1Brownian Dynamics 2----------------- 3 4In the limit of high friction, stochastic dynamics reduces to Brownian 5dynamics, also called position Langevin dynamics. This applies to 6over-damped systems, *i.e.* systems in which the inertia effects are 7negligible. The equation is 8 9.. math:: {{\mbox{d}}\mathbf{r}_i \over {\mbox{d}}t} = \frac{1}{\gamma_i} \mathbf{F}_i(\mathbf{r}) + {\stackrel{\circ}{\mathbf{r}}}_i 10 :label: eqnbrowniandyn 11 12where :math:`\gamma_i` is the friction coefficient 13:math:`[\mbox{amu/ps}]` and 14:math:`{\stackrel{\circ}{\mathbf{r}}}_i\!\!(t)` is a noise 15process with 16:math:`\langle {\stackrel{\circ}{r}}_i\!\!(t) {\stackrel{\circ}{r}}_j\!\!(t+s) \rangle = 2 \delta(s) \delta_{ij} k_B T / \gamma_i`. 17In |Gromacs| the equations are integrated with a simple, explicit scheme 18 19.. math:: \mathbf{r}_i(t+\Delta t) = \mathbf{r}_i(t) + 20 {\Delta t \over \gamma_i} \mathbf{F}_i(\mathbf{r}(t)) 21 + \sqrt{2 k_B T {\Delta t \over \gamma_i}}\, {\mathbf{r}^G}_i, 22 :label: eqnbrowniandynint 23 24where :math:`{\mathbf{r}^G}_i` is Gaussian distributed 25noise with :math:`\mu = 0`, :math:`\sigma = 1`. The friction 26coefficients :math:`\gamma_i` can be chosen the same for all particles 27or as :math:`\gamma_i = m_i\,\gamma_i`, where the friction constants 28:math:`\gamma_i` can be different for different groups of atoms. Because 29the system is assumed to be over-damped, large timesteps can be used. 30LINCS should be used for the constraints since SHAKE will not converge 31for large atomic displacements. BD is an option of the :ref:`mdrun <gmx mdrun>` program. 32