Non-Equilibrium Continuous Description

Appendices

On the Extension of the Gibbs Relation to Non-Equilibrium

For the specific Helmholtz energy the ordinary Gibbs relation in equilibrium was found to be

$$ \delta f(T,v,\xi)=-s({{\mathbf{r}}})\,\delta T+ {\sum\limits_{i=1}^{n-1}{\psi_{i}\,\delta\xi_{i}({{\mathbf{r}}})}}- p({{\mathbf{r}}})\,\delta v({{\mathbf{r}}}) $$

(3.62)

Spatial Gibbs relations in equilibrium was found to be

$$\nabla f({{\mathbf{r}}})={\sum\limits_{i=1}^{n-1}{\psi_{i}\,\nabla\xi_{i}({{\mathbf{r}}})}}-p({{\mathbf{r}}})\,\nabla v({{\mathbf{r}}})+v({{\mathbf{r}}})\,{\frac{{\partial \gamma_{{\alpha \beta }}({\mathbf{r}})}}{{\partial x_{\alpha } }}}$$

(3.63)

One may wonder why to use the Gibbs relations for the specific internal energy, not for the specific Helmholtz energy, to extend them to non-equilibrium analysis. Following the same procedure, as in Sect. 3.3, we can extend the Gibbs relations for the specific Helmholtz energy to non-equilibrium in the following way

$${\frac{\partial f({{\mathbf{r}}}, t)}{\partial t}}=-s({{\mathbf{r}}}, t)\,{\frac{\partial T({{\mathbf{r}}},t)}{\partial t}} +{\sum\limits_{i=1}^{n-1}{\psi_{i}({{\mathbf{r}}},t)\,{\frac{\partial \xi_{i}({{\mathbf{r}}}, t)}{\partial t}}}}-p({{\mathbf{r}}}, t)\,{\frac{\partial v({{\mathbf{r}}}, t)}{\partial t}}$$

(3.64)

$$\nabla f({{\mathbf{r}}}, t)={\sum\limits_{i=1}^{n-1}{\psi_{i}({{\mathbf{r}}}, t)\,\nabla\xi_{i}({{\mathbf{r}}},t)}}-p({{\mathbf{r}}}, t)\,\nabla v({{\mathbf{r}}},t)+v({{\mathbf{r}}}, t)\,{\frac{{\partial \gamma _{{\alpha \beta }}({\mathbf{r}},t)}}{{\partial x_{\alpha} }}}$$

(3.65)

For the ordinary Gibbs relations, like in homogeneous description, there is no preference in the thermodynamic potential. Provided Eq. 3.6, equilibrium ordinary Gibbs relations 3.9 and 3.62 are equivalent. The non-equilibrium relation between these potentials

$$ u({{\mathbf{r}}},\,t)=f({{\mathbf{r}}},\,t)+s({{\mathbf{r}}},\,t) T({{\mathbf{r}}},\,t) $$

(3.66)

makes non-equilibrium ordinary Gibbs relations (3.11) and (3.64) to be equivalent as well.

The situation is different for the spatial Gibbs relations, however. Provided Eq. 3.6, equilibrium spatial Gibbs relations Eqs. 3.10 and 3.63 are equivalent. The non-equilibrium relation Eq. 3.66 between these potentials makes, however, non-equilibrium spatial Gibbs relations (3.12) and (3.65) to be $not$ equivalent. The reason for that is that the equilibrium spatial Gibbs relation for the specific Helmholtz energy, as given in Eq. 3.63, does not contain the term, proportional to ${\nabla}T,$ since ${\nabla}T=0$ in equilibrium. In non-equilibrium ${\nabla}T({\bf r}, t)\neq0$ and, as one can see from Eqs. 3.12 and 3.66, $\nabla f({\bf r}, t)$ contains such a term.

