynamics [37–40], available in the LAMMPs codebase [41], and others [42].

The statistical mechanics formulation of microscopic fluxes (stress and heat flux) using the singlet density in terms of Dirac delta was pioneered by Irving and Kirkwood (IK) in 1950 in their classical paper ‘Statistical Mechanical Theory of Transport Processes IV. The Equations of Hydrodynamics’ [6]. IK defined local densities of mass, linear momentum, and energy as point functions ‘since mass or momentum of any molecule may be considered as localized at that molecule’ [6]. The stress tensor and heat flux vector are then obtained through the molecular description of the transport equations as point functions. Over the past 68 years, more than two thousand papers whose works were inspired by IK's formalism of fluxes have been published; many of these replacing the Dirac delta with a spatial-averaging functions such as Hardy [13,14] and several others [43–46]. Additionally, a form of the stress formula in terms of pair density can be noted in the Appendix [6], in which IK ultimately expressed the pair density stress formula using Taylor series expansion with the intent to connect the pair density expression for stress to that in terms of singlet density. A rigorous derivation of the relationship that connects the two types of flux formulae has been, however, lacking.

From the viewpoint of MD simulation, formulations of fluxes that employ pair density are considerably more impractical to compute, even after assumptions such as the ergodic hypothesis to avoid ensemble averaging and replacing the Dirac delta with a spatial-averaging kernel. This is because this class of flux formulae involves multiple integrals even when foregoing the ensemble average and the Dirac delta through spatial averaging. Computation of these integrals requires numerical approximations that can be expensive and challenging, as the functions to be computed are highly nonlinear and non-local at the atomic scale. This makes existing numerical integration methods such as Gaussian quadrature expensive or inappropriate for many such computations that would need to be carried out in a MD simulation. Consequently, up to date, such formulae have found little practical applicability and their value or accuracy remains rather enigmatic in the context of MD simulation, or continuum mechanics-based simulations, results.

The objective of this work is to reformulate fluxes in momentum and energy transport processes in terms of pair density, by using distributional calculus [47], to connect them to the flux formalisms represented as singlet densities. Following the introduction, In §2, we introduce the Irving and Kirkwood (IK) formalism for deriving ensemble-averaged microscopic fluxes, in terms of singlet densities, as well as the later developments that express surface averages of the IK fluxes without the ensemble average; in §3, we review the development of mathematical representations of flux formulae in terms of pair density; in §4, we reformulate pair density fluxes by using the properties of the Dirac delta distribution to demonstrate that the resulting fluxes are equivalent to existing flux formulae derived from the IK formalism's singlet density, with the mathematic proofs being detailed in the appendix; the paper ends with a summary in §5.

2. The Irving and Kirkwood formalism and later developments employing singlet density

Irving and Kirkwood derived the hydrodynamical equations based on molecular variables using the method of non-equilibrium statistical mechanics [6]. The Irving and Kirkwood (IK) formalism involves the use of the Dirac δ [48] to define the local densities of mass, momentum, and total energy as ensemble-averaged ‘point’ functions. Considering a system of N particles with the positions of the particles being denoted by ${\mathit{r}}_1,{\mathit{r}}_2,\cdots ,{\mathit{r}}_N$ and their momentum by ${\mathit{p}}_1,{\mathit{p}}_2,\cdot \cdot \cdot ,{\mathit{p}}_N$, the probability per unit volume that the kth particle is located at x, at time t, is expressed with the Dirac delta as

$$⟨\delta ({\mathit{r}}_k-\mathit{x});f⟩={\int }_{\text{6N}}\dots {\int }_{\text{fold}}\delta ({\mathit{r}}_k-\mathit{x})f({\mathit{r}}_1,\cdots ,{\mathit{p}}_1,\cdots ;t)\text{d}{\mathit{r}}_1\cdots \text{d}{\mathit{r}}_N\text{d}{\mathit{p}}_1\cdots d{\mathit{p}}_N,$$ 2.1

where f denotes the probability density function (PDF). The local density of a dynamic quantity, $a({\mathit{r}}_1,\cdots ,{\mathit{p}}_1,\cdots )$, at point x is thus given by

$$\begin{array}{l}⟨a\delta ({\mathit{r}}_k-\mathit{x});f⟩={\int }_{\text{6N}}\dots {\int }_{\text{fold}}a({\mathit{r}}_1,\cdots ,{\mathit{p}}_1,\cdots )\delta ({\mathit{r}}_k-\mathit{x})f({\mathit{r}}_1,\cdots ,{\mathit{p}}_1,\cdots ;t)\text{d}{\mathit{r}}_1\cdots \text{d}{\mathit{r}}_N\text{d}{\mathit{p}}_1\cdots \text{d}{\mathit{p}}_N.\end{array}$$ 2.2

In later literature, the notation of an ensemble-averaged local density as expressed in equation (2.2) is often simplified as $⟨a\delta ({\mathit{r}}_k-\mathit{x});f⟩\equiv ⟨a\delta ({\mathit{r}}_k-\mathit{x})⟩$.

It follows from equation (2.2) that the total mass density, ρ, the linear momentum density, ρv, and energy density, e, at x, at time t, due to all particles are defined as [6]:

$$\begin{array}{rl}\rho (\mathit{x},t)& =⟨\sum _{k}m{}_k\delta ({\mathit{r}}_k-\mathit{x})⟩,\\ \rho (\mathit{x},t)\mathit{v}(\mathit{x},t)& =⟨\sum _{k}m{}_k\mathit{v}{}_k\delta ({\mathit{r}}_k-\mathit{x})⟩\\ \text{and}e(\mathit{x},t)& =⟨\sum _{k}E{}_k\delta ({\mathit{r}}_k-\mathit{x})⟩,\end{array}\}$$ 2.3

where m_k, ${\mathit{r}}_k$, ${\mathit{v}}_k$, E_k are respectively the mass, position, velocity, total energy of kth particle. Equation (2.3) links the microscopic dynamic function of conserved quantities in phase space to their local densities in physical space–time. Fluxes are then obtained by equating the time rate of change of the conserved quantities to the differential form of the conservation laws of hydrodynamics. The IK formulation is mathematically rigorous; its use of the infinitely-peaked Dirac delta is fully consistent with the mathematical theory of distributions. However, Irving and Kirkwood did not derive a closed-form expression for the fluxes that are directly computable; rather they expressed them as a power series. Closed-form expressions appeared in later developments, for example, in the paper by Miller [49], and in the statistical mechanics book by Kreuzer [50], with the difference between two δ functions being expressed as a line integral of the distribution's action, as:

$$\delta ({\mathit{r}}_k-\mathit{x})-\delta ({\mathit{r}}_l-\mathit{x})=-{\mathrm{\nabla }}_{\mathit{x}}\cdot {\int }_0^1{\mathit{r}}_{kl}\delta (\alpha {\mathit{r}}_{lk}+{\mathit{r}}_k-\mathit{x})\text{d}\mathrm{\lambda },$$ 2.4

where ${\mathit{r}}_{kl}={\mathit{r}}_k-{\mathit{r}}_{\mathit{l}}$. Recently, using the scaling property of Dirac delta listed by Paul Dirac in his Principles of Quantum Mechanics [48], or simply using the fundamental theorem of line integrals, equation (2.4) has been rewritten as a path integral with the path L_kl being a line segment from r_k to r_l [15,16], as

$$\delta ({\mathit{r}}_k-\mathit{x})-\delta ({\mathit{r}}_l-\mathit{x})={\mathrm{\nabla }}_{\mathit{x}}\cdot {\int }_{{\mathit{L}}_{kl}}\delta (\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }.$$ 2.5

The potential part of stress ${\mathit{\sigma }}_{\text{pot}}$ is then obtained in terms of singlet density by equating its divergence to the internal force density ${\mathit{f}}_{\mathrm{int}}$ as

$$\begin{array}{rl}{\mathrm{\nabla }}_{\mathit{x}}\cdot {\mathit{\sigma }}_{\text{pot}}(\mathit{x})& ={\mathit{f}}_{\mathrm{int}}(\mathit{x})=-⟨\sum _{k,l}\frac{\mathrm{\partial }{\mathit{\Phi }}_{kl}}{\mathrm{\partial }{\mathit{r}}_{kl}}\delta ({\mathit{r}}_k-\mathit{x})⟩=-⟨\frac{1}{2}\sum _{k,l}\frac{\mathrm{\partial }{\mathit{\Phi }}_{kl}}{\mathrm{\partial }{\mathit{r}}_{kl}}(\delta ({\mathit{r}}_k-\mathit{x})-\delta ({\mathit{r}}_l-\mathit{x}))⟩\\ & =-{\mathrm{\nabla }}_{\mathit{x}}\cdot ⟨\frac{1}{2}\sum _{k,l}{\int }_{L_{kl}}\frac{\mathrm{\partial }{\mathit{\Phi }}_{kl}}{\mathrm{\partial }{\mathit{r}}_{kl}}\delta (\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }⟩,\end{array}$$ 2.6

where Φ_kl is the two-body potential energy between particles k and l. The stress field can thus be derived from equation (2.4) and equation (2.5), respectively, and be expressed as a sum of kinetic and potential parts in the following two equivalent forms using singlet density:

$$\begin{array}{rl}\mathit{\sigma }(\mathit{x},t)& =-⟨\sum _k{m}_k({\mathit{v}}_{\mathit{k}}-\mathit{v})\otimes ({\mathit{v}}_{\mathit{k}}-\mathit{v})\delta ({\mathit{r}}_k-\mathit{x})+\sum _{k<l}\frac{\mathrm{\partial }{\mathit{\Phi }}_{kl}}{\mathrm{\partial }{\mathit{r}}_{kl}}\otimes {\mathit{r}}_{kl}{\int }_0^1\delta (\mathrm{\lambda }{\mathit{r}}_{kl}+{\mathit{r}}_k-\mathit{x})\text{d}\mathrm{\lambda }⟩\\ & =-⟨\sum _k{m}_k({\mathit{v}}_{\mathit{k}}-\mathit{v})\otimes ({\mathit{v}}_{\mathit{k}}-\mathit{v})\delta ({\mathit{r}}_k-\mathit{x})+\sum _{k<l}\frac{\mathrm{\partial }{\mathit{\Phi }}_{kl}}{\mathrm{\partial }{\mathit{r}}_{kl}}\otimes {\int }_{L_{kl}}\delta (\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }⟩.\end{array}$$ 2.7

