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 .. math:: {{\mbox{d}}\mathbf{r}_i \over {\mbox{d}}t} = \frac{1}{\gamma_i} \mathbf{F}_i(\mathbf{r}) + {\stackrel{\circ}{\mathbf{r}}}_i :label: eqnbrowniandyn where :math:\gamma_i is the friction coefficient :math:[\mbox{amu/ps}] and :math:{\stackrel{\circ}{\mathbf{r}}}_i\!\!(t) is a noise process with :math:\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 .. math:: \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, :label: eqnbrowniandynint where :math:{\mathbf{r}^G}_i is Gaussian distributed noise with :math:\mu = 0, :math:\sigma = 1. The friction coefficients :math:\gamma_i can be chosen the same for all particles or as :math:\gamma_i = m_i\,\gamma_i, where the friction constants :math:\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 :ref:mdrun  program.