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 can be activated by using
integrator=bd
and the simulations are run using the
mdrun program.
In BD there are no velocities, so there is also no kinetic energy. Still gmx mdrun will report a kinetic energy and temperature based on atom displacements per step \(\Delta x\). This can be used to judge the quality of the integration. A too high temperature is an indication that the time step chosen is too large. The formula for the kinetic energy term reported is: