# Thermal fluctuations in orientation of point particles

I am modeling group of point particles with 6 degrees of freedom each - 3 positional degrees of freedom and 3 orientational degrees of freedom. So, each particle has 3 position coordinates and a unit quaternion that represents a rotation from the global z-axis to the orientation associated with the particle. I want to study effects of thermal fluctuations on the particle. The particles also interact via a potential ( Morse, Lennard Jones etc.) I can write a Brownian motion force balance for a collection of such particles as follows:

$$\mathbf{F}\left(\mathbf{x}^{(n+1)},\mathbf{n}^{n+1}\right) + \frac{k_BT}{D}\frac{\mathbf{x}^{(n+1)}-\mathbf{x}^{(n)}}{\Delta t} - k_BT\sqrt{\frac{2}{D\Delta t}}{\xi}= 0$$

In this equation $\mathbf{x}^{(n)}$ represents position of all the particles at time $n$. $F$ is a force obtained from a potential function through which the particles interact. This potential depends on both the positions and the orientations of the particles. The last two terms are just the viscous and Brownian forces, where I use $(\mathbf{x}^{(n+1)} - \mathbf{x}^{(n)})/\Delta t$ as a velocity and $\xi$ is a random vector drawn from a standard normal distribution representing random motion in the positional degrees of freedom.

Question: How can I modify the equation above to include thermal fluctuations in the orientational degrees of freedom? Please let me know if you need any more details to answer this question.

Edit: The potential from which the force $\mathbf{F}$ is derived looks like this $$V = \phi_N(\mathbf{n_i},\mathbf{n_j},\mathbf{r_{ij}}) +\phi_C(\mathbf{n_i},\mathbf{n_j},\mathbf{r_{ij}})$$ where $$\phi_N = \sum_{i\neq j}||\mathbf{n_i} - \mathbf{n_j}||^2\psi\left(||\mathbf{r}_{ij}||\right)$$ and $$\phi_C = \sum_{i\neq j}\left( \frac{ \left( \mathbf{n_i} + \mathbf{n_j} \right) \cdot\mathbf{r_{ij}}} {||\mathbf{r_{ij}}||}\right)^2\psi\left(||\mathbf{r}_{ij}||\right)$$ and finally, $$\psi(x,y,z) = Ke^{-\frac{x^2+y^2+z^2}{2b^2}}.$$ $\mathbf{r_{ij}} = \mathbf{x}_j - \mathbf{x}_i$. $r_e, D_e$ and $K$ are just scalar parameters which I can change.

• In what way do the orientational DOFs interact with the positions? – lemon Dec 5 '17 at 21:50
• Thank you for your time and interest. I have added the information to my question. – user67800 Dec 5 '17 at 22:04