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

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

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

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

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.