1. Introduction
Over the past few decades, Feynman diagrams have significantly contributed to the formulation of various theoretical predictions. They emerged as an important tool in quantum field theory calculations, but they have also contributed to the development of computational methods for many other problems of mathematical modeling of physical processes, with a special emphasis on their automation. A comprehensive overview of the present state of algorithmic Feynman diagram evaluation from a theoretical perspective is given in [
1]. Two primary factors explain their high importance in modern science: an algorithmic framework developed by physicists for manipulating these diagrams and efficient algebraic programs designed to perform Feynman diagram calculations.
Feynman diagrams are a powerful research tool arising in theoretical physics over the past hundred years. Initially developed within quantum electrodynamics to compute physical process amplitudes, they have evolved into a core mathematical concept in quantum field theory. Owing to their abstract nature, they find use in areas extending beyond physical applications. A detailed analysis of Feynman diagrams and their application in scientific models as well as a description of other diagrammatic methods in mathematical physics that are used when working with series are given in [
2,
3,
4,
5]. The example of application of Feynman diagrams shows that models are understood as mathematical objects that correspond to the axioms of a certain theory and correspond to the empirical phenomenon.
Diagrammatic approaches are used as a component in the theory of analogical reasoning in mathematics, where general conditions are formulated to determine when an analogy can serve as legitimate inductive support for a mathematical hypothesis in empirical sciences [
6]. The development of Feynman diagrams arose due to the ability to visualize phenomena and processes, and such visualization contributed to the formation of well-grounded mathematical modeling of many phenomena through physical interpretation [
7]. Diagrammatic approaches have proven effective in computational biology and structural bioinformatics problems [
8,
9].
The study of the properties of Feynman diagrams has also inspired mathematicians to develop many new approaches to mathematical modeling. Utilizing response and correlation functions, the relevant Feynman diagrams were constructed and analyzed, which facilitated the formulation of a systematic perturbation theory applicable to stochastically controlled nonlinear oscillators under Gaussian white noise excitation [
10].
Research on singular partial differential equations using methods of theoretical physics, in particular quantum field theory, such as Feynman diagrams, was carried out in [
11]. In [
12], moments of the Gaussian integral were found using the diagrammatic approach. The development of the wave functions of a harmonic oscillator using the Feynman diagram technique was found in [
13]. In [
14], a number of Feynman algorithmic rules enabling analytical study of large fluctuations in stochastic systems are presented, which allows for the calculation of multi-time correlation functions. Special diagram rules have been developed to simplify transformation. Mellin–Barnes representations have been used to obtain the systems of linear homogeneous differential equations corresponding to the original Feynman diagrams without the need for integration by parts [
15]. The authors of [
16] obtained the results of the summation of Euler series that appear in the calculations of the Feynman diagram.
The above-described methods and approaches of the Feynman diagram theory occupy a special place in the mathematical modeling of diffusion transfer processes and in convection models. When analyzing the equations and structures of solutions of a stochastic velocity field that obeys the Navier–Stokes equation describing incompressible fluid flow under the action of both regular and random external forces, the Feynman wireframe diagram method is used [
17]. Using this equation, one can calculate multiple statistical properties describing the velocity field, including the variance of velocity pulsations (pair correlation function) or the average reaction of the velocity field under the influence of external forces [
18]. Using the Green’s function expansion, an exact expression is presented in [
19] for determining the number of distinct connected Feynman diagrams. In [
20], the Green’s function expansions under perturbations at time points far from the initial state, which is formulated in terms of the Feynman diagram, are obtained. The work [
21] is dedicated to theoretical modeling of heat transfer in multiphase systems exhibiting random structural inhomogeneity. The Feynman diagram technique is applied to analyze fields of temperature averaged over the ensemble of phase configurations in bodies with multiple stochastic phases. The study of averaged diffusion fields in the physical and mathematical modeling of mass transfer processes in multiphase stochastically inhomogeneous bodies using the Feynman diagram technique was carried out in [
22]. A method for improving the convergence of Neumann integral series for diffusion processes has been developed. An exact reaction–diffusion equation for describing chemical reactions occurring in polymer systems was derived in [
23] using diagrams similar to Feynman diagrams. It was found that, when interactions between molecules are absent, the obtained equation reduces to the classical reaction–diffusion equation, and the diagram approach can be used to obtain a more precise dynamic equation in which diffusion and reaction manifest themselves more deeply than the diffusion of small molecules.
