On the Symmetric Character of the Thermal Conductivity Tensor via Tensor Analysis

Abstract

In this paper, the symmetric character of the thermal conductivity tensor for anisotropic materials is established based on arguments from tensor analysis and the physical constraints on the domain of definition of the conductivity tensor. The non-singular nature of the conductivity tensor plays a fundamental role in demonstrating not only its symmetry but also its positive definiteness.

1. Introduction

Joseph Fourier [1] developed the theory of the heat equation, in which he introduced the concept of a thermal conductivity property to describe the linear heat conduction in isotropic materials. In anisotropic materials, the conductivity of matter varies with direction. To generalize the linear heat conduction law for anisotropic materials, Duhamel [2] introduced the thermal conductivity tensor. By using non-equilibrium statistical mechanics, Onsager [3] has shown that the thermal conductivity tensor is symmetric. For more insight, based on non-equilibrium thermodynamics and macroscopic transport theory, see [4,5,6]. Since classical continuum mechanics did not provide any direct reasoning for this property for a long time, it was generally believed that the symmetry condition can only be derived based on additional physical assumptions. For instance, Day and Gurtin [7] have introduced the requirement that the thermal work functional attain a weak relative minimum at equilibrium.

Recently, the symmetric character of the conductivity tensor has been established using arguments from tensor analysis and linear algebra [8]. The proof relies on the consistency of the system of linear equations that represent the heat conduction law in different coordinate systems. Notably, this approach demonstrates that classical continuum mechanics offers a mathematical justification for the symmetry of the conductivity tensor, which is an essential condition for ensuring consistent and physically meaningful tensorial relationships in classical heat conduction theory. Nevertheless, it remains possible that alternative methods could be used to establish this symmetric property.

Here the symmetric character of the thermal conductivity tensor is established more fundamentally by focusing on the physicality of its domain of definition. Interestingly, the non-singularity of the conductivity tensor is still the main argument in this establishment, which immediately shows that the conductivity tensor cannot be skew-symmetric. By using this argument, it is demonstrated that a nearly skew-symmetric conductivity tensor is non-physical. This in turn requires that the conductivity tensor be symmetric. Subsequently, the fact that the symmetric conductivity tensor needs to represent the conductivity of an isotropic material proves that it is a positive definite tensor. Remarkably, these developments show the subtle character of physical tensors in three-dimensional space, which has not been fully recognized previously.

The paper is organized as follows. Section 2 provides an overview of the classical heat conduction relations for linear anisotropic materials. Section 3 discusses the non-singular nature of the conductivity tensor and its direct implication that the tensor cannot be skew-symmetric. In Section 4, the symmetric character of the conductivity tensor is established using arguments from tensor analysis and considerations regarding the physicality of its domain of definition. Section 5 presents proof of the positive-definite nature of the symmetric conductivity tensor. Finally, Section 6 offers a summary and general conclusions. Appendix A outlines several properties of second-order tensors.

2. Linear Heat Conduction Theory

Consider the three-dimensional orthogonal coordinate system $x_1{x}_2{x}_3$ as the reference frame. Fourier’s heat conduction law for linear isotropic material [1] is

$$q_i=-k{T}_{,i},$$

(1)

where $k$ is the material’s conductivity, which relates the heat flux vector $q_i$ to the gradient of the temperature field $T$. Note that the scalar conductivity $k$ as a material property is always a positive number. Therefore, the minus sign in Equation (1) assures that heat flows from a higher to a lower temperature. This satisfies the second law of thermodynamics, which was formulated after Fourier’s development of heat conduction law.