In a similar manner, the atomistic formula for heat flux also has two equivalent forms in terms of singlet density as

$$\begin{array}{rl}\mathit{q}(\mathit{x},t)& =-⟨\sum _k({\mathit{v}}_{\mathit{k}}-\mathit{v})[\frac{1}{2}{m}^k{({\mathit{v}}_{\mathit{k}}-\mathit{v})}^2+{\mathit{\Phi }}_k]\delta ({\mathit{r}}_k-\mathit{x})+\sum _{k<l}\frac{\mathrm{\partial }{\mathit{\Phi }}_{kl}}{\mathrm{\partial }{\mathit{r}}_{kl}}\cdot ({\mathit{v}}_{\mathit{k}}-\mathit{v}){\mathit{r}}_{kl}{\int }_0^1\delta (\mathrm{\lambda }{\mathit{r}}_{lk}+{\mathit{r}}_k-\mathit{x})\text{d}\mathrm{\lambda }⟩\\ & =-⟨\sum _k({\mathit{v}}_{\mathit{k}}-\mathit{v})[\frac{1}{2}{m}^k{({\mathit{v}}_{\mathit{k}}-\mathit{v})}^2+{\mathit{\Phi }}_k]\delta ({\mathit{r}}_k-\mathit{x})-\sum _{k<l}\frac{\mathrm{\partial }{\mathit{\Phi }}_{kl}}{\mathrm{\partial }{\mathit{r}}_{kl}}\cdot ({\mathit{v}}_{\mathit{k}}-\mathit{v}){\int }_{{\mathit{L}}_{kl}}\delta (\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }⟩.\end{array}$$ 2.8

It is noted that equations (2.4) and (2.5) hold in the distributional sense. The flux operators, in terms of singlet density, defined inside the ensemble averages of equations (2.7) and (2.8) have been shown to be formally equal to the stress and heat flux operators that satisfy the conservation laws in the instantaneous sense [23,51]. These operators can be integral averaged over a surface element and a time interval to calculate fluxes per unit area and time in MD simulations without ensemble average [15,16]. The resulting space and time averages of equations (2.6)–(2.8) provide regular functions for the calculation of instantaneous stress and heat flux in MD that are free of any integral operator. For example, for a surface element A³ with area A, centred at point x, where the local coordinate system is always chosen such that the normal is the coordinate axis ${\mathit{e}}^3$, the αth component of the potential part of the stress vector, $t_{\text{pot}}^{\alpha }(\mathit{x},t,{\mathit{e}}^3)$, can be written as

$$t_{\text{pot}}^{\alpha }(\mathit{x},t,{\mathit{e}}^3)=\frac{1}{A}{\int }_{A^3(\mathit{x})}{\int }_{L_{kl}}\sum _{k<l}{F}_{kl}^{\alpha }\delta (\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }=\frac{1}{A}{\int }_{L_{kl}}\sum _{k<l}{F}_{kl}^{\alpha }{\overline{\delta }}_A^3(\mathit{\phi }-\mathit{x}){\mathit{e}}^3\cdot \text{d}\mathit{\phi }{\text{d}}^2{x}^{\mathrm{\prime }},$$ 2.9

where L_kl represents a line segment from r_k to r_l, and $F_{kl}^{\alpha }=-\mathrm{\partial }{\mathit{\Phi }}_{kl}/\mathrm{\partial }{r}_{kl}^{\alpha }$ for a two body potential. Denote ${\mathit{n}}_{kl}=(n_{kl}^1,n_{kl}^2,n_{kl}^3)=n_{kl}^{\alpha }{\mathit{e}}^{\alpha }$ as the unit direction vector of L_kl and introduce a scalar ϕ such that $\mathit{\phi }-{\mathit{r}}_k=\mathrm{\lambda }({\mathit{r}}_l-{\mathit{r}}_k)=\varphi {\mathit{n}}_{kl}$; the path integral in equation (2.9), as well as that in equations (2.7) and (2.8), can then be parametrized by

$${\int }_{{\mathit{L}}_{kl}}\delta (\mathit{\phi }-{\mathit{x}}^{\mathbf{\prime }})\text{d}\mathit{\phi }={\mathit{n}}_{kl}{\int }_0^{|{\mathit{r}}_{kl}|}\delta (\varphi {\mathit{n}}_{kl}+{\mathit{r}}_k-{\mathit{x}}^{\mathbf{\prime }})\text{d}\varphi .$$ 2.10

The surface-averaged path integral of the Dirac delta over surface element A³, centred at x, can then be expressed as

$$\begin{array}{rl}& {\int }_{{\mathit{L}}_{kl}}{\overline{\delta }}_A^3(\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }\\ & \equiv \frac{1}{A^3}\int {\int }_{{\mathit{A}}^3}{\int }_{{\mathit{L}}_{kl}}\delta (\mathit{\phi }-{\mathit{x}}^{\mathbf{\prime }}){\mathit{e}}^3\cdot \text{d}\mathit{\phi }\text{d}{x}^{\mathrm{\prime }}\text{d}{y}^{\mathrm{\prime }}=\frac{1}{A^3}\int {\int }_{{\mathit{A}}^3}{\int }_0^{|{\mathit{r}}_{kl}|}{\mathit{e}}^3\cdot {\mathit{n}}_{kl}\delta (\varphi {\mathit{n}}_{kl}+{\mathit{r}}_k-{\mathit{x}}^{\mathbf{\prime }})\text{d}\varphi \text{d}{x^{\mathrm{\prime }}}^1\text{d}{x^{\mathrm{\prime }}}^2\\ & =\frac{{\mathit{e}}^3\cdot {\mathit{n}}_{kl}}{A^3}\int {\int }_{{\mathit{A}}^3}{\int }_0^{|{\mathit{r}}_{kl}|}\delta (\varphi n_{kl}^3+r_k^3-x^{\mathrm{\prime }})\delta (\varphi n_{kl}^1+r_k^1-{x^{\mathrm{\prime }}}^1)\delta (\varphi n_{kl}^2+r_k^2-{x^{\mathrm{\prime }}}^2)\text{d}\varphi \text{d}{x^{\mathrm{\prime }}}^1\text{d}{x^{\mathrm{\prime }}}^2\\ & =\frac{n_{kl}^3}{A^3||n_{kl}^3||}{\int }_0^{|{\mathit{r}}_{kl}|}\delta \left(\varphi +\frac{r_k^3}{n_{kl}^3}-\frac{{x^{\mathrm{\prime }}}^3}{n_{kl}^3}\right)\left[\int {\int }_{{\mathit{A}}^z}\delta (\varphi n_{kl}^1+r_k^1-{x^{\mathrm{\prime }}}^1)\delta (\varphi n_{kl}^2+r_k^2-{x^{\mathrm{\prime }}}^2)\text{d}{x^{\mathrm{\prime }}}^1\text{d}{x^{\mathrm{\prime }}}^2\right]\text{d}\varphi \\ & =\frac{1}{A^3}\{\begin{array}{ll}1& \text{if}{r}_{kl}\text{intersects }{\mathit{A}}^3\text{and }{\mathit{e}}^3\cdot {\mathit{n}}_{kl}\ge 0\\ -1& \text{if}{r}_{kl}\text{intersects }{\mathit{A}}^3\text{and }{\mathit{e}}^3\cdot {\mathit{n}}_{kl}<0\\ 0& \text{otherwise}\end{array}\end{array}$$ 2.11

This then leads to the formula for the potential part of the stress vector defined as a surface density, as shown previously [15], in terms of a line-plane intersection problem, consistent with the averaged Dirac delta defined in equation (2.11), as follows:

$$F_{kl}^{\alpha }{\int }_{{\mathit{L}}_{kl}}{\overline{\delta }}_A^3(\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }=\frac{F_{kl}^{\alpha }}{A^3}\{\begin{array}{ll}1& \text{if}{r}_{kl}\text{intersects }\hspace{0.17em}{\mathit{A}}^3(\mathit{x})\text{ and }{\mathit{e}}^3\cdot {\mathit{n}}_{kl}>0\\ -1& \text{if}{r}_{kl}\text{intersects }\hspace{0.17em}{\mathit{A}}^3(\mathit{x})\text{ and }{\mathit{e}}^3\cdot {\mathit{n}}_{kl}<0\\ 0& \text{otherwise}\end{array}$$ 2.12

that is also consistent with the intuition that the forces of interaction through the surface of a body are responsible for changing the body's linear momentum as portrayed in other works [16,52]. It is also worth noting that the result of equation (2.9) is consistent with the method of planes (MOP) when the surface in question is an infinite plane, when the surface is finite the local aspect of the relationship between flux and the line-plane intersection problem, that is distinct from MOP, becomes evident as seen in figure 1.

