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\documentclass[12pt]{article}
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\begin{document}
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$$
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{\vec F}_i = - \partial U / \partial {\vec r}_i + {\vec F}_{langevin} - \nabla P_e/n_{ion}
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$$
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\end{document}
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\documentclass[12pt]{article}
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\begin{document}
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$$
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\nabla_x P_e = \left[\frac{C_e{}T_e(x)\lambda}{(x+\lambda)^2} + \frac{x}{x+\lambda}\frac{(C_e{}T_e)_{x+\Delta x}-(C_e{}T_e)_{x}}{\Delta x} \right]
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$$
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\end{document}
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\documentclass[12pt]{article}
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\begin{document}
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$$
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C_e = C_0 + (a_0 + a_1 X + a_2 X^2 + a_3 X^3 + a_4 X^4) \exp (-(AX)^2)
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$$
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\end{document}
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\documentclass[12pt]{article}
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\begin{document}
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$$
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C_e \rho_e \frac{\partial T_e}{\partial t} =
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\bigtriangledown (\kappa_e \bigtriangledown T_e) -
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g_p (T_e - T_a) + g_s T_a' + \theta (x-x_{surface})I_0 \exp(-x/l_{skin})
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$$
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\end{document}
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