We see, that the Gibbs relations for the specific internal energy describe the system more adequately since they do not suffer from the unaccounted effect of possible temperature changes. Because of this reason we should use the Gibbs relations for the specific internal energy, not the specific Helmholtz energy, to extend them to non-equilibrium. Of course, this does not make the internal energy to be somehow preferred thermodynamic potential in non-equilibrium. If one does not disregards terms with ${\nabla}T$ or others which are zero in equilibrium, one will see the fully equivalence between all the thermodynamic potentials, as it should be.

Two-Dimensional Isotropic Components in the 3D Tensorial Quantities

As in Sect. 3.6 we shall use the special notation for the tensorial quantities of different order and different behavior in this section. Any tensorial quantity is denoted as $Q^{(d\,{\rm{r}})}.$ Here $d$ indicates the dimensionality of the space, in which the quantity is being considered, and can be either 3 or 2 here. $\hbox{r}$ indicates the rank of the tensorial quantity, and can be $\hbox{s}$ for scalar, $\hbox{v}$ for vectorial or $\hbox{t}$ for tensorial quantities. For example, $Q^{(2\,{\rm t})}$ indicates the two-dimensional tensor, i.e. the quantity $\left(\begin{array}{ll} q_{11} & q_{12}\\q_{21} & q_{22}\end{array}\right),$ where $q_{ij}$ are numbers, and $Q^{(3\,{\rm v})}$ indicates the three-dimensional vector, i.e. the quantity $(q_{1}, q_{2}, q_{3}),$ where $q_{i}$ are numbers. Scalars are the numbers irrespectively of the dimensionality of the space, so they will be denoted simply by $Q^{({\rm s})}.$

Some quantities reveal the tensorial behavior of a some rank in $d$-dimensional space only under some specified transformations, while in general they do not. In this section we are interested only in rotations around and reflections with respect to some constant vector $N^{(3\,{\rm v})}$ in three-dimensional space. We will denote quantities which reveal the tensorial behavior of rank $\hbox{r}$ under these transformations by $Q^{(d\,{\rm r}_{\,N})}.$

We show how in presence of the constant vector $N^{(3\,{\rm v})}$ one can split the tensorial quantity $Q^{(3\,{\rm r})}$ into a combination of the tensorial quantities $Q^{(2\,{\rm r}_{\,N})}.$ Without loss of generality we will assume that $N^{{\rm (3\,v)}}=(1,0,0).$

From 3D vector $V^{(3\,{\rm v})}$ one can construct the following quantities, which are linear in $V^{(3\,{\rm v})}$: one scalar quantity

$$ V^{({\rm s}_{\,N})}\equiv V^{(3\,{\rm v})}\cdot N^{(3\,{\rm v})}= V^{(3\,{\rm v})}_{1} $$

and one vectorial quantity

$$ V^{(3\,{\rm v}_N)}\equiv V^{(3\,{\rm v})}- V^{({\rm s}_{\,N})}N^{(3\,{\rm v})}=\left(0,V^{(3\,{\rm v})}_{2}, V^{(3\,{\rm v})}_{3}\right) $$

which is perpendicular to the $N^{(3\,{\rm v})}.$ Denoting $V^{(2\,{\rm v}_{\,N})}\equiv(V^{(3\,{\rm v})}_{2}, V^{(3\,{\rm v})}_{3})$ we can write that $V^{(3\,{\rm v}_N)}=(0,V^{(2\,{\rm v}_{\,N})}).$ Thus,

$$ V^{(3\,{\rm v})}=V^{({\rm s}_{\,N})}N^{(3\,{\rm v})}+ V^{(3\,{\rm v}_{\,N})}=(V^{({\rm s}_{\,N})},V^{(2\,{\rm v}_{\,N})}) $$

(3.67)