Figure 1.

Example system with a plane bisection two groups of atoms. Lines of interaction, for interatomic forces, between atoms are shown solid if they intersect the plane and contribute; dashed lines denote interactions that do not contribute to the flux through the plane. Points of intersection between a line of interaction and the plane are denoted by an x. (Online version in colour.)

Similarly, the potential part of heat flux for a two-body interaction force can also be obtained as a surface density, and line-plane intersection problem, as

$${\mathit{q}}_{\text{pot}}=\sum _{k<\hspace{0.17em}l}{\mathit{F}}_{kl}\cdot ({\mathit{v}}_k-\mathit{v})\hspace{0.17em}{\int }_{{\mathit{L}}_{kl}}{\overline{\delta }}_A^3(\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }=\sum _{k<l}{\mathit{F}}_{kl}\cdot \left(\frac{{\mathit{v}}_k+{\mathit{v}}_l}{2}-\mathit{v}\right){\int }_{{\mathit{L}}_{kl}}{\overline{\delta }}_A^3(\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }.$$ 2.13

Equation (2.12) demonstrates that the potential part of the stress vector defined in equation (2.9) is identical to the physical concept of mechanical stress as the force per unit area acting on a surface element, with each of the α components of force per unit area acting on the corresponding coordinate plane. Using equation (2.9), the αβ component of the potential part of the stress tensor, ${\sigma }_{\text{pot}}^{\alpha \beta }(\mathit{x},t)$, can also be defined; this is consistent with the definition of constructing the stress tensor as the stress vector defined on the three orthogonal planes [53,54]. On the other hand, equation (2.13) shows that the potential part of heat flux on a surface element expressed in (2.13) is the rate of work done per unit area by the interaction forces between particles on opposing ends of the surface through the thermal motion of the particles. Both are intuitively consistent with the physical definitions of stress and heat flux. It is also worth asserting that these instantaneous definitions of flux require no numerical integration to be carried out since the result is a simple selection process.

3. Pair density flux formulae

The pair density formulae for fluxes emerged when it was realized that momentum and energy in liquids should be considered as transmitted by the action of intermolecular forces rather than merely the movement of molecules [55]. It was argued that if momentum and energy are transmitted by the intermolecular forces, then it is no longer possible to speak of the contribution of single molecules but rather only pairs of molecules [55]. A general kinetic theory of transport processes that admits the pair density definition of fluxes was developed by Born and Green [1,2]; prior to the landmark papers by Born and Green, a kinetic theory-based derivation of the pressure tensor due to interparticle forces in two phase fluids can be found in the work by Fowler [4]. This was followed by a more rigorous statistical mechanical theory of surface tensions formulated by Kirkwood and Buff in 1949 [7]; a different procedure with such a density was used to derive the surface tension in a single phase and two phase fluid description as well, in agreement with the work by Kirkwood and Buff, by MacLellan [12].

It is worth mentioning that the very same paper that introduced the well-known statistical mechanical theory of transport processes and derived the hydrodynamical equations, i.e. the Irving–Kirkwood formalism, also introduced, in its Appendix, a general pair-density operator for the stress as:

$${\mathit{\sigma }}_{\text{pot}}^{\text{IK-pair}}(\mathit{x},t)=\frac{1}{2}\int \int {\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\mathit{z}}{|\mathit{z}|}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}{\rho }^{(2)}(\mathit{x}-\mathrm{\lambda }\mathit{z},\mathit{x}-\mathrm{\lambda }\mathit{z}+\mathit{z},t)\text{d}\mathrm{\lambda }\text{d}\mathit{z},$$ 3.1

where the pair density

$${\rho }^{(2)}(\mathit{x},\mathit{y},t)=\sum _{k\ne l}⟨\delta (\mathit{x}-{\mathit{r}}_k)\delta (\mathit{y}-{\mathit{r}}_l)⟩$$ 3.2

is the probability (per unit volume) that one particle will be located at x and another will be at y .

This form of the stress operator arose yet again in the work of Walter Noll, in his paper published in 1955 [9,32]. Using the following lemma for integrable and differentiable $f(\mathit{x},\mathit{y})$ such that $f(\mathit{x},\mathit{y})=-f(\mathit{y},\mathit{x})$:

$${\int }_{\mathit{y}}f(\mathit{x},\mathit{y})\text{d}\mathit{y}=-\frac{1}{2}{\mathrm{\nabla }}_{\mathit{x}}\cdot {\int }_{\mathit{z}}\left[\mathit{z}{\int }_0^{1}f(\mathit{x}+\mathrm{\lambda }\mathit{z},\mathit{x}-[1-\mathrm{\lambda }]\mathit{z})\text{d}\mathrm{\lambda }\right]\text{d}z.$$ 3.3

Noll obtained the same expression for the potential part of stress using the relationship between stress and internal force density. The procedure was similar to the development of the IK formalism, but the internal force density was instead expressed in terms of pair density as opposed to singlet density as we've described in §2. The resulting stress operator is described in terms of $⟨W|{\mathit{r}}_k=\mathit{x},{\mathit{r}}_l=\mathit{y}⟩$, the probability that particles k and l be at x and y respectively, as:

$${\mathit{\sigma }}_{\text{pot}}^{\text{Noll}}(\mathit{x},t)=\frac{1}{2}\sum _{k\ne l}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\mathit{z}}{|\mathit{z}|}{\int }_0^1⟨W|{\mathit{r}}_k=\mathit{x}+\mathrm{\lambda }\mathit{z},\hspace{0.17em}{\mathit{r}}_l=\mathit{x}-[1-\mathrm{\lambda }]\mathit{z}⟩\text{d}\mathrm{\lambda }\text{d}\mathit{z}.$$ 3.4

Similarly, the potential part of heat flux is

$${\mathit{q}}_{\text{pot}}^{\text{Noll}}(\mathit{x},t)=-\frac{1}{2}\sum _{k\ne l}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}}{|\mathit{z}|}{\int }_0^1⟨\mathit{z}\cdot \left(\frac{{\mathit{v}}_{\mathit{k}}+{\mathit{v}}_l}{2}-\mathit{v}\right)W|{\mathit{r}}_k=\mathit{x}+\mathrm{\lambda }\mathit{z},\hspace{0.17em}{\mathit{r}}_l=\mathit{x}-[1-\mathrm{\lambda }]\mathit{z}⟩\text{d}\mathrm{\lambda }\text{d}\mathit{z}.$$ 3.5

It should be noted that Noll's notation of $⟨W|{\mathit{r}}_k=\mathit{x},{\mathit{r}}_l=\mathit{y}⟩,$ the probability of particle k and l being located at x and y respectively, is identical to the action of the Dirac delta since $⟨\delta ({\mathit{r}}_k-\mathit{x})\delta ({\mathit{r}}_l-\mathit{y}),W⟩$ computes that very same probability. This means that the following are equivalent:

$$\begin{array}{rl}{\rho }^{(2)}(\mathit{x}-\mathrm{\lambda }\mathit{z},\mathit{x}-\mathrm{\lambda }\mathit{z}+\mathit{z},t)& =\sum _{k\ne l}⟨W|{\mathit{r}}_k=\mathit{x}+\mathrm{\lambda }\mathit{z},\hspace{0.17em}{\mathit{r}}_l=\mathit{x}-[1-\mathrm{\lambda }]\mathit{z}⟩\\ & =\sum _{k\ne l}⟨\delta (\mathit{x}-\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}-[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)⟩.\end{array}$$ 3.6

The advantage the r.h.s. of equation (3.6) possesses is that the expression for pair density required by the flux formulae is directly expressed as the product of two singlet densities; A similar expression of the pair density into singlets can also be found in IK's work where they argue that the stress function expanded as a Taylor series can be equivalently expressed in terms of pair or singlet density above their Eq. 5.15 [6].