For linear anisotropic material, Duhamel has generalized Fourier’s heat conduction law as [2]

$$q_i=-k_{ij}{T}_{,j}.$$

(2)

Here $k_{ij}$ is the representation of the second-order material thermal conductivity tensor $k$ in the coordinate system $x_1{x}_2{x}_3$, which relates the heat flux vector $q_i$ to the gradient of the temperature field $T$. In terms of components, the second-order conductivity tensor $k_{ij}$ can be written as

$$\left[k_{ij}\right]=\left[\begin{array}{ccc}{k}_{11}& k_{12}& k_{13}\\ k_{21}& k_{22}& k_{23}\\ k_{31}& k_{32}& k_{33}\end{array}\right].$$

(3)

It is important to note that, at this stage, the conductivity tensor is characterized by nine independent components in the general case. Consequently, $k_{ij}$ can be represented as a point in an abstract nine-dimensional Euclidean space $E^9$. Let $D$ denote the domain of definition of the conductivity tensor $k_{ij}$ within this space, where the conductivity tensor is physically acceptable. As will be shown, certain constraints apply to the form of the conductivity tensor, which restricts its domain of definition $D$. Therefore, $D$ may be a nine-dimensional or lower-dimensional topological subset of $E^9$.

By decomposing the thermal conductivity tensor $k_{ij}$ into symmetric $k_{\left(ij\right)}$ and skew-symmetric $k_{\left[ij\right]}$ parts, one obtains

$$k_{ij}=k_{\left(ij\right)}+k_{\left[ij\right]},$$

(4)

where

$$k_{\left(ij\right)}={\displaystyle \frac{1}{2}}\left(k_{ij}+k_{ji}\right)=k_{\left(ji\right)},$$

(5)

$$k_{\left[ij\right]}={\displaystyle \frac{1}{2}}\left(k_{ij}-k_{ji}\right)=-k_{\left[ji\right]}.$$

(6)

Notice that here we have introduced parentheses surrounding a pair of indices to denote the symmetric part of a second-order tensor, whereas square brackets are associated with the skew-symmetric part. Since the general conductivity tensor $k_{ij}$ is specified by nine independent components, the tensors $k_{\left(ij\right)}$ and $k_{\left[ij\right]}$ are specified by six and three independent components, respectively.

3. Non-Singular Character of the Conductivity Tensor and Its Direct Consequence

From a physical standpoint, it is postulated that there is a one-to-one relationship between the temperature gradient $T_{,i}$ and the heat flux $q_i$ in Equation (2). This means for any arbitrary temperature gradient $T_{,i}$, there is one and only one heat flux vector $q_i$. This requires that the conductivity tensor $k_{ij}$ be invertible, i.e., that it has an inverse tensor $k_{ij}^{\left(-1\right)}$ such that

$$k{k}^{-1}=k^{-1}k=1,k_{ij}{k}_{jk}^{\left(-1\right)}=k_{ij}^{\left(-1\right)}{k}_{jk}={\delta }_{ij}.$$

(7)

Here ${\delta }_{ij}$ is Kronecker delta in three-dimensional physical space. It is noted that the condition (7) requires that the conductivity tensor $k_{ij}$ be non-singular, that is

$$\det \left(k\right)=\det \left[k_{ij}\right]\ne 0.$$

(8)

One notices that $\det \left(k\right)=0$ specifies an eight-dimensional hyper-surface $S^9$ in the abstract nine-dimensional space $E^9$. This hyper-surface divides the space $E^9$ into two exclusive subsets $E^{9+}$ and $E^{9-}$, where

$$\det \left[k_{ij}\right]>0\mathrm{in}{E}^{9+},$$

(9a)

$$\det \left[k_{ij}\right]<0\mathrm{in}{E}^{9-},$$

(9b)

$$\det \left[k_{ij}\right]=0\mathrm{on}{S}^9.$$

(9c)

It is obvious

$$E^{9+}\cup E^{9-}\cup S^9=E^9,$$

(10a)

$$D\cap S^9=Ø.$$

(10b)

The immediate consequence of the non-singular character of the conductivity tensor is as follows:

Theorem 1.

The conductivity tensor cannot be skew-symmetric.

In Appendix A, the well-known fact is demonstrated that a three-dimensional skew-symmetric second-order tensor is singular. As a result, the non-singular conductivity tensor $k_{ij}$ cannot be skew-symmetric, such that $k_{ij}=k_{\left[ij\right]}$. This means there is no material with a purely skew-symmetric conductivity tensor $k_{ij}=-k_{ji}$.