The symmetrical properties of the diffusion process refer to the homogeneity and balance in the movement of particles from areas of high concentration to areas of low concentration. In a symmetric diffusion process, particles are distributed uniformly in all directions, resulting in a homogeneous distribution over time. Key aspects of symmetry in diffusion include diffusion isotropy. This occurs when the speed of diffusion is the same in all directions and the medium within which the particles diffuse has no impact on the speed, resulting in a symmetrical distribution. Mathematical models of diffusion based on Fick’s laws assume symmetry to simplify the calculations, and the corresponding solutions to the parabolic equations reflect homogeneous concentration changes in space and time. In practical scenarios, symmetry may be affected by boundaries or constraints. For example, in a closed medium, diffusion can be symmetric until it reaches the walls, and after that point in time, the symmetry can be broken due to reflection or absorption. Thus, symmetry in diffusion is an important concept that helps to analyze and predict the behavior of particles as they move and propagate in different media.
Symmetry in diffusion and Feynman diagrams can be understood through various contexts in physics. Many physical theories demonstrate symmetries that are represented in Feynman diagrams. The results of this article can be used to analyze how diffusion processes and the Feynman diagrams used to visualize them reflect the fundamental symmetries that govern the behavior of systems in physics and to analyze the symmetry of Green’s functions, being an important aspect in the context of field theory.
The analysis of applying diagrammatic approaches to the mathematical modeling of processes of a probabilistic nature shows the effectiveness (in particular, computational) of such methods. Therefore, the purpose of the research conducted in this paper is to apply the Feynman diagram technique to the mathematical modeling of impurity diffusion processes in a multiphase body with random inhomogeneities. To achieve this aim, we follow the steps below:
The problems for the impurity diffusion are formulated for each phase separately and include non-ideal conditions for the concentration function at random phase contact boundaries.
The contact mass transfer problem, which takes into consideration the jumps of the searched function and its derivative at stochastic interphases, is reduced to a parabolic partial differential equation for the entire body domain.
The obtained problem is reduced to an integro-differential equation with a random kernel.
The Feynman diagram technique is developed to study the mass transfer processes described by such parabolic equations.
The solution of the problem is constructed using Feynman diagrams.
The Dyson equation is obtained in graphical and analytical forms.
The resulting solution under uniform distribution of the phases is compared with solutions of mass transfer problems in a homogeneous layer with base phase coefficients and in a homogeneous layer with characteristics averaged over the body volume.
3. Equation of Mass Transfer for the Entire Body
Let us reduce the contact problem of diffusion (1), (3), and (4) to the mass transfer equation for a body in general. For this purpose, let us define a random function of the spatial coordinate
that characterizes concentration field in the body as a whole:
Let us seek
, considering that the concentration field
has first kind jump discontinuities (conditions (3), (4)) on the boundaries of contact between regions
. Therefore, we obtain [
29]
where
is the radius vector of the boundary
points;
are the regions of continuity of the function;
is the function jump at the boundary
; and
stands for the delta function.
We observe that the jump () is unit normal vector to the contact boundary) is in fact a vector quantity and varies with the radius vector .
The quantity
is found in the same way as (7), considering that the function
has first kind jump discontinuities on the surfaces of phase division (conditions (4)). Then, we have
Coefficients
and
are designated for open regions, namely
At the same time, at the contact boundaries of , a jump of these coefficients takes place , .
Note that the relation for the stochastic field of impurity concentration in the whole body (6) completely describes the function for , .