Expressing the pair density in terms of two Dirac deltas enables its redefinition in terms of a spatial-averaging function, $w(\mathit{x}-{\mathit{r}}_k)$, to define local density. This is done by simply replacing each of the two Dirac deltas with said functions and absconding with the ensemble average. As a result fluxes are represented with a pair density that is the product of two spatial-averaging functions [5,35]. Since pair densities expressed using a spatial-averaging function for each respective particle are admissible to Noll's lemma when multiplied by the pair interaction force, these works may then derive the following form of instantaneous spatially averaged fluxes in terms of pair density:

$${\mathit{\sigma }}_{\text{pot}}^w(\mathit{x},t)=\frac{1}{2}\sum _{k\ne l}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\mathit{z}}{|\mathit{z}|}{\int }_0^{1}w(\mathit{x}-\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)w(\mathit{x}-[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)\text{d}\mathrm{\lambda }\text{d}\mathit{z}$$ 3.7

and

$${\mathit{q}}_{\text{pot}}^w(\mathit{x},t)=-\frac{1}{2}\sum _{k\ne l}{\int }_z{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}}{|\mathit{z}|}{\int }_0^1\mathit{z}\cdot \left(\frac{{\mathit{v}}_{\mathit{k}}+{\mathit{v}}_l}{2}-\mathit{v}\right)w(\mathit{x}-\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)w(\mathit{x}-[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)\text{d}\mathrm{\lambda }\text{d}\mathit{z}.$$ 3.8

This kind of formula is analogous to the volume-averaged flux formulae developed by Hardy using a line integral, called the bond function, of the singlet density [13]. However, unlike Hardy's work, these forms of fluxes now involve an additional integration over the space of z to carry out. The requisite quadrature rules to approximate these with a spatial-averaging function has made the computation of such fluxes dramatically less popular over its lifespan than the analogous method by Hardy; which even has an implementation in the Atomistic to Continuum (AtC) package [56] of the popular molecular dynamics simulator, LAMMPS. One is then left to wonder, with the lack of popularity and usage, if the formalisms for flux in terms of pair density retain additional merit or application over their singlet density counterparts. Can a practical selection process free of integration, such as the surface-averaged instantaneous flux shown previously, be derived for these formulae just as in the case of singlet density? Will the answer always be fundamentally different, or will the two flux formulations grant the same answer?

4. A distributional reformulation of pair density fluxes

Although closed-form expressions for the fluxes in terms of pair density now exist, none are yet practical to compute due to the higher order of integration with respect to singlet formulations; this is so even after replacing the Dirac delta with a spatial-averaging function and absconding with the ensemble average. Conversely, equations (2.12) and (2.13), the surface-averaged flux formulae, enable instantaneous definitions to be implemented in MD simulations without any integral calculations due to the simplicity of the analytical result.

It is then in our interest to mathematically analyse the forms of fluxes that use pair density for any analytical solution that may simplify their computation. Recall that in the Appendix of their paper, Irving and Kirkwood proposed the pair density formula, equation (A 6) in their paper [6], for the potential part of stress in terms of the Dirac delta as

$$\begin{array}{rl}{\mathit{\sigma }}_{\text{pot}}^{\text{IK-pair}}(\mathit{x},t)& =\frac{1}{2}{\int }_{3\text{fold}}-{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\mathit{z}}{|\mathit{z}|}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}{\rho }^{(2)}(\mathit{x}-\mathrm{\lambda }\mathit{z},\mathit{x}-\mathrm{\lambda }\mathit{z}+\mathit{z},t)\text{d}\mathrm{\lambda }\text{d}\mathit{z}\\ & =\frac{1}{2}{\int }_{3\text{fold}}^{}-{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\mathit{z}}{|\mathit{z}|}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}⟨\delta (\mathit{x}-\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}-[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)⟩\text{d}\mathrm{\lambda }\text{d}\mathit{z},\end{array}$$ 4.1

where the equivalence in terms of two Dirac deltas is shown. We can then use the distributional properties of the Dirac delta to remove one delta function, together with the integral over the infinite space of z, in the formula for the potential part of stress; see the proof in appendix A for the necessary convolutional identity [47] to prove the form of the first step in the following equation. The potential part of the stress tensor in equation (4.1) can thus be further derived as:

$$\begin{array}{rl}{\mathit{\sigma }}_{\text{pot}}^{\text{pair}}(\mathit{x},t)& =-\sum _{k\ne l}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\mathit{z}}{|\mathit{z}|}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}⟨\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)⟩\text{d}\mathrm{\lambda }\text{d}\mathit{z}\\ & =-⟨{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}\sum _{k\ne l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(\left|\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\right|)\frac{{\mathit{r}}_k-\mathit{x}}{|{\mathit{r}}_k-{\mathit{x}}^{\mathbf{\prime }}|}\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\delta (\mathit{x}+[\mathrm{\lambda }-1]\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}-{\mathit{r}}_l)\frac{1}{{\mathrm{\lambda }}^3}\text{d}\mathrm{\lambda }⟩\\ & =-⟨{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}\sum _{k\ne l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(\left|\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\right|)\frac{{\mathit{r}}_k-\mathit{x}}{|{\mathit{r}}_k-\mathit{x}|}\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\delta ({\mathit{r}}_{kl}-\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }})\frac{1}{{\mathrm{\lambda }}^3}\text{d}\mathrm{\lambda }⟩\\ & =-⟨{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}\sum _{k\ne l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|{\mathit{r}}_{kl}|)\frac{{\mathit{r}}_{kl}}{|{\mathit{r}}_{kl}|}{\mathit{r}}_{kl}\delta ({\mathit{r}}_{kl}+\frac{{\mathit{r}}_k}{\mathrm{\lambda }}-\frac{\mathit{x}}{\mathrm{\lambda }})\frac{1}{{\mathrm{\lambda }}^3}\text{d}\mathrm{\lambda }⟩\\ & =-⟨{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}\sum _{k\ne l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|{\mathit{r}}_{kl}|)\frac{{\mathit{r}}_{kl}}{|{\mathit{r}}_{kl}|}{\mathit{r}}_{kl}\delta (\mathrm{\lambda }{\mathit{r}}_{kl}+{\mathit{r}}_k-\mathit{x})\text{d}\mathrm{\lambda }⟩.\end{array}$$ 4.2

In deriving the last two steps of equation (4.2), the identity f(x)δ(x − a) = f(a)δ(x − a) and the scaling property of the Dirac delta over ${\mathit{r}}_l$ in the phase integral is used.

It is seen that the ensemble-averaged potential stress in terms of pair density has now become identical to the potential stress derived by Miller [49] and Kreuzer [50], as shown in equation (2.7). A similar derivation would link the definitions of heat flux as well. It is worth noting that the equivalence between pair and singlet density expressions for the stress field was argued in the IK paper based on a dubious Taylor series expansion of the Dirac delta, that does not converge even in the sense of distributions as seen in appendix C, of both the singlet and pair density operators. Equation (4.2) is thus the rigorous proof of the equivalence of the two foundational formulations of atomistic fluxes in transport processes.

Since ensemble averages involving 6N fold integrals, where N is the number of particles in the computer model, have never been calculated in a MD simulation, we need flux formulae that can be defined instantaneously. Such a need is well argued in the non-equilibrium statistical mechanics book by Evans and Morriss: ‘Firstly, the fluxes are governed by conservation laws and these laws are valid instantaneously and hence they do not require ensemble averaging to be true; secondly, the ergodic hypothesis, that the result obtained by ensemble averaging is equal to that obtained by time averaging, implies that one should be able to develop expressions for the fluxes which do not require ensemble averaging’ [23,51].

It should be noted that the operators for local stress and heat flux expressed in terms of the Dirac delta, the so-called point functions for flux, only exist as integral operators and are not valid instantaneous solutions until they are appropriately averaged. It is well known that the physical concept of a flux is as the flow of a physical property across a surface element per unit area, and the area is of the surface element through which the property is flowing. This means that fluxes are surface densities. The appropriate space average for a flux operator is thus over the surface element for which the physical quantity is measured. Using this physical concept, the surface-averaged formula for instantaneous stress vector acting on the surface element at x with normal e³ was obtained by surface averaging the stress operator as shown in §2.

Next, it remains to demonstrate that an instantaneous form of the pair density stress formula can also be derived directly from the operator of equation (4.1), which is written in terms of the Dirac delta pair density$\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)$. By Surface averaging the operator form, we expect to obtain a regular computable function as the operator within the distributional/ensemble integral; this regular function can then be used directly as the instantaneous solution for stress and heat flux. Following this procedure, we first assume the existence of the following surface-averaged instantaneous formula for the stress vector in terms of pair density:

$${\mathit{t}}_{\text{pot}}^{\text{pair}}(\mathit{x},t,{\mathit{e}}^3)=-\frac{1}{A}\sum _{k\ne l}{\text{∯}}_{\mathit{x}}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}}{|\mathit{z}|}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)\mathit{z}\cdot \text{d}\mathit{S}\text{d}\mathrm{\lambda }\text{d}\mathit{z}.$$ 4.3

We are then motivated to obtain an expression of equation (4.3) that is readily computable such as the surface-averaged stress in §2 in terms of the line-plane intersection problem. We must first express the distribution $\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)$ as a convolution of two Dirac deltas through coordinate transformation. It can then be shown that the area average of equation (4.3) is equivalent to equation (2.9), as proven within §B of the appendix, to grant:

$$\begin{array}{rl}{\mathit{t}}_{\text{pot}}^{\text{pair}}(\mathit{x},t,{\mathit{e}}^3)& =-\frac{1}{A}\sum _{k\ne l}{\text{∯}}_x{\int }_{\mathit{z}}^{}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}}{|\mathit{z}|}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)\mathit{z}\cdot \text{d}\mathit{S}\text{d}\mathrm{\lambda }\text{d}\mathit{z}\\ & =\frac{1}{A}{\int }_{L_{kl}}\sum _{k<l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}\frac{{\mathit{r}}_{kl}}{|{\mathit{r}}_{kl}|}{\overline{\delta }}_A^3(\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }.\end{array}$$ 4.4

