# 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

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

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.