Let us introduce the random operator
as the “structure function”, which is defined as follows:
which satisfies the condition of body solidity
Note that the probability law of the structural function matches the given probability law for the distribution of phases in the body.
Then, the kinetic coefficient
and the density
can be represented using a random structure function (10) as follows:
For the body as a whole, the mass balance equation holds
Here,
is the impurity flow.
Let us take into consideration that
is a piecewise-constant function and has first kind jump discontinuities at the contact surfaces; then, the action of the nabla operator on it leads to
Since we assumed the problem coefficients to be constant in each phase, Formula (14) reduces to the form
Taking into account the ratios (7), (8), and (15), we present the Equation (13) as follows:
It is taken into consideration here that the function is a piecewise continuous one in time together with the first derivative. It is also accepted that the regions of continuity of the kinetic coefficient and the function coincide:.
Using the condition of body solidity (11) and representations (12), the equation for the entire body is written as
In Equation (17), let us add and subtract the deterministic operator
with coefficients that are characteristics of the base phase (matrix)
Then, we will obtain
Here,
is the following:
Thus, a stochastic differential equation of the mass transfer of an impurity for a medium with randomly located inclusions was obtained in the form of (17), and a “perturbed” differential equation was obtained in the form of (18) with a random operator (19).
5. Feynman Diagram Technique for Mass Transfer Problems in a Stochastically Inhomogeneous Medium
The Neumann series (29) expresses the random concentration field as a series of perturbations resulting from inclusions with diffusion characteristics distinct from those of the matrix. To examine the structure of this series, a diagrammatic representation of its terms based on R. Feynman diagrams [
31] is introduced. Here, the typical notations are used [
32].
Let us associate the Green’s function
with a straight line segment, to whose ends the points
and
are assigned [
32]:
Let us associate the operator with a vertical segment with dots at its ends:
Let us associate the functions of the random concentration field and the concentration field in a homogeneous body with a tick and a wavy line, respectively:
The space–time positions , , where the lines visualizing , , , and merge together, are the vertices of a diagram. The integration is performed over the internal vertices’ coordinates. The total number of such vertices in the diagram determines its order. Such notations allow us to associate each term of the series (29) with a Feynman diagram. The first right-hand side element of (29) is represented by the diagram
and the second term of this formula corresponds to the diagram
In the general case, a term of the order
of the Neumann series for
has the form
Therefore, the diagrams of the order
contain
lines of the Green’s function and
lines of internal vertices
, …,
. At the same time, they always end in a wavy line
[
30]:
Then, the series (29) in diagrammatical form will take the following form:
Averaging the random concentration field (29) over the ensemble of phase configurations yields the following analytical expression:
Considering the structure of the operator
(19), one needs to calculate the expressions of the type
,
,
,
,
, etc. That is, to calculate the averaged concentration field, it is required to determine the moments
,
, and
of all orders. With a general statistical distribution, this becomes a complex problem, and additionally, the issue of choosing an appropriate method for summing the averaged series arises. For example, in the case of a small volume fraction of inclusions for small diffusion coefficients, one can limit oneself to the “Born approximation” [
33], i.e., considering only the first two terms of the Neumann series. Then, the problem is reduced to the case of perturbation of concentration fields by specified random sources.
In this case, the random field (first approximation) is a linear functional of fluctuations , . Therefore, each moment of is expressed linearly through the moments and , having the same order.
The field of the averaged concentration in the Born approximation is defined as follows:
In this case,
.
The correlation function (also known as autocorrelation), by its definition, is [
34]
Here, if we use
, then we will obtain the variance of the random field
at the point
:
Relation (34) together with analogous expressions for higher-order moments of the field
yield a complete statistical solution of the problem. The statistical characteristics are described by cumulant (or correlation) functions of arbitrary order. We associate these functions with dashed lines, where the order of the cumulant function
[
34] matches the diagram order.
Because the concentration field is time deterministic, time appears as a parameter in cumulative functions.