From 3D tensor $T^{(3\,{\rm t})}$ one can construct the following quantities, which are linear in $T^{(3\,{\rm t})}$: 2 scalar quantity,

$$ T^{(\hbox{s})}_{0}\equiv\hbox{Tr}\,T^{(3\,{\rm t})}= T^{(3\,{\rm t})}_{11}+T^{(3\,{\rm t})}_{22}+T^{(3\,{\rm t})}_{33} $$

and

$$ T^{({\hbox{s}}_{\,N})}_{1}\equiv N^{(3\,{\rm v})}\cdot T^{(3\,{\rm t})} \cdot N^{(3\,{\rm v})}=T^{(3\,{\rm t})}_{11} $$

two vectorial quantities

$$ T^{(3\,{\hbox{v}}_{\,N})}_{l}\equiv N^{(3\,{\rm v})}\cdot T^{(3\,{\rm t})}- T^{{(\hbox{s}}_{\,N})}_{1} N^{(3\,{\rm v})}=\left(0,T^{(3\,{\rm t})}_{12}, T^{(3\,{\rm t})}_{13}\right) $$

and

$$ T^{(3\,{\hbox{v}}_{\,N})}_{r}\equiv T^{(3\,{\rm t})} \cdot N^{(3\,{\rm v})} - T^{({\rm s}_{\,N})}_{1}N^{(3\,{\rm v})}=\left(0,T^{(3\,{\rm t})}_{21}, T^{(3\,{\rm t})}_{31}\right) $$

(which are equal, if $T^{(3\,{\rm t})}$ is symmetric); and tensorial quantity

$$ \begin{aligned} T^{(3\,{\hbox{t}}_{\,N})}&\equiv T^{(3\,{\rm t})} - T^{{\rm s}_{\,N}}_{1}N^{(3\,{\rm v})}N^{(3\,{\rm v})}- T^{(3\,{{\rm v}}_{\,N})}_{l}N^{(3\,{\rm v})}- N^{(3\,{\rm v})}T^{(3\,{{\rm {v}\,}_{N}})}_{r}\\ &=\left(\begin{array}{lll} 0 & 0 & 0 \\ 0 & T^{(3\,{\rm t})}_{22} & T^{(3\,{\rm t})}_{23} \\ 0 & T^{(3\,{\rm t})}_{32} & T^{(3\,{\rm t})}_{33} \end{array}\right) \end{aligned} $$

Denoting

$$ \begin{aligned} T^{(2\,{\hbox{v}}_{\,N})}_{l}&\equiv\left(T^{(3\,{\rm t})}_{12}, T^{(3\,{\rm t})}_{13}\right)\quad T^{(2\,{\rm t}_{\,N})}\equiv \left(\begin{array}{ll} T^{(3\,{\rm t})}_{22} & T^{(3\,{\rm t})}_{23} \\ T^{(3\,{\rm t})}_{32} & T^{(3\,{\rm t})}_{33} \end{array}\right)\\ T^{(2\,{\rm v}_{\,N})}_{r}&\equiv\left(T^{(3\,{\rm t})}_{21}, T^{(3\,{\rm t})}_{31}\right) \end{aligned} $$

we can write that

$$ \begin{aligned} T^{(3\,{\rm v}_{\,N})}_{l}&=\left(0,T^{(2\,{\rm v}_{\,N})}_{l}\right)\quad T^{(3\,{\rm t}_{\,N})}=\left(\begin{array}{ll} 0 & 0 \\ 0 & T^{(2\,{\rm t}_{\,N})} \end{array}\right)\\ T^{(3\,{\rm v}_{\,N})}_{r}&=\left(0,T^{(2\,{\rm v}_{\,N})}_{r}\right) \end{aligned} $$