Let $D_{skew}$ denote the domain in $E^9$ corresponding to skew-symmetric tensors. It is noticed that $D_{skew}$ is a subset of the hyper-surface $S^9$, that is

$$D_{skew}\subset S^9.$$

(11)

It is also obvious that the domains $D$ and $D_{skew}$ are disjoint, that is

$$D\cap D_{skew}=Ø.$$

(12)

In the following sections, it is demonstrated based exclusively on tensor analysis and the physicality of the domain of definition of the conductivity tensor that the conductivity tensor $k_{ij}$ is symmetric positive-definite. It turns out that Theorem 1 provides the methodology direction.

4. Symmetric Character of the Conductivity Tensor

Since, based on Theorem 1, the conductivity tensor $k_{ij}$ cannot be skew-symmetric, it must have a non-zero symmetric part $k_{\left(ij\right)}$.

Theorem 2.

The conductivity tensor is symmetric.

The method of proof is by contradiction. If the conductivity tensor $k_{ij}$ of a material were not symmetric, it would have a non-zero skew-symmetric part $k_{\left[ij\right]}$ as

$$k_{ij}=k_{\left(ij\right)}+k_{\left[ij\right]},k_{\left[ij\right]}\ne 0.$$

(13)

However, this is not consistent with the fact that the domain of definition of the conductivity tensor $D$ is physical. This important point is demonstrated in more detail as follows.

The relation (13) shows that the symmetric part $k_{\left(ij\right)}$ can become as arbitrarily small as one wishes as long as the conductivity tensor $k_{ij}$ remains non-singular, that is

$$k_{ij}=k_{\left(ij\right)}+k_{\left[ij\right]},k_{\left[ij\right]}\ne 0,k_{\left(ij\right)}\to 0.$$

(14)

As a result, the tensor $k_{ij}$ in $D$ can approach the non-zero limit value $k_{\left[ij\right]}$ in many arbitrary ways, but it cannot become equal to $k_{\left[ij\right]}$. This means the general non-singular tensor $k_{ij}$ can become almost skew-symmetric by making the symmetric part $k_{\left(ij\right)}$ increasingly small, but it cannot become exactly skew-symmetric.

For convenience, let the skew-symmetric tensor $k_{\left[ij\right]}$ be represented by

$$\left[k_{\left[ij\right]}\right]=\left[\begin{array}{ccc}0& -{\kappa }_3& {\kappa }_2\\ {\kappa }_3& 0& -{\kappa }_1\\ -{\kappa }_2& {\kappa }_1& 0\end{array}\right],$$

(15)

where

$${\kappa }_1=k_{\left[32\right]},{\kappa }_2=k_{\left[13\right]}.{\kappa }_3=k_{\left[21\right]}.$$

(16)

Now, let us take the symmetric part $k_{\left(ij\right)}$ to be the diagonal tensors

$$\left[k_{\left(ij\right)}^{\epsilon I}\right]=\left[\begin{array}{ccc}\epsilon & 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[k_{\left(ij\right)}^{\epsilon II}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& \epsilon & 0\\ 0& 0& 0\end{array}\right],\left[k_{\left(ij\right)}^{\epsilon III}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& \epsilon \end{array}\right],$$

(17)

where $\epsilon$ is an arbitrary non-zero number, that is $\epsilon \ne 0$. The corresponding non-singular conductivity tenors are

$$\left[k_{ij}^{\epsilon I}\right]=\left[\begin{array}{ccc}\epsilon & -{\kappa }_3& {\kappa }_2\\ {\kappa }_3& 0& -{\kappa }_1\\ -{\kappa }_2& {\kappa }_1& 0\end{array}\right],\left[k_{ij}^{\epsilon II}\right]=\left[\begin{array}{ccc}0& -{\kappa }_3& {\kappa }_2\\ {\kappa }_3& \epsilon & -{\kappa }_1\\ -{\kappa }_2& {\kappa }_1& 0\end{array}\right],\left[k_{ij}^{\epsilon III}\right]=\left[\begin{array}{ccc}0& -{\kappa }_3& {\kappa }_2\\ {\kappa }_3& 0& -{\kappa }_1\\ -{\kappa }_2& {\kappa }_1& \epsilon \end{array}\right],$$

