1Brownian Dynamics
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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