In the general case, the moments from the operator
that contains fluctuations
and
can be written down as [
35]
or in the form of diagrams
Note that expression (35) decomposes into the sum of terms, in which the arguments , , …, connected in every possible way. Therefore, at the time of averaging, we obtain of diagrams of the order, in which vertices connect with each other in all possible ways. Note that, since integration occurs over the coordinates of the interior vertices, the analytical expression represented by the diagram has no dependency on the interior vertices’ coordinates. Thus, such coordinates are not indicated in the subsequent diagrams.
Let us introduce a graphical representation for the averaged concentration field in an inhomogeneous medium:
Then, the series (32) can be presented graphically as follows
Here, in addition to the terms written in (32), seven fourth-order terms (diagrams with four vertices) are also given.
Each Feynman diagram uniquely corresponds to an analytical expression and vice versa.
Certain diagrams comprising (37) include lower-order diagrams as fragments. For example, diagram 3 contains, as a fragment, diagram 2, and diagram 6 includes diagrams 2 and 4. This makes it possible to shorten the analytical expression.
Let us consider the physical interpretation of Feynman diagrams. Diagram
1 from the series (37) describes the propagation of a concentration field from a source or from a surface (depending on the boundary conditions and the presence of internal mass sources) in a homogeneous medium. Diagram
2 describes the following process: the concentration field propagates from the source to point
as in a homogeneous medium. At point
, the concentration field is perturbed because this point belongs to the inclusion or its boundary (
belonging to the inclusion is determined by the averaged operator
, otherwise it is zero). Then, the perturbed field reaches point
, where the observation takes place (
Figure 2a).
Let us now consider the second-order diagrams. Diagram 3 represents the spread of the concentration field from the source to point
, which is in the inclusion (operator
), where it experiences a perturbation; then, the perturbation field extends to point
, which belongs to another inclusion (operator
in analytical form) and experiences a second perturbation, after which, the twice perturbation field extends to the observation point
(
Figure 2b). Diagram 4 differs from diagram 3 by the presence of the correlation function
, which indicates that the two perturbation points
and
are correlated, i.e., both perturbations occurred in the same heterogeneity (
Figure 2c). All third-order diagrams 5–9 contain the functions
,
,
, and
. This means that the concentration field has spread to the point
after perturbation in point
, to point
after perturbation in point
, and so on. Therefore, all these diagrams describe a triple perturbation of the concentration field. However, diagrams 5–9 are topologically different. Diagram 5 does not contain correlation functions, that is, three perturbations of the concentration field occurred in different inhomogeneities. Diagrams 6–8 contain correlation functions
,
, and
, respectively. This means that the field is perturbed three times in two inhomogeneities (
Figure 3a–e).
Diagram 9 contains the cumulative function , that is, all three concentration field perturbations occur in a single inclusion.
Representing the solution of the initial-boundary value problem (17), (2), as a set of diagrams (37),allows for the transformationof the Neumann series using the topological features of the diagrams that contain solution.
The sum of series (32) can be represented as a sum involving a certain infinite subsequence of the same series. For this purpose, we classify the diagrams involved in (37) [
31].
A diagram is termed weakly connected if, by breaking a certain one line , it can be split into two separate diagrams. In expression (37), diagrams 3, 5–7, 10–13, 15, and 17–20 are weakly connected. The other diagrams, i.e., 1, 2, 4, 8, 9, 14, 16, and 21, are strongly connected. The diagrams obtained as a result of breaking lines can, in turn, be either strongly or weakly connected. If the set of “secondary” diagrams contains weakly connected diagrams, they can be split down into simpler diagrams. Following this procedure, we will end up with a certain number of strongly connected diagrams. The quantity of strongly connected diagrams resulting from the decomposition of a weakly connected diagram defines the “connectivity index” of the original diagram. Thus, in relation (37), the connectivity index of diagrams 3, 6, 7, 12, 13, 15, 19, and 20 is 2, and the connectivity index of diagrams 5, 11, 17, and 18 is 3. Let us assign the connectivity index 1 to the strongly connected diagrams.