Likewise, for the heat flux:

$$\begin{array}{rl}{q}_{\text{pot}}^{\text{pair}}(\mathit{x},t,{\mathit{e}}^3)& =-\frac{1}{A}\sum _{k\ne l}{\text{∰}}_{\mathit{x}}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}}{|\mathit{z}|}\cdot \left(\frac{{\mathit{v}}_{\mathit{k}}+{\mathit{v}}_l}{2}-\mathit{v}\right)\\ & \times {\int }_{\alpha =0}^{\alpha =1}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)\mathit{z}\cdot \text{d}\mathit{S}\text{d}\mathrm{\lambda }\text{d}\mathit{z}\\ & =\frac{1}{A}{\int }_{L_{kl}}\sum _{k<l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}\frac{{\mathit{r}}_{kl}}{|{\mathit{r}}_{kl}|}\cdot \left(\frac{{\mathit{v}}_{\mathit{k}}+{\mathit{v}}_l}{2}-\mathit{v}\right){\overline{\delta }}_A^3(\mathit{\phi }-\mathit{x})\text{d}\mathit{\phi }\end{array}$$ 4.5

demonstrating that pair and singlet density fluxes are equivalent as both ensemble-averaged point fluxes and area-averaged fluxes.

Thus, although there are numerous formulae for fluxes in the literature in terms of pair and singlet density, fluxes that have been derived to satisfy the conservation law of linear momentum can be expressed either in terms of a line integral of one Dirac delta by, e.g., Miller [49] and Kreuzer [50], or in terms of two Dirac deltas by, e.g., Irving & Kirkwood [6]. The equations derived above show that these representations can lead to equivalent formulae for local stress and heat flux that are defined instantaneously and can be readily computed in MD simulations using the resulting line plane intersection formula, as depicted in equation (2.12).

It is worth noting that many analytical identities unique to the Dirac delta's sifting property were needed to arrive at the analytical equivalence of the two formalisms in both the ensemble and surface-averaged cases; the arbitrary nature of the spatial-averaging functions provides no such properties to analytically simplify one form to the other and the integrals of each formalism may potentially evaluate to different local flux results. The Dirac delta also serves the purpose of informing us that, when expressed with well-behaved averaging functions, the two formalisms converge to within some error tolerance once the support of the averaging functions becomes vanishingly small [47,57].

The advantage of having the Dirac delta in the formulation, as opposed to spatial-averaging functions, is also its ability to produce a much simpler calculation, free of numerical integration, that is consistent with both flux representations. In addition to its mathematically rigorous concept within the theory of distributions, the Dirac delta also facilitates physical interpretations of the flux formulae. The Dirac delta can represent the mapping between variables in two different spaces or descriptions, it also represents a selection process such as the line-plane intersection used to describe the flux across or through a surface element as demonstrated in the works by Chen and Diaz [15,16].

5. Summary and discussions

The definition of fluxes in terms of pair density emerged in the formulation of kinetic theories of transport in the 1940s. This is the consequence of a drive to formalize the momentum and energy transport due to the action of inter-molecular forces between pairs of particles. Such a definition aimed to statistically represent the fluxes in transport processes as the direct consequence of the interactions between pairs of particles. The IK formalism, on the other hand, statistically defines the densities of conserved quantities, i.e. mass, momentum, and energy, as a point function; the fluxes are then obtained mathematically, as point functions, by satisfying the differential form of the conservation laws of hydrodynamics. The resulting flux formulae are thus consistent with the local conservation laws. Although numerous formulae for local fluxes have been developed in the past half centuries, these formulae can be categorized into either one of the two types of formulae in terms of their mathematical structure or molecular distribution. However, a mathematical link between the two types of formulae was yet to be established.

One distinct feature of the IK formalism is its use of the Dirac delta to link the local densities in physical space to the dynamical functions in phase space. The Dirac delta is a generalized function or distribution, first introduced by Paul Dirac in his Principles of Quantum Mechanics [48], to model discrete systems with point mass or point charge. Contrary to the belief of those who advocate avoiding the use of the Dirac delta in flux formulations, the distributional solutions to the conservation equations formulated using the Dirac delta are mathematically rigorous. Using the Dirac delta distribution in this work, we have reformulated flux formulae to find the equivalent distributional representations of the stress in terms of the pair density operator expression proposed by IK and later Noll for both stress and heat flux. Specifically, this work has demonstrated that

• (1)The closed form formulae for ensemble-averaged singlet density fluxes are identical to the ensemble-averaged pair density fluxes when retaining the rigorous mathematical description of fluxes afforded by the Dirac delta.
• (2)The surface averages of pair density fluxes expressed with the Dirac delta are identical to the surface averages of fluxes expressed in terms of singlet density [15,16]; these surface-averaged fluxes are simple to compute in MD simulations and are applicable to transient fluxes in inhomogeneous materials or transient transport processes.
• (3)The sifting property of the Dirac delta enables the removal of the additional Dirac delta together with the integration over the infinite Euclidian space in the pair density formulae, thereby simplifying the formulae for the potential parts of fluxes to forms that are simple to compute. On the other hand, the use of a spatial-averaging kernel, in addition to complicating computation, did not possess the distributional properties to elucidate this relationship.

The equivalence of the two types of flux formulae is thus rigorously proved based on the theory of distributions. The long-standing confusion about the relationship between the two types of formalisms is therefore removed. This mathematical equivalence of the flux formulae promotes the consistency of the atomistic representation of fluxes derived based on the physical concept of fluxes or the conservation laws.

Appendix A: the convolution of the Dirac delta with itself

The following is a demonstration of the operator equivalence of:

$$\begin{array}{rl}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta ({\mathit{r}}_l-(\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z}))\text{d}\mathit{z}& =\frac{1}{{\mathrm{\lambda }}^3}\left[{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(\frac{|{\mathit{r}}_k-\mathit{x}|}{\mathrm{\lambda }})\frac{{\mathit{r}}_k-\mathit{x}\otimes {\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }|{\mathit{r}}_k-\mathit{x}|}\right]\\ & \times \delta \left({\mathit{r}}_l-\left[\mathit{x}+{\mathit{r}}_k-\mathit{x}-\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\right]\right).\end{array}$$ A 1

This equivalence is utilized to re-express the ensemble-averaged equations in §4. The proof is motivated by the same set of operations that define other convolutional identities in the distributional sense, such as:

$$\int \int \delta (x-a)\delta (x-b)\text{d}x\text{d}a=\int \delta (a-b)\text{d}a,$$ A 2

which is typically found as a lemma in discussions about convoluting two Dirac delta distributions. To construct an explicit demonstration, denote ${⟨f⟩}_{{\mathit{r}}_l}$ as the ensemble average of the pdf that omits the integral on the subscripted d.f.s. We then have:

$$\begin{array}{rl}{\mathit{\sigma }}_{\text{pot}}^{\text{Noll}}(\mathit{x},t)& =-\frac{1}{2}\sum _{k\ne l}{\int }_0^1{\int }_{\mathit{z}}{\int }_{{\mathit{r}}_k}{\int }_{{\mathit{r}}_l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z}-{\mathit{r}}_l){⟨f⟩}_{{\mathit{r}}_k,{\mathit{r}}_l}\\ & \times ({\mathit{r}}_l,{\mathit{r}}_k)\text{d}\mathit{z}\text{d}\mathrm{\lambda }\text{d}{\mathit{r}}_l\text{d}{\mathit{r}}_k.\end{array}$$ A 3

We may then express a distribution, $g({\mathit{r}}_l,{\mathit{r}}_k,\mathrm{\lambda })$, in terms of this direct product of Dirac deltas:

$$\begin{array}{rl}& {\int }_{\mathit{z}}{\int }_{{\mathit{r}}_l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z}-{\mathit{r}}_l){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l\text{d}\mathit{z}\\ & ={\int }_{\mathit{z}}{\int }_{{\mathit{r}}_l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta ({\mathit{r}}_l-(\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z})){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l\text{d}\mathit{z}\\ & ={\int }_{{\mathit{r}}_l}\left[\underset{\mathit{z}}{\int }{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta ({\mathit{r}}_l-(\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z}))\text{d}\mathit{z}\right]{⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l\\ & ={\int }_{{\mathit{r}}_l}g({\mathit{r}}_l,{\mathit{r}}_k,\mathrm{\lambda }){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l.\end{array}$$ A 4

