Merge remote-tracking branch 'origin/master' into cmake_improvements

\documentclass[12pt]{article}

\begin{document}

$$

  v(t+\frac{\Delta t}{2}) = v(t) + \frac{\Delta t}{2}\cdot a(t),

$$

$$

  r(t+\Delta t) = r(t) + \Delta t\cdot v(t+\frac{\Delta t}{2}),

$$

$$

  a(t+\Delta t) = \frac{1}{m}\cdot F\left[ r(t+\Delta t), v(t) +\lambda \cdot \Delta t\cdot a(t)\right],

$$

$$

  v(t+\Delta t) = v(t+\frac{\Delta t}{2}) + \frac{\Delta t}{2}\cdot a(t+\Delta t)

$$

\end{document}

\documentclass[12pt]{article}

\begin{document}

$$

  \mathbf{F}_{ij}^{C} = \alpha_{ij}{\omega_{C}}(r_{ij})\mathbf{e}_{ij},

$$

$$

  \mathbf{F}_{ij}^{D} = -\gamma {\omega_{D}}(r_{ij})(\mathbf{e}_{ij} \cdot \mathbf{v}_{ij})\mathbf{e}_{ij},

$$

$$

  \mathbf{F}_{ij}^{R} = \sigma {\omega_{R}}(r_{ij}){\xi_{ij}}\Delta t^{-1/2} \mathbf{e}_{ij},

$$

$$

  \omega_{C}(r) = 1 - r/r_c,

$$

$$

  \alpha_{ij} = A\cdot k_B(T_i + T_j)/2,

$$

$$  

  \omega_{D}(r) = \omega^2_{R}(r) = (1-r/r_c)^s,

$$

$$  

  \sigma_{ij}^2 = 4\gamma k_B T_i T_j/(T_i + T_j),

$$

\end{document}