(18)

with the following determinants:

$$\det \left[k_{ij}^{\epsilon I}\right]={\kappa }_1^2\epsilon ,\det \left[k_{ij}^{\epsilon II}\right]={\kappa }_2^2\epsilon ,\det \left[k_{ij}^{\epsilon III}\right]={\kappa }_3^2\epsilon .$$

(19)

Let us assume that the individual components of the skew-symmetric tensor $k_{\left[ij\right]}$ given by Equation (15) are non-zero, that is

$${\kappa }_1\ne 0,{\kappa }_2\ne 0.{\kappa }_3\ne 0.$$

(20)

Therefore, the determinant of all three conductivity tensors $k_{ij}^{\epsilon I}$, $k_{ij}^{\epsilon II}$, and $k_{ij}^{\epsilon III}$ given in Equation (19) are non-zero. It is noticed that for positive values of $\epsilon$ these determinants are positive and for negative values of $\epsilon$ they are negative, that is

$$\mathrm{for}\epsilon >0,\det \left[k_{ij}^{\epsilon }\right]>0,k_{ij}^{\epsilon }\mathrm{in}{E}^{9+},$$

(21a)

$$\mathrm{for}\epsilon <0,\det \left[k_{ij}^{\epsilon }\right]<0,k_{ij}^{\epsilon }\mathrm{in}{E}^{9-}.$$

(21b)

Therefore, the conductivity tensors $k_{ij}^{\epsilon I}$, $k_{ij}^{\epsilon II}$, and $k_{ij}^{\epsilon III}$ in the forms of Equation (19) are acceptable conductivity tensors for both values of $\epsilon >0$ and $\epsilon <0$. This is true even when the positive or negative $\epsilon$ become infinitesimally small. This means materials with conductivity tensors $k_{ij}^{\epsilon I}$, $k_{ij}^{\epsilon II}$, and $k_{ij}^{\epsilon III}$ given by Equation (18) can exist for $\epsilon \to 0+$ and $\epsilon \to 0-$, but a material with skew-symmetric conductivity tensor $k_{\left[ij\right]}$ given by Equation (15) corresponding to $\epsilon =0$ cannot. However, this contradicts physical reality. The conductivity tensor $k_{ij}$ is a material property. If materials with conductivity tensors $k_{ij}^{\epsilon I}$, $k_{ij}^{\epsilon II}$, and $k_{ij}^{\epsilon III}$ given by Equation (18) can exist for $\epsilon \to 0+$ and $\epsilon \to 0-$, the material with the limiting skew-symmetric tensor $k_{ij}=k_{\left[ij\right]}$ given by Equation (15) corresponding to $\epsilon =0$ in Equation (18) should also exist. However, this contradicts Theorem 1, which states that the conductivity tensor $k_{ij}$ cannot be skew-symmetric. This contradiction indicates that the initial assumption regarding the existence of a skew-symmetric part $k_{\left[ij\right]}$ is invalid. Therefore, the conductivity tensor $k_{ij}$ must be symmetric, meaning that its skew-symmetric part $k_{\left[ij\right]}$ vanishes; that is

$$k_{ij}=k_{\left(ij\right)},k_{\left[ij\right]}=0.$$

(22)

This result establishes that the conductivity tensor is always symmetric, that is

$$k_{ji}=k_{ij}.$$

(23)

Therefore, the symmetric conductivity tensor is specified by six independent components.

It is worth noting that, in establishing all the theorems presented thus far, no reference has been made to the second law of thermodynamics or the Clausius–Duhem inequality. While the Clausius–Duhem inequality is commonly used to demonstrate the positive definiteness of the conductivity tensor, this property can also be deduced directly from the assumption of its non-singularity. This alternative approach will be discussed in the following section.