From the series (37), we extract all strongly connected diagrams, i.e., those that cannot be partitioned into two separate diagrams by cutting a single line
. As each diagram begins with a straight line and terminates with a wavy
line, the sum of all strongly connected diagrams may be represented as
Analytically, we obtain
where
is the mass operator kernel:
The mass operator kernel (40) can also be presented graphically:
Let us focus on the sum of all the strongly connected diagrams with connectivity index = 1. Each of those diagrams possesses the following form:
where
![Symmetry 17 00920 i025]()
and
![Symmetry 17 00920 i026]()
denote arbitrary diagrams that are on the right side of (41).
Since, when the series (37) is being constructed, all variants of pairwise connection of the vertices are considered, the sum of all possible terms of the form (42) is
Here,
![Symmetry 17 00920 i028]()
is the total sum of the mass operator kernel (40).
Similarly, the sum of all diagrams with a connectivity index of 3 has the form
and so on. Thus, we can present the averaged concentration field as a diagrammatic series:
Such a representation is distinct from the diagrammatic series (37), solely in the rearrangement of its elements.
Let us verify that the series (41) is the solution of the following equation:
which is called the Dyson equation. We present the Dyson Equation (44) in analytical form:
Let us show that the series (43) is a solution of equation (45). Let us apply the method of successive iterations. Relation (44) is substituted into the expression for the averaged concentration field
into the right-hand side of (44). We obtain
Again, by substituting the right-hand side of the resulting relation into the right side of (44), we obtain
If we continue the iterations, we will obtain the series (43). Similar calculations can be carried out in analytical form if we proceed from Equation (45). In this case, we will obtain the expansion (43) in analytical form, namely
Equation (45), assuming is known, is an integral equation in relation to , which can be solved in some cases. At that, we will obtain an explicit expression of the averaged concentration field through the mass operator kernel , i.e., the sum of the series (32) is given using the quantity (39), which is a certain subsequence of the same series.
, as the operator in the general case and as the function in a given instance, is not known exactly. In the approximate case, for example, the sum of the first few terms of the series (46) can be used as this operator (function). In the case of the Bourret approximation [
31],representing the first-order term in the series expansion of the mass operator, we have
Considering the structure of the operator
(19), the expression for
can be presented at non-ideal mass contact conditions as
and at ideal contact conditions as
The Bourret approximation for the averaged concentration field
is graphically designated as
If, in the diagrammatic Formula (43), we put diagram (47) instead of
, this leads to the diagrammatic representation of
as
and the analytical form as
If a randomly inhomogeneous medium is not only statistically homogeneous but also statistically isotropic, then Formula (51) can be simplified by transitioning to spherical coordinates and performing integration over angular variables [
31].
7. Impurity Diffusion in a Two-Phase Randomly Inhomogeneous Layer
To study the influence of inhomogeneities on the behavior and the values of the averaged concentration field, we consider the case of impurity mass transfer in a two-phase randomly inhomogeneous layer. The phases in the body are arranged according to a uniform probability distribution. The obtained averaged concentration field will be compared with solutions of similar problems of impurity diffusion in a homogeneous layer with matrix characteristics and in a homogeneous layer with parameters averaged over the body volume.
The diffusion of impurity in a layer of thickness
is described by an initial-boundary value problem based on Equation (61) in the one-dimensional case, i.e.,
The diffusion of an impurity in a layer with volume-averaged characteristics is governed by the following equation:
with the boundary conditions (64). Here,
,
.
The equation of the diffusion of an impurity in a homogeneous layer with matrix characteristics has the form
Boundary conditions remain the same.
Note that the partial differential equation for the averaged concentration (63) can be considered as a certain homogenized diffusion equation, the coefficients of which differ significantly from the averaged ones of Equation (65). At the same time, the operators of these equations have the same structure.
Let us investigate the solution to the problem (63) and (64) and compare it with the solutions of the initial-boundary value problems (64), (65) and (60), (64).
The solution to the problem (63) and (64) is the following [
25]:
where
,
.
