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Example

We consider in the framework of classical mechanics the system of Brownian particles and fluid with the Hamiltonian

<math>\mathcal{H} = \mathcal{H}_1 + \mathcal{H}_2</math>

with

<math>\mathcal{H}_1 = \sum_\alpha \left[ \frac{\mathbf{P}_\alpha^2}{2M} + \sum_q V(|\mathbf{R}_\alpha-\mathbf{r}_q|) \right]</math>

and

<math>\mathcal{H}_2 = \sum_i \left[ \frac{\mathbf{p}_i^2}{2m} + \sum_{f(<i)} V(|\mathbf{r}_i-\mathbf{r}_j|) \right]</math>

where<math>\mathbf{P}_\alpha, \mathbf{R}_\alpha, M, \mathbf{p}_i, \mathbf{r}_i</math> and <math>m</math> and the momentum, position and mass of the <math>\alpha</math>-th Brownian particle and the <math>i</math>-th fluid particle respectively.

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<math> s_k \equiv 0 \pmod{m} n\ \bmod M = r</math>

<math>\int_{-N}^{N} e^x\, dx</math>

<math>A \xleftarrow{n+\mu-1} B\xrightarrow[T]{n\pm i-1} C</math>

<math>\oint_{C} x^3\, dx + 4y^2\, dy</math>

<math>\begin{cases}

   3x + 5y +  z = 1\\
   7x - 2y + 4z = 2\\
  -6x + 3y + 2z = 3

\end{cases}</math>

<math>\phi_n(\kappa)

= \frac{1}{4\pi^2\kappa^2} \int_0^\infty
\frac{\sin(\kappa R)}{\kappa R}
\frac{\partial}{\partial R}
\left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR</math>

<math>{}_pF_q(a_1,...,a_p;c_1,...,c_q;z)

= \sum_{n=0}^\infty
\frac{(a_1)_n\cdot\cdot\cdot(a_p)_n}{(c_1)_n\cdot\cdot\cdot(c_q)_n}
\frac{z^n}{n!}</math>

<math> E=mc^2 </math>

( ^ω^)