5. The Positive-Definite Character of the Symmetric Conductivity Tensor

Based on Appendix A, the three eigenvalues ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ of the symmetric conductivity tensor $k_{ij}$ are all real and their corresponding real eigenvectors are mutually orthogonal for distinct eigenvalues or can be taken mutually orthogonal for repeated eigenvalues. The following theorem is established using these facts.

Theorem 3.

The signs of eigenvalues of the symmetric conductivity tensor are invariants.

This means there must be a restriction on the signs of eigenvalues ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ in the physical domain of definition $D$. As mentioned in Appendix A, the symmetric conductivity tensor $k_{ij}$ can be always diagonalized by using the mutually orthogonal eigenvectors. As a result, without loss of generality the symmetric conductivity tensor $k_{ij}$ can be assumed diagonal, such that

$$\left[k_{ij}\right]=\left[\begin{array}{ccc}{k}_1& 0& 0\\ 0& k_2& 0\\ 0& 0& k_3\end{array}\right],$$

(24)

where the diagonal values are the corresponding real eigenvalues

$${\lambda }_1=k_1,{\lambda }_2=k_2,{\lambda }_3=k_3.$$

(25)

Note that

$$\det \left(k\right)=\det \left[k_{ij}\right]=k_1{k}_2{k}_3.$$

(26)

Since the conductivity tensor $k_{ij}$ in (24) is non-singular, the diagonal components $k_1$, $k_2$, and $k_3$ are non-zero. Therefore, it is enough to establish the following:

Lemma 1.

The signs of components of a diagonal conductivity tensor are invariants.

This means there must be a restriction on the signs of diagonal values $k_1$, $k_2$, and $k_3$. This lemma is established by contradiction. If there were no restriction on the sign of these components, for example component $k_1$, the determinant $\det \left[k_{ij}\right]=k_1{k}_2{k}_3$ could be negative or positive depending on the sign of $k_1$. This means the diagonal conductivity tensor given by Equation (24) is acceptable for both values of $k_1>0$ and $k_1<0$. Accordingly, this would also be true even when the positive or negative $k_1$ becomes infinitesimally small. Therefore, if there were no restriction on the sign of $k_1$, materials with the diagonal conductivity tensor could, in principle, exist for $k_1\to 0+$ and $k_1\to 0-$, where the limiting tensor

$$\left[k_{ij}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& k_2& 0\\ 0& 0& k_3\end{array}\right],$$

(27)

corresponding to $k_1=0$ in Equation (24), cannot exist. This means $k_1$ in the diagonal conductivity tensor $k_{ij}$ (24) can become increasingly small, but it cannot become exactly zero.

This outcome is inconsistent with physical reality. The conductivity tensor is a material property; thus, if materials with a diagonal conductivity tensor as in Equation (24) can exist for both positive and negative $k_1>0$ and $k_1<0$, then a material with $k_1=0$ as in Equation (27), should also be possible. However, this contradicts with the fundamental requirement that $k_1$ is non-zero, that is the conductivity tensor must be non-singular. This contradiction implies that the original assumption that there is no restriction on the sign of component $k_1$ is invalid. Therefore, $k_1$ cannot change sign. By similar reasoning, the other diagonal components, $k_2$ and $k_3$, must also maintain a consistent sign. In other words, the diagonal elements—or equivalently, the eigenvalues—of a diagonal conductivity tensor cannot change sign.

This result in turn proves Theorem 3: that the eigenvalues of the symmetric conductivity tensor do not change sign. The important consequence of Theorem 3 is as follows:

Theorem 4.

The symmetric conductivity tensor is positive-definite.

For an isotropic material, the symmetric conductivity tensor $k_{ij}$ reduces to

$$k_{ij}=k{\delta }_{ij}.$$

(28)

Since the scalar conductivity $k$ for an isotropic material is positive, the eigenvalues of the isotropic tensor (28) are positive:

$${\lambda }_1={\lambda }_2={\lambda }_3=k>0.$$

(29)

This clearly shows that, for the general anisotropic conductivity tensor to reduce to the isotropic case, it is necessary that the eigenvalues ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ always be positive. Therefore, the strictly non-singular symmetric conductivity tensor $k_{ij}$ is positive-definite.

