B. Appendix 2: The Gay-Berne potential

In order to evaluate the above expressions for the irreversible work in shear flow, elongational flow, and heat flow, we have simulated a coarse grained model system composed of molecules interacting via a purely repulsive version of the commonly used Gay-Berne potential [16, 17, 21],

$$U(\mathbf{r}\_{12}, \hat{\mathbf{u}}\_1, \hat{\mathbf{u}}\_2) = 4e(\hat{\mathbf{r}}\_{12}, \hat{\mathbf{u}}\_1, \hat{\mathbf{u}}\_2) \left( \frac{\sigma\_0}{r\_{12} - \sigma(\hat{\mathbf{r}}\_{12}, \hat{\mathbf{u}}\_1, \hat{\mathbf{u}}\_2) + \sigma\_0} \right)^{18},\tag{A.7}$$

where r<sup>12</sup> ¼ r<sup>2</sup> � r<sup>1</sup> is the distance vector from the center of mass of molecule 1 to the center of mass of molecule 2, ^r<sup>12</sup> is the unit vector in the direction of r12, r<sup>12</sup> is the length of the vector r12, and u^<sup>1</sup> and u^<sup>2</sup> are the unit vectors parallel to the axes of revolution of molecule 1 and molecule 2. The parameter σ<sup>0</sup> is the length of the axis perpendicular to the axis of revolution, that is, the minor axis of a calamitic ellipsoid of revolution and the major axis of a discotic ellipsoid of revolution. The strength and range parameters are given by

$$\begin{aligned} \varepsilon(\hat{\mathbf{r}}\_{12}, \hat{\mathbf{u}}\_{1}, \hat{\mathbf{u}}\_{2}) &= \varepsilon\_{0} \left[ 1 - \chi^{2} (\hat{\mathbf{u}}\_{1} \cdot \hat{\mathbf{u}}\_{2})^{2} \right]^{-1/2} \\ & \left\{ 1 - \frac{\chi'}{2} \left[ \frac{(\hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{1} + \hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{2})^{2}}{\mathbf{1} + \chi' \hat{\mathbf{u}}\_{1} \cdot \hat{\mathbf{u}}\_{2}} + \frac{(\hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{1} - \hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{2})^{2}}{\mathbf{1} - \chi' \hat{\mathbf{u}}\_{1} \cdot \hat{\mathbf{u}}\_{2}} \right] \right\}^{2} \quad \text{(A.8a)} \end{aligned}$$

and

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics… DOI: http://dx.doi.org/10.5772/intechopen.80977

$$\sigma(\hat{\mathbf{r}}\_{12}, \hat{\mathbf{u}}\_{1}, \hat{\mathbf{u}}\_{2}) = \sigma\_{0} \left\{ 1 - \frac{\chi}{2} \left[ \frac{(\hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{1} + \hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{2})^{2}}{\mathbf{1} + \chi \hat{\mathbf{u}}\_{1} \cdot \hat{\mathbf{u}}\_{2}} + \frac{(\hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{1} - \hat{\mathbf{r}}\_{12} \cdot \hat{\mathbf{u}}\_{2})^{2}}{\mathbf{1} - \chi \hat{\mathbf{u}}\_{1} \cdot \hat{\mathbf{u}}\_{2}} \right] \right\}^{-1/2},\tag{A.8b}$$

where the parameter <sup>χ</sup> is equal to <sup>κ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>=</sup> <sup>κ</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> and <sup>κ</sup> is the ratio of the axis of revolution and the axis perpendicular to this axis, χ<sup>0</sup> is equal to <sup>κ</sup>01=<sup>2</sup> � <sup>1</sup> � �<sup>=</sup> <sup>κ</sup>01=<sup>2</sup> <sup>þ</sup> <sup>1</sup> � � and <sup>κ</sup><sup>0</sup> is the ratio of the potential energy minima of the side by side and end to end configurations of calamitic ellipsoids or the ratio of the edgeto-edge and face-to-face configurations of discotic ellipsoids, and ε<sup>0</sup> denotes the depth of the potential minimum in the cross configuration, where ^r12, u^1, and u^<sup>2</sup> are perpendicular to each other. The parameters κ and κ<sup>0</sup> have been given the values 3 and 5, respectively, for the calamitic ellipsoids and 1/3 and 1/5 for the discotic ellipsoids.

The denominators in Eqs. (A.8a) and (A.8b) are never equal to zero because the absolute value of the scalar product u^<sup>1</sup> � u^<sup>2</sup> is less than or equal to one since u^<sup>1</sup> and u^<sup>2</sup> are unit vectors, and the absolute values of the parameters χ and χ<sup>0</sup> are less than one. The ordinary Lennard-Jones potential is recovered in the limit when κ and κ<sup>0</sup> go to one. Note that, the potential is purely repulsive, so there are no potential minima but the value of κ<sup>0</sup> optimized for the attractive Gay-Berne potential has been retained. The transport properties of this system of purely repulsive soft ellipsoids are similar to those of a system where the molecules interact according to the conventional Gay-Berne potential with attraction as well, so the results are still relevant.