Thus ⁵

$$ \begin{aligned} T^{(3\,{\rm t})}&=T^{({\rm s}_{\,N})}_{1}N^{(3\,{\rm v})} N^{(3\,{\rm v})}+T^{(3\,{\rm v}_{\,N})}_{l} N^{(3\,{\rm v})}+N^{(3\,{\rm v})}T^{(3\,{\rm v}_{\,N})}_{r} +T^{(3\,{\rm t}_{\,N})}\\ &=\left(\begin{array}{ll} T^{({\rm s}_N)}_{1} & T^{(2\,{\rm v}_{\,N})}_{l} \\ T^{(2\,{\rm v}_{\,N})}_{r} & T^{(2\,{\rm t}_{\,N})}\end{array}\right) \end{aligned} $$

(3.68)

Tensor $T^{(2\,{\rm t}_{\,N})}$ still contains the scalar part

$$ T^{({\rm s}_{\,N})}_{2}\equiv\hbox{Tr}\,T^{(2\,{\rm t}_{\,N})}= T^{(3\,{\rm t})}_{22}+T^{(3\,{\rm t})}_{33} $$

which obeys the relation

$$ T^{(\hbox{s})}_{0}=T^{({\hbox{s}}_{\,N})}_{1}+T^{({\hbox{s}}_{\,N})}_{2} $$

(3.69)

Two of these three scalar quantities are linearly independent, and one can use any pair. Since we want to reduce all the quantities to the form $Q^{(2\,{\rm r}_{\,N})}$ we will use $T^{({\rm s}_{\,N})}_{1}$ and $T^{({\rm s}_{\,N})}_{2}$ as independent pair. Introducing the traceless tensor

$$ {\mathring{T}}^{(2\,{\rm t}_{\,N})}\equiv T^{(2\,{\rm t}_{\,N})}-{\frac{1}{2}} T^{({\rm s}_{\,N})}_{2}U^{(2\,{\rm t})}=\left(\begin{array}{ll} T^{(3\,{\rm t})}_{22}-{\frac{1}{2}}T^{({\rm s}_{\,N})}_{2} & T^{(3\,{\rm t})}_{23} \\ T^{(3\,{\rm t})}_{32} & T^{(3\,{\rm t})}_{33}-{\frac{1}{2}}T^{({\rm s}_{\,N})}_{2} \end{array}\right) $$

and

$${\mathring{T}}^{(3\,{\hbox{t}}_{\,N})} \equiv T^{(3\,{\hbox{t}}_{\,N})}- {\frac{1}{2}}T^{{({\hbox{s}}}_{\,N})}_{2}U^{(3\,{\hbox{t}}_{\,N})}= \left(\begin{array}{ll} 0 & 0 \\ 0& {\mathring{T}}^{(2\,{\hbox{t}}_{\,N})} \end{array}\right)$$

where

$$U^{(2\,\hbox{t})}\equiv\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)\quad U^{(3\,\hbox{t}_{\,N})}\equiv \left(\begin{array}{ll} 0 & 0 \\ 0& U^{(2\,\hbox{t})} \end{array}\right) $$

we can write a 3D tensor as

$$\begin{aligned} T^{(3\,{\rm t})}&=T^{({\rm s}_{\,N})}_{1}N^{(3\,{\rm v})} N^{(3\,{\rm v})}+{\frac{1}{2}}T^{({\rm s}_{\,N})}_{2} U^{(3\,{\rm t}_{\,N})} +{\mathring{T}}^{(3\,{\rm t}_{\,N})} \\ &\quad+ T^{(3\,{\rm v}_{\,N})}_{l}N^{(3\,{\rm v})} + N^{(3\,{\rm v})}T^{(3\,{\rm v}_{\,N})}_{r}\\ &=\left(\begin{array}{ll} T^{({\rm s}_{\,N})}_{1} & T^{(2\,{\rm v}_{\,N})}_{l} \\ T^{(2\,{\rm v}_{\,N})}_{r} & {\frac{1}{2}} T^{({\rm s}_{\,N})}_{2}U^{(2\,{\rm t})} +{\mathring{T}}^{(2\,{\rm t}_{\,N})} \end{array}\right) \end{aligned} $$

(3.70)