For which we then try to express a distributional equivalent through the action of the two deltas:

$$\begin{array}{rl}& {\int }_{\mathit{z}}{\int }_{{\mathit{r}}_l}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z}-{\mathit{r}}_l){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l\text{d}\mathit{z}\\ & ={\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}(\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z},{\mathit{r}}_k)\text{d}\mathit{z}\\ & =\frac{1}{{\mathrm{\lambda }}^4}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(\frac{|\mathit{z}|}{\mathrm{\lambda }})\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathit{z}-{\mathit{r}}_k){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}(\mathit{x}+\mathit{z}-\frac{\mathit{z}}{\mathrm{\lambda }},{\mathit{r}}_k)\text{d}\mathit{z}\\ & =\frac{1}{{\mathrm{\lambda }}^4}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(\frac{|{\mathit{r}}_k-\mathit{x}|}{\mathrm{\lambda }})\frac{{\mathit{r}}_k-\mathit{x}\otimes {\mathit{r}}_k-\mathit{x}}{|{\mathit{r}}_k-\mathit{x}|}{⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}(\mathit{x}+{\mathit{r}}_k-\mathit{x}-\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }},{\mathit{r}}_k)\\ & ={\int }_{{\mathit{r}}_l}\frac{1}{{\mathrm{\lambda }}^2}\left[{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(\frac{|{\mathit{r}}_k-\mathit{x}|}{\mathrm{\lambda }})\frac{{\mathit{r}}_k-\mathit{x}\otimes {\mathit{r}}_k-\mathit{x}}{|{\mathit{r}}_k-\mathit{x}|}\right]\delta ({\mathit{r}}_l-\left[\mathit{x}+{\mathit{r}}_k-\mathit{x}-\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\right]){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l\\ & ={\int }_{{\mathit{r}}_l}h({\mathit{r}}_l,{\mathit{r}}_k,\mathrm{\lambda }){⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l.\end{array}$$ A 5

Comparing g and h leads to their equivalence in the distributional sense when integrated against any test function such as the ${⟨f⟩}_{{\hspace{0.17em}}_{{\mathit{r}}_k,{\mathit{r}}_l}}({\mathit{r}}_l,{\mathit{r}}_k)$:

$$\begin{array}{rl}h& =g\to {\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\otimes \mathit{z}}{|\mathit{z}|}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta ({\mathit{r}}_l-(\mathit{x}+\mathrm{\lambda }\mathit{z}-\mathit{z}))\text{d}\mathit{z}\\ & =\frac{1}{{\mathrm{\lambda }}^4}\left[{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(\frac{|{\mathit{r}}_k-\mathit{x}|}{\mathrm{\lambda }})\frac{{\mathit{r}}_k-\mathit{x}\otimes {\mathit{r}}_k-\mathit{x}}{|{\mathit{r}}_k-\mathit{x}|}\right]\delta \left({\mathit{r}}_l-\left[\mathit{x}+{\mathit{r}}_k-\mathit{x}-\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\right]\right).\end{array}$$ A 6

Using this in equation (A 3) grants:

$$\begin{array}{rl}{\mathit{\sigma }}_{\text{pot}}^{\text{Noll}}(\mathit{x},t)& =-\frac{1}{2}\sum _{k\ne l}{\int }_0^1{\int }_{{\mathit{r}}_k}{\int }_{{\mathit{r}}_l}\frac{1}{{\mathrm{\lambda }}^4}\left[{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}\left(\frac{|{\mathit{r}}_k-\mathit{x}|}{\mathrm{\lambda }}\right)\frac{{\mathit{r}}_k-\mathit{x}\otimes {\mathit{r}}_k-\mathit{x}}{|{\mathit{r}}_k-\mathit{x}|}\right]\\ & \delta \left({\mathit{r}}_l-\left[\mathit{x}+{\mathit{r}}_k-\mathit{x}-\frac{{\mathit{r}}_k-\mathit{x}}{\mathrm{\lambda }}\right]\right){⟨f⟩}_{{\mathit{r}}_k,{\mathit{r}}_l}({\mathit{r}}_l,{\mathit{r}}_k)\text{d}{\mathit{r}}_l\text{d}{\mathit{r}}_k\text{d}\mathrm{\lambda },\end{array}$$ A 7

which is the first equality used in equation (4.2) after reintroducing the two isolated d.f. into the ensemble operator. This is then equal to the ensemble average at the end of equation (4.2) after applying a few more properties of the Dirac delta. Thus, using the distributional properties of the Dirac delta there exists a mathematical link between the pair density first proposed by IK and the density in terms of a single Dirac delta.

Appendix B: The line-plane intersection derived from the pair density

The following is the expression of the surface-averaged stress in terms of the pair density:

$${\mathit{t}}_{\text{pot}}^{\text{pair}}(\mathit{x},t,\mathit{n})=-\int {\int }_{S^n(x)}\sum _{k\ne l}{\int }_{\mathit{z}}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}\mathit{z}}{|\mathit{z}|}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}\delta (\mathit{x}+\mathrm{\lambda }\mathit{z}-{\mathit{r}}_k)\delta (\mathit{x}+[\mathrm{\lambda }-1]\mathit{z}-{\mathit{r}}_l)\text{d}\mathrm{\lambda }\text{d}\mathit{z}\cdot \text{d}\mathit{S}.$$ B 1

In order to formally derive the result of this distribution we must first expand the direct products of each Dirac delta; this is due to the substantial difference between the distribution defined by the alpha integral and the surface average. We then define, with little loss of generality due to the ability to exhaust complex surfaces with simple planes, the surface as a plane with unit normal in the x³ direction to grant each integral in the series over k,l as:

$$\begin{array}{rl}& {\int }_{x_1^1}^{x_2^1}{\int }_{x_1^2}^{x_2^2}{\int }_{z_1^1}^{z_2^1}{\int }_{z_1^2}^{z_2^2}{\int }_{\mathrm{\lambda }=0}^{\mathrm{\lambda }=1}{\int }_{z_1^3}^{z_2^3}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}(\mathit{z}\cdot n)}{|\mathit{z}|}\delta (x^1+\mathrm{\lambda }{z}^1-{\mathit{r}}_k^1)\delta (x^1+[\mathrm{\lambda }-1]z^1-{\mathit{r}}_l^1)\delta (x^2+\mathrm{\lambda }{z}^2-{\mathit{r}}_k^2)\dots \\ & \times \delta (x^2+[\mathrm{\lambda }-1]z^2-{\mathit{r}}_l^2)\delta (x^3+\mathrm{\lambda }{z}^3-{\mathit{r}}_k^3)\delta (x^3+[\mathrm{\lambda }-1]z^3-{\mathit{r}}_l^3)\text{d}\mathrm{\lambda }\text{d}\mathit{z}\text{d}\mathit{S}.\end{array}$$ B 2

Start by making the following coordinate transformations:

$$\begin{array}{rl}{u}^1& =x^1+\mathrm{\lambda }{z}^1\\ u^2& =x^2+\mathrm{\lambda }{z}^2\end{array}\}$$ B 3

to grant:

$$\begin{array}{rl}& {\int }_{z_1^3}^{z_2^3}{\int }_0^1\delta (x^3+\mathrm{\lambda }{z}^3-r_k^3)\delta (x^3+[\mathrm{\lambda }-1]z^3-r_l^3){\int }_{z_1^2}^{z_2^2}{\int }_{x_1^2+\mathrm{\lambda }{z}^2}^{x_2^2+\mathrm{\lambda }{z}^2}\delta (u^2-r_k^2)\delta (u^2-z^2-r_l^2)\\ & \times {\int }_{z_1^1}^{z_2^1}{\int }_{x_1^1+\mathrm{\lambda }{z}^1}^{x_2^1+\mathrm{\lambda }{z}^1}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}(z^3)}{|\mathit{z}|}\delta (u^1-r_k^1)\delta (u^1-z^1-r_l^1)\text{d}\mathit{u}\text{d}\mathrm{\lambda }\text{d}\mathit{z}.\end{array}$$ B 4

To evaluate the distributions in the first and second set of coordinates readily we must first re-express the bounds of each iterated integral so that uⁱ determines the upper and lower bounds on the span of zⁱ, rather than the reverse which is the current expression, such as:

$${\int }_{z_1^i}^{z_2^i}{\int }_{x_1^i+\mathrm{\lambda }{z}^i}^{x_2^i+\mathrm{\lambda }{z}^i}\delta (u^i-r_k^i)\delta (u^i-z^i-r_l^i)\text{d}{u}^i\text{d}{z}^i={\int }_{u_1^i}^{u_2^i}{\int }_{g(\mathrm{\lambda },u^i)}^{f(\mathrm{\lambda },u^i)}\delta (u^i-r_k^i)\delta (u^i-z^i-r_l^i)\text{d}{u}^i\text{d}{z}^i.$$ B 5

