\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}
