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

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

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

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

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.