For each of these the original domain can be plotted as the region bounded between the four lines $u^i=x_2^i+\mathrm{\lambda }{z}^i,$ $u^i=x_1^i+\mathrm{\lambda }{z}^i,$ $z^i=z_1^i,$ $z^i=z_2^i.$ In the case of $z_1^i\to -\mathrm{\infty }$and $z_2^i\to \mathrm{\infty }$ the expression (B 5) becomes:

$${\int }_{z_1^i}^{z_2^i}{\int }_{x_1^i+\mathrm{\lambda }{z}^i}^{x_2^i+\mathrm{\lambda }{z}^i}\delta (u^i-r_k^i)\delta (u^i-z^i-r_l^i)\text{d}{u}^i\text{d}{z}^i={\int }_{u_1^i}^{u_2^i}{\int }_{u^i/\mathrm{\lambda }-x_2^i/\mathrm{\lambda }}^{u^i/\mathrm{\lambda }-x_1^i/\mathrm{\lambda }}\delta (u^i-r_k^i)\delta (u^i-z^i-r_l^i)\text{d}{u}^i\text{d}{z}^i,$$ B 6

where $u_1^i\to -\mathrm{\infty }$ and $u_2^i\to \mathrm{\infty }$. The integral (B 4) is then:

$$\begin{array}{rl}& {\int }_{z_1^3}^{z_2^3}{\int }_0^1\delta (x^3+\mathrm{\lambda }{z}^3-r_k^3)\delta (x^3+[\mathrm{\lambda }-1]z^3-r_l^3){\int }_{u_1^2}^{u_2^2}{\int }_{u^2/\mathrm{\lambda }-x_2^2/\mathrm{\lambda }}^{u^2/\mathrm{\lambda }-x_1^2/\mathrm{\lambda }}\delta (u^2-r_k^2)\delta (u^2-z^2-r_l^2)\\ & \times {\int }_{u_1^1}^{u_2^1}{\int }_{u^1/\mathrm{\lambda }-x_2^1/\mathrm{\lambda }}^{u^1/\mathrm{\lambda }-x_1^1/\mathrm{\lambda }}{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|\mathit{z}|)\frac{\mathit{z}(z^3)}{|\mathit{z}|}\delta (u^1-r_k^1)\delta (u^1-z^1-r_l^1)\text{d}\mathit{u}\text{d}\mathrm{\lambda }\text{d}\mathit{z}.\end{array}$$ B 7

Completing the sifts on z¹ and z² grants us the following form of (B 2):

$$\begin{array}{rl}& {\int }_{z_1^3}^{z_2^3}{\int }_0^1\delta (x^3+\mathrm{\lambda }{z}^3-r_k^3)\delta (x^3+[\mathrm{\lambda }-1]z^3-r_l^3)\\ & \times {\int }_{u_1^2}^{u_2^2}\left[\theta \left(\frac{u^2}{\mathrm{\lambda }}-\frac{x_1^2}{\mathrm{\lambda }}-r_l^2+u^2\right)-\theta \left(\frac{u^2}{\mathrm{\lambda }}-\frac{x_2^2}{\mathrm{\lambda }}-r_l^2+u^2\right)\right]\delta (u^2-r_k^2)\dots \\ & {\int }_{u_1^1}^{u_2^1}\left[\theta \left(\frac{u^1}{\mathrm{\lambda }}-\frac{x_1^1}{\mathrm{\lambda }}-r_l^1+u^1\right)-\theta \left(\frac{u^1}{\mathrm{\lambda }}-\frac{x_2^1}{\mathrm{\lambda }}-r_l^1+u^1\right)\right]z^3{{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_l^1-u^1,r_l^1-u^1,z^3⟩|)\\ & \times \frac{⟨r_l^1-u^1,r_l^2-u^2,z^3⟩}{|⟨r_l^1-u^1,r_l^2-u^2,z^3⟩|}\delta (u^1-r_k^1)\text{d}\mathit{u}\text{d}\mathrm{\lambda }\text{d}\mathit{z}.\end{array}$$ B 8

Performing the next set of sifts on u¹ and u² grants

$$\begin{array}{r}{\int }_{z_1^3}^{z_2^3}{\int }_0^1\begin{array}{c}\left[\theta \left(\frac{r_k^2}{\mathrm{\lambda }}-\frac{x_1^2}{\mathrm{\lambda }}-r_{lk}^2\right)-\theta \left(\frac{r_k^2}{\mathrm{\lambda }}-\frac{x_2^2}{\mathrm{\lambda }}-r_{lk}^2\right)\right]\left[\theta \left(\frac{r_k^1}{\mathrm{\lambda }}-\frac{x_1^1}{\mathrm{\lambda }}-r_{lk}^1\right)-\theta \left(\frac{r_k^1}{\mathrm{\lambda }}-\frac{x_2^1}{\mathrm{\lambda }}-r_{lk}^1\right)\right]\dots \\ {{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,z^3⟩|)z^3\frac{⟨r_{lk}^1,r_{lk}^2,z^3⟩}{|⟨r_{lk}^1,r_{lk}^2,z^3⟩|}\delta (x^3+\mathrm{\lambda }{z}^3-r_k^3)\delta (x^3+[\mathrm{\lambda }-1]z^3-r_l^3)\end{array}\text{d}\mathrm{\lambda }d{z}^3.\end{array}$$ B 9

Since $\theta ({\mathit{r}}_k^i/\mathrm{\lambda }-x_1^i/\mathrm{\lambda }-{\mathit{r}}_{lk}^i)=\theta ({\mathit{r}}_k^i-x_1^i-\mathrm{\lambda }{\mathit{r}}_{lk}^i)$ for λ > 0 we have:

$$\begin{array}{rl}& {\int }_{z_1^3}^{z_2^3}{\int }_0^1\begin{array}{c}[\theta (r_k^2-x_1^2-\mathrm{\lambda }{r}_{lk}^2)-\theta (r_k^2-x_2^2-\mathrm{\lambda }{r}_{lk}^2)][\theta (r_k^1-x_1^1-\mathrm{\lambda }{r}_{lk}^1)-\theta (r_k^1-x_2^1-\mathrm{\lambda }{r}_{lk}^1)]\dots \\ {{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,z^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,z^3⟩}{|⟨r_{lk}^1,r_{lk}^2,z^3⟩|}{z}^3\delta (x^3+\mathrm{\lambda }{z}^3-r_k^3)\delta (x^3+[\mathrm{\lambda }-1]z^3-r_l^3)\end{array}\text{d}\mathrm{\lambda }\text{d}{z}^3.\end{array}$$ B 10

Consider

$$P(\alpha )=[\theta (r_k^2-x_1^2-\mathrm{\lambda }{r}_{lk}^2)-\theta (r_k^2-x_2^2-\mathrm{\lambda }{r}_{lk}^2)][\theta (r_k^1-x_1^1-\mathrm{\lambda }{r}_{lk}^1)-\theta (r_k^1-x_2^1-\mathrm{\lambda }{r}_{lk}^1)],$$ B 11

which determines whether the 1 and 2 components of the line between particle l and k project onto the region of the plane we defined. This is part of the line-plane intersection; it remains to compute the final component. First, we define the coordinate transformation:

$$u^3=\mathrm{\lambda }{z}^3\to \text{d}\mathrm{\lambda }=\frac{1}{z^3}\text{d}{u}^3.$$ B 12

To grant

$${\int }_{z_1^3}^{z_2^3}{\int }_0^{z^3}P\left(\frac{u^3}{z^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,z^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,z^3⟩}{|⟨r_{lk}^1,r_{lk}^2,z^3⟩|}\delta (x^3+u^3-r_k^3)\delta (x^3+u^3-z^3-r_l^3)\text{d}{u}^3\text{d}{z}^3.$$ B 13

We must again re-express the bounds of the iterated integral in order to sift z³ first. Expressing limits on z³ in terms of u³ gives:

$$\begin{array}{rl}& {\int }_{u_1^3}^0{\int }_{z_1^3}^{u^3}P\left(\frac{u^3}{z^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,z^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,z^3⟩}{|⟨r_{lk}^1,r_{lk}^2,z^3⟩|}\delta (x^3+u^3-r_k^3)\delta (x^3+u^3-z^3-r_l^3)\text{d}{z}^3\text{d}{u}^3\\ & +{\int }_0^{u_2^3}{\int }_0^{z_2^3-u^3}P\left(\frac{u^3}{z^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,z^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,z^3⟩}{|⟨r_{lk}^1,r_{lk}^2,z^3⟩|}\delta (x^3+u^3-r_k^3)\delta (x^3+u^3-z^3-r_l^3)\text{d}{z}^3\text{d}{u}^3,\end{array}$$ B 14

where $u_1^3,z_1^3\to -\mathrm{\infty }$ and $u_2^3,z_2^3\to \mathrm{\infty }$. Carrying out the first sifts on z³ grants:

$$\begin{array}{rl}& {\int }_{u_1^3}^0[\theta (r_l^3-x^3)-\theta (z_1^3-x^3-u^3+r_l^3)]P\left(\frac{u^3}{r_l^3-x^3-u^3}\right)\\ & \times {{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_l^3-x^3-u^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_l^3-x^3-u^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_l^3-x^3-u^3⟩|}\delta (x^3+u^3-r_k^3)\text{d}{u}^3\\ & +{\int }_0^{u_2^3}[\theta (z_2^3-2u^3-x^3+r_l^3)-\theta (r_l^3-x^3-u^3)]P\left(\frac{u^3}{r_l^3-x^3-u^3}\right)\\ & \times {{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_l^3-x^3-u^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_l^3-x^3-u^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_l^3-x^3-u^3⟩|}\delta (x^3+u^3-r_k^3)\text{d}{u}^3.\end{array}$$ B 15

Performing $\int f(x)\delta (x-a)\text{d}x=\int f(a)\delta (x-a)\text{d}x$ on u³ grants:

$$\begin{array}{rl}& {\int }_{u_1^3}^0[\theta (r_l^3-x^3)-\theta (z_1^3+r_{lk}^3)]P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|}\delta (x^3+u^3-r_k^3)\text{d}{u}^3\\ & +{\int }_0^{u_2^3}[\theta (z_2^3-r_k^3+r_{lk}^3)-\theta (r_{lk}^3)]P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|}\delta (x^3+u^3-r_k^3)\text{d}{u}^3.\end{array}$$ B 16