As mentioned previously, no reference has been made thus far to the second law of thermodynamics or the Clausius–Duhem inequality. As is known, the Clausius–Duhem inequality is derived by combining the first and second law of thermodynamics [9], and is expressed as

$$q_i{T}_{,i}\le 0.$$

(30)

By using the relation Equation (2) for heat flux, the Clausius–Duhem inequality Equation (30) can be written as

$$k_{ij}{T}_{,i}{T}_{,j}\ge 0.$$

(31)

Therefore, the Clausius–Duhem inequality in the form of Equation (31) is consistent with the established symmetric positive-definite character of the conductivity tensor. Note that the positive character of the quadratic form in Equation (31) does not prove the symmetry of the general conductivity tensor $k_{ij}$ in Equation (2). It only establishes the positive-definite character of its symmetric part $k_{\left(ij\right)}$.

Although the second law of thermodynamics or the Clausius–Duhem inequality did not have any direct role in the establishment of the symmetric and positive-definite characters of the conductivity tensor, this law was already satisfied by Fourier’s heat conduction law for linear isotropic material, as stated in Equation (2). This indicates that the second law has been indirectly used in the establishment of Theorem 4.

6. Conclusions

Using arguments from tensor analysis and the physical constraints on the domain of definition of the conductivity tensor, the symmetric character of the conductivity tensor for linear anisotropic material has been established. Notably, the non-singular character of the conductivity tensor plays a fundamental role in this result. It has been shown that assuming a fully general form of the conductivity tensor leads to a contradiction with its required non-singularity, thereby necessitating that the tensor be both symmetric and positive-definite. This proof demonstrates that classical continuum mechanics itself provides a sufficient mathematical foundation for the symmetry of the conductivity tensor.

The methodology employed here highlights the subtle and often overlooked characteristics of physical tensors in three-dimensional space. This mathematical perspective, which is rooted in the physicality of a tensor’s domain of definition, can yield significant results that are not accessible through traditional approaches in classical physics. As demonstrated in this work, the tensorial approach challenges the widespread belief that the symmetry of the conductivity tensor must be derived from additional physical postulates, such as the requirement that the thermal work functional attain a weak relative minimum at equilibrium.

It should be noted that the results presented in this article remain valid under all conditions as long as the Duhamel heat conduction law, Equation (2), holds and a one-to-one relationship exists between the temperature gradient $T_{,j}$ and the heat flux $q_i$. This is characteristic of a true continuum, in which the conductivity tensor $k_{ij}$ is non-singular. However, if such a one-to-one relationship does not exist, the conductivity tensor $k_{ij}$ may become singular and, consequently, is not necessarily symmetric. This situation can be regarded as a deviation from a true continuum mechanical system, as observed in materials with a significant distribution of micro-discontinuities in their structure or with point-like distributions of molecules. Powers [10] investigated a general form for the conductivity tensor that encompassed both symmetric and skew-symmetric cases.

Remarkably, the presented proof also reveals a philosophical aspect connected to the geometry of physical space. Specifically, the inherent singularity of skew-symmetric second-order tensors in three-dimensional space of our universe plays a critical role in ensuring the symmetry of the conductivity tensor. As explained in Appendix A, such tensors are necessarily singular in any odd-dimensional space, including our universe. By contrast, in even-dimensional spaces, second-order skew-symmetric tensors are not necessarily singular. This implies that, in a hypothetical even-dimensional universe, such as one with two or four dimensions, the symmetry of the conductivity tensor would not be guaranteed without additional physical assumptions.

As expected, the same reasoning can be extended to other physical systems. The symmetry of the resistivity tensor in Ohm’s law for electrical conduction and the diffusion coefficient tensor in Fick’s law for mass transfer, and other diffusive systems, can be established using analogous arguments.