「メインページ」の版間の差分
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== Hamiltonian == | == Hamiltonian == | ||
− | We consider in the framework of classical mechanics the system of Brownian | + | We consider in the framework of classical mechanics the system of [[Brownian particle]]s and fluid with the [[Hamiltonian]] |
:<math>\mathcal{H} = \mathcal{H}_1 + \mathcal{H}_2</math> | :<math>\mathcal{H} = \mathcal{H}_1 + \mathcal{H}_2</math> |
2007年4月5日 (木) 02:03時点における版
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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^)
Hamiltonian
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</math>, <math>\mathbf{R}_\alpha</math>, <math>M</math>, <math>\mathbf{p}_i</math>, <math>\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; <math>V(|\mathbf{R}_\alpha-\mathbf{r}_i|)</math> is the interaction potential of the Brownian particle with the fluid particle; <math>u(|\mathbf{r}_i-\mathbf{r}_j|)</math> is the interaction energy of two fluid particles.