Since $z_1^3\to -\mathrm{\infty }$ and $z_2^3\to \mathrm{\infty }$:

$$\begin{array}{r}\theta (r_l^3-x^3)-\theta (z_1^3+r_{lk}^3)=\theta (r_l^3-x^3)\\ \theta (z_2^3-r_k^3+r_{lk}^3)-\theta (r_{lk}^3)=[1-\theta (r_{lk}^3)].\end{array}\}$$ B 17

Giving:

$$\begin{array}{rl}& {\int }_{u_1^3}^0\theta (r_l^3-x^3)P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|}\delta (x^3+u^3-r_k^3)\text{d}{u}^3\\ & +{\int }_0^{u_2^3}[1-\theta (r_{lk}^3)]P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|}\delta (x^3+u^3-r_k^3)\text{d}{u}^3.\end{array}$$ B 18

Evaluating further:

$$\begin{array}{rl}& [\theta (x^3-r_k^3)-\theta (x^3+u_1^3-r_k^3)]\theta (r_l^3-x^3)P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3>}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3>|}\\ & +[\theta (x^3+u_2^3-r_k^3)-\theta (x^3-r_k^3)]\hspace{0.17em}[1-\theta (r_{lk}^3)]P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|}.\end{array}$$ B 19

Since $u_1^3\to -\mathrm{\infty }$ and $u_2^3\to \mathrm{\infty }$ this simplifies to:

$$\begin{array}{rl}& [\theta (x^3-r_k^3)]\theta (r_l^3-x^3)P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|}\\ & +[1-\theta (x^3-r_k^3)]\hspace{0.17em}[1-\theta (r_{lk}^3)]P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|}\\ & =(\theta (x^3-r_k^3)\theta (r_l^3-x^3)+[1-\theta (x^3-r_k^3)][1-\theta (r_{lk}^3)])P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right){{\mathit{\Phi }}^{\mathrm{\prime }}}_{kl}(|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|)\frac{⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩}{|⟨r_{lk}^1,r_{lk}^2,r_{lk}^3⟩|},\end{array}$$ B 20

where the last line above is the line plane intersection problem, shown in equation (2.11), when it is fully expressed and evaluated in terms of the Heaviside function for a single pair l,k. Thus;

$$\begin{array}{rl}& (\theta (x^3-r_k^3)\theta (r_l^3-x^3)+[1-\theta (x^3-r_k^3)][1-\theta (r_{lk}^3)])P\left(\frac{r_k^3-x^3}{r_{lk}^3}\right)\\ & =\{\begin{array}{ll}1& \text{if}{r}_{lk}\text{intersects }\hspace{0.17em}{\mathit{A}}^3(\mathit{x})\text{ and }{\mathit{e}}^3\cdot {\mathit{n}}_{lk}>0\\ -1& \text{if}{r}_{lk}\text{intersects }\hspace{0.17em}{\mathit{A}}^3(\mathit{x})\text{ and }{\mathit{e}}^3\cdot {\mathit{n}}_{lk}<0\\ 0& \text{otherwise}\end{array}\end{array}$$ B 21

Summing over all l,k would then grant the surface-averaged stress formula posed in §2. The procedure is virtually identical for heat-flux. The connection between surface-averaged fluxes expressed in terms of the pair density and the line plane intersection problem is thus demonstrated.

Appendix C: Taylor series expansion found in the work of Irving and Kirkwood

Consider IK's definition of a Taylor series expansion in the difference between two Dirac deltas as seen in his Eq. (5.9) [6]:

$$\delta ({\mathit{r}}_k-\mathit{x})-\delta ({\mathit{r}}_l-\mathit{x})=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{({\mathit{r}}_{kl}\cdot {\mathrm{\nabla }}_{\mathit{x}})}^n\delta ({\mathit{r}}_l-\mathit{x}).$$ C 1

To simplify the demonstration of why such a series expansion is not defined, we shall use the case of a one-dimensional space without loss of generality in the inherit analytical limitation; in addition, the mathematical operations taken all work identically when performed with the ensemble average. It is clear from IK's description that in the context of the standard form of 1D Taylor series, $f(x)={\sum }_{n=0}^{\mathrm{\infty }}((f^{(n)}(a))/n!){(x-a)}^n,$ we have x = r_k, a = r_l, and$f^{(n)}(a)={(-1)}^{n-1}(\mathrm{\partial }/\mathrm{\partial }x)\delta (r_l-x).$ The derivatives of the Dirac delta of course only exist in the distributional sense, i.e. $\int ({\mathrm{\partial }}^n/\mathrm{\partial }{x}^n)\delta (r_l-x)\phi (r_l)d{r}_l={(-1)}^n\int \delta (r_l-x)({\mathrm{\partial }}^n/\mathrm{\partial }{x}^n)\phi (r_l)d{r}_l,$ where φ belongs to the space of infinitely differentiable functions. This formula is the rigorous consequence of taking the limit of some differentiable function, $\text{li}{\text{m}}_{\epsilon \to 0}{f}_{\epsilon }(r_l)\to \delta (r_l),$ that approaches the action of the Dirac delta and conducting integration by parts [47]. Similarly, we can assess the existence of IK's proposed Taylor series expansion, in 1D, in the distributional sense by placing the differentiable function whose action converges to the action of the delta in equation (C 1):

$$\underset{\epsilon \to 0}{lim}\sum _{n=1}^{\mathrm{\infty }}\int \frac{{(-1)}^{n-1}}{n!}{(x-r_l)}^n\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{x}^n}{f}_{\epsilon }(r_l-x)\phi (r_l)\text{d}{r}_l=\underset{\epsilon \to 0}{lim}\sum _{n=1}^{\mathrm{\infty }}\int \frac{1}{n!}{(x-r_l)}^n\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{r_l}^n}{f}_{\epsilon }(r_l-x)\phi (r_l)\text{d}{r}_l,$$ C 2

where we must perform integration by parts to grant:

$$\begin{array}{rl}& \underset{\epsilon \to 0}{lim}\sum _{n=1}^{\mathrm{\infty }}\int \frac{1}{n!}{(x-r_l)}^n\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{r_l}^n}{f}_{\epsilon }(r_l-x)\phi (r_l)\text{d}{r}_l\\ & =\underset{\epsilon \to 0}{lim}\left[\sum _{n=1}^{\mathrm{\infty }}\int \frac{{(-1)}^n}{n!}{(x-r_l)}^n{f}_{\epsilon }(r_l-x)\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{r_l}^n}\phi (r_l)\text{d}{r}_l+\sum _{n=1}^{\mathrm{\infty }}\int \frac{{(-1)}^n}{n!}\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{r_l}^n}{(x-r_l)}^n{f}_{\epsilon }(r_l-x)\phi (r_l)\text{d}{r}_l\right]\\ & =\underset{\epsilon \to 0}{lim}\left[\sum _{n=1}^{\mathrm{\infty }}\int \frac{{(-1)}^n}{n!}{(x-r_l)}^n{f}_{\epsilon }(r_l-x)\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{r_l}^n}\phi (r_l)\text{d}{r}_l+\sum _{n=1}^{\mathrm{\infty }}\int f_{\epsilon }(r_l-x)\phi (r_l)\text{d}{r}_l\right].\end{array}$$ C 3

We then move the limit into each individual term to judge if the result will converge:

$$\begin{array}{rl}& \underset{\epsilon \to 0}{lim}\left[\sum _{n=1}^{\mathrm{\infty }}\int \frac{{(-1)}^n}{n!}{(x-r_l)}^n{f}_{\epsilon }(r_l-x)\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{r_l}^n}\phi (r_l)\text{d}{r}_l+\sum _{n=1}^{\mathrm{\infty }}\int f_{\epsilon }(r_l-x)\phi (r_l)\text{d}{r}_l\right]\\ & =\left[\sum _{n=1}^{\mathrm{\infty }}\int \frac{{(-1)}^n}{n!}{(x-r_l)}^n\delta (r_l-x)\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{r_l}^n}\phi (r_l)\text{d}{r}_l+\sum _{n=1}^{\mathrm{\infty }}\int \delta (r_l-x)\phi (r_l)\text{d}{r}_l\right].\end{array}$$ C 4

It becomes immediately evident that the second summation term in (C 4) could not converge for any possible differentiable function φ since the action of the Dirac delta would produce an infinite summation of finite evaluations of φ. Thus, the Taylor series expansion proposed in the work of IK is either undefined or inadmissible as an operator due to lack of convergence.

Ethics

This work did not involve any active collection of human data, only theoretical analysis.

Data accessibility

This work does not have any experimental data.

Authors' contributions

A.D. initiated the project, led the effort in the mathematical formulation and writing of the paper; D.D. discussed the work in the context of existing works based on Noll's lemma and contributed to the writing of this paper; Y.C. initiated the writing of the paper and contributed to the mathematical formulation.

Computational solution

This work does not have any computational data.

Competing interests

We declare we have no competing interests.

Funding

This material is based upon research supported by the U.S. DOE, Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering under Award # DE-SC0006539. D.D. was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), grant DA 1664/2-1.