# Brownian Dynamics#

In the limit of high friction, stochastic dynamics reduces to Brownian dynamics, also called position Langevin dynamics. This applies to over-damped systems, i.e. systems in which the inertia effects are negligible. The equation is

(117)#${{\mbox{d}}\mathbf{r}_i \over {\mbox{d}}t} = \frac{1}{\gamma_i} \mathbf{F}_i(\mathbf{r}) + {\stackrel{\circ}{\mathbf{r}}}_i$

where $$\gamma_i$$ is the friction coefficient $$[\mbox{amu/ps}]$$ and $${\stackrel{\circ}{\mathbf{r}}}_i\!\!(t)$$ is a noise process with $$\langle {\stackrel{\circ}{r}}_i\!\!(t) {\stackrel{\circ}{r}}_j\!\!(t+s) \rangle = 2 \delta(s) \delta_{ij} k_B T / \gamma_i$$. In GROMACS the equations are integrated with a simple, explicit scheme

(118)#$\mathbf{r}_i(t+\Delta t) = \mathbf{r}_i(t) + {\Delta t \over \gamma_i} \mathbf{F}_i(\mathbf{r}(t)) + \sqrt{2 k_B T {\Delta t \over \gamma_i}}\, {\mathbf{r}^G}_i,$

where $${\mathbf{r}^G}_i$$ is Gaussian distributed noise with $$\mu = 0$$, $$\sigma = 1$$. The friction coefficients $$\gamma_i$$ can be chosen the same for all particles or as $$\gamma_i = m_i\,\gamma_i$$, where the friction constants $$\gamma_i$$ can be different for different groups of atoms. Because the system is assumed to be over-damped, large timesteps can be used. LINCS should be used for the constraints since SHAKE will not converge for large atomic displacements. BD is an option of the mdrun program.

In BD there are no velocities, so there is also no kinetic energy. Still gmx mdrun will report a kinetic energy and temperature based on atom displacements per step $$\Delta x$$. This can be used to judge the quality of the integration. A too high temperature is an indication that the time step chosen is too large. The formula for the kinetic energy term reported is:

(119)#$\frac{1}{2} \sum_i \frac{\gamma_i \Delta x_i^2}{2 \, \Delta t}$