4.1.1. Non-Equilibrium Thermodynamics of Conformation Tensor

First, let the state variables be density of matter *ρ*, momentum density **u**, entropy density *s* and conformation tensor *cij*, which expresses correlations of dumbbell orientation and prolongation. Poisson bracket expressing kinematics of these state variables is

$$\begin{split} \{A,B\}^{\{\epsilon\}} &= \quad \{A,B\}^{\{FM\}} + \int \mathrm{d}\mathbf{r} \boldsymbol{\varepsilon}\_{ij} \left( \partial\_{k} A\_{\mathcal{E}\_{ij}} B\_{u\_{k}} - \partial\_{k} B\_{\mathcal{E}\_{ij}} A\_{u\_{k}} \right) \\ &+ \int \, \mathrm{d}\mathbf{r} \boldsymbol{\varepsilon}\_{ij} \left( \left( A\_{\mathcal{E}\_{kj}} + A\_{\mathcal{E}\_{jk}} \right) \partial\_{i} B\_{u\_{k}} - \left( B\_{\mathcal{E}\_{kj}} + B\_{\mathcal{E}\_{jk}} \right) \partial\_{i} A\_{u\_{k}} \right), \end{split} \tag{31}$$

where *<sup>c</sup>ij* was identified with *cij* for simplicity of notation. Bracket {•, •}(*FM*) is the fluid mechanics Poisson bracket (A3). Note that the conformation tensor is related to the left Cauchy–Green tensor **B** by **c** = *ρ***B**. Reversible evolution equations for state variables **x** = (*ρ*, **u**,*s*, **c**) implied by bracket (31) are

$$\begin{cases} \frac{\partial \rho}{\partial t} &= -\partial\_i \left( \rho E\_{\overline{u\_i}} \right), \\\\ \end{cases} \tag{32a}$$

*∂ui <sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*∂<sup>j</sup> uiEuj* − *ρ∂iE<sup>ρ</sup>* − *uj∂iEuj* − *s∂iEs* − 

$$-\mathfrak{c}\_{jk}\partial\_{\bar{i}}E\_{\varepsilon^{jk}} + \partial\_{k}\left(\mathfrak{c}\_{kj}\left(E\_{\varepsilon\_{ij}} + E\_{\varepsilon\_{ji}}\right)\right),\tag{32b}$$

$$\begin{aligned} \frac{\partial s}{\partial t} &= -\partial\_i \left( s E\_{\mu\_i} \right), \\ \sim & \end{aligned} \tag{32c}$$

$$\frac{\partial \mathfrak{c}\_{ij}}{\partial t} = -\eth\_{\mathbf{k}} \left( \mathfrak{c}\_{i\not\!j} E\_{u\_{\mathbf{k}}} \right) + \mathfrak{c}\_{k\not\!j} \eth\_{\mathbf{k}} E\_{u\_{\mathbf{i}}} + \mathfrak{c}\_{k\not\!i} \eth\_{\mathbf{k}} E\_{u\_{\mathbf{j}}} \tag{32d}$$

where *E* is the total energy of the system.

With energy

$$E = \int \mathrm{d}\mathbf{r} \left[ \frac{\mathbf{u}^2}{2\rho} + \varepsilon(\rho, s, \mathbf{c}) \right],\tag{33}$$

where internal energy *ε* still remains unspecified, evolution equation (32d) can be rewritten in terms of the upper-convected time-derivative as

$$\stackrel{\nabla}{\mathbf{c}} = -\mathbf{c} \nabla \cdot \mathbf{v}, \qquad \mathbf{v} = \frac{\mathbf{u}}{\rho'} \tag{34}$$

which is the reversible part of Maxwell rheological model [29].

Considering a suspension of Hookean dumbbells, entropy is

$$S = \int \mathrm{d}\mathbf{r} \left[ \frac{1}{2} n k\_B \ln \det \mathbf{c} + s \left( n, \mathbf{c} - \frac{\mathbf{u}^2}{2\rho} - \frac{1}{2} H \, \mathrm{Tr} \mathbf{c} \right) \right],\tag{35}$$

as derived for instance in [14]. Parameter *H* is the spring constant of the dumbbells, and *n* represents concentration of dumbbells. Derivative of this entropy with respect to **c** is

$$\mathbf{c}\_{ij}^{\*} = \frac{\partial S}{\partial \mathbf{c}\_{ij}} = \frac{1}{2} k\_B n \mathbf{c}\_{ij}^{-1} - \frac{1}{2} \frac{H}{T} \delta\_{ij\prime} \tag{36}$$

which is equal to zero for

$$
\overline{\mathbf{c}} = \frac{k\_B T n}{H} \mathbf{I}.\tag{37}
$$

This is the MaxEnt value of **c**. Note that inverse temperature *T*−<sup>1</sup> is identified as derivative of entropy with respect to energy density.

Dissipation potential can be prescribed as

$$
\Delta^{(\mathsf{c})} = \int d\mathbf{r} \Lambda\_{\mathsf{c}} c\_{i\bar{j}} c\_{i\bar{k}}^{\*} c\_{j\bar{k}\prime}^{\*} \tag{38}
$$

the derivative of which is

$$
\Delta\_{c\_{ij}^\*}^{(\mathfrak{c})} = 2\Lambda\_c c\_{ik} c\_{kj}^\*.\tag{39}
$$

The evolution equation of **c** then gains an irreversible term,

$$\frac{\partial \Xi^{(\mathbf{c})}}{\partial c\_{ij}^{\*}}\Big|\_{\mathbf{c}\_{\*} = \mathbf{c}\_{\mathbf{c}}} = -\frac{\Lambda\_{\mathbf{c}}H}{T} \left(c\_{ij} - \frac{k\_{B}Tn}{H}\delta\_{ij}\right). \tag{40}$$

After the transformation to the energetic representation (conjugates with respect to energy rather than entropy), the sum of the reversible and irreversible contributions to evolution equations for (*ρ*, **u**, **c**,*s*) is prepared for the DynMaxEnt reduction.