Let us show the graphs of the average concentration of impurity migrating in a layer. Numerical calculations were carried out in dimensionless variables
The following parameter values were used as baselines: , , , and , , , , and . Calculations were carried out using Formula (67). The accuracy when summing rows in formulas was . Along the ordinate axis, the function , that is the concentration function normalized to its boundary value (64), was laid down.
Figure 4 illustrates the distributions of concentration
averaged over the ensemble of phase configurations (curves a, solid lines), concentrations in a layer with volume-averaged characteristics (curves b, dash-dotted lines), and concentrations in a homogeneous layer with matrix characteristics (c curves, dashed lines) at the moments of time
(curves 1),
(curves 2), and
(curves 3). Here,
Figure 4a is constructed for
and
Figure 4b for
. The same applies to the figures a and b below unless stated otherwise.
Figure 5 presents the graphs of the average particle concentration for different values of the kinetic coefficient in inclusions
= 0.01, 0.5, 1.5, 5, and 20 (curves 1–5, respectively) for
(
Figure 5a) and
(
Figure 5b). Here, and further on, the dashed lines correspond to the impurity concentration in the body possessing the characteristics of the base phase.
Figure 6 demonstrates the behavior of the function
depending on different values of the inclusion’s volume fraction of
0.01, 0.05, 0.1, 0.2, and 0.3 (curves 1–5) for
(
Figure 6a) and
(
Figure 6b).
Figure 7 illustrates the graphs of the averaged concentration
and particle concentrations in the matrix
for different values of the searched function on the bottom surface of the layer
0.01, 0.05, 0.25, and 0.5 (curves 1–4),
(
Figure 7a), and
(
Figure 7b).
It should be noted that the use of the impurity diffusion model to calculate the concentration averaged over the ensemble of phase configurations in a two-phase layer in which the phases are uniformly distributed leads to results that are significantly different from those obtained using diffusion models with volume-averaged characteristics and with matrix characteristics (
Figure 4). At the same time, the difference between the solutions of problems, (64), (65) and (64), (66) is insignificant and amounts to 7% for the time moment
. With increasing diffusion time, the concentration functions in all models increase until they reach a steady state (curves 3b, c in
Figure 4a and curves 3a in
Figure 4b), and the difference between them is decreasing. If the values of the diffusion coefficient in inclusions are smaller than in the matrix, i.e.,
, the values of the function
are always smaller than those of the concentration functions calculated using other models (
Figure 4a,b). For
, the opposite situation is observed—the values of the concentration averaged over the ensemble of phase configurations are always higher than the solutions of problems (64), (65) and (64), (66) (
Figure 4b).
Let us also remark that the greater the ratio
, the faster the function
reaches a steady state (
Figure 5). Thus, the time needed to reach the steady state regime of the concentration function for
is
and for
is
.
Changing the volume fraction of inclusions has a negligible effect on the value of the function
(
Figure 6). Therefore, the difference between
and
, that is, when
is increased by 30 times, reaches 25%. The value of the concentration function
at the bottom surface of the layer affects its behavior in the lower half of the body (
Figure 7).The largest differences between the average impurity concentration in the layer and the matrix characteristics are observed in the interval
(
Figure 7).
As an example of a real diffusion process that can be simulated by the methods developed in this article, we present graphs of the averaged hydrogen concentration field in a two-phase iron–copper body. Consider the problem of hydrogen migration in the composite material
, taking iron as the base phase. The diffusion coefficients of hydrogen were assumed as [
36,
37]: in iron
m
2/s, in copper
m
2/s; the densities of iron
kg/m
3 and copper
kg/m
3 From these values, one obtains the dimensionless parameters
and
.
Illustrated in
Figure 8 is the behavior of the averaged hydrogen-concentration field for
(
Figure 8a) and
(
Figure 8b) in the composite material
. Curves 1–3 correspond to the dimensionless times
, respectively.
It should be noted that, as the volume fraction of copper increases, the averaged hydrogen-concentration field reaches its steady-state regime more rapidly (
Figure 8).