# Brownian dynamics

Brownian dynamics (BD) can be used to describe the motion of molecules in molecular simulation. It is a simplified version of Langevin dynamics and corresponds to the limit where no average acceleration takes place during the simulation run. This approximation can also be described as 'overdamped' Langevin dynamics, or as Langevin dynamics without inertia.

In Langevin dynamics, the equation of motion is

$M\ddot{X} = - \nabla U(X) - \gamma M\dot{X} + \sqrt{2 \gamma k_B T M} R(t)$

where $U(X)$ is the particle interaction potential; $\nabla$ is the gradient operator such that $- \nabla U(X)$ is the force calculated from the particle interaction potentials; the dot is a time derivative such that $\dot{X}$ is the velocity and $\ddot{X}$ is the acceleration; T is the temperature, kB is Boltzmann's constant; and $R(t)$ is a delta-correlated stationary Gaussian process with zero-mean, satisfying

$\left\langle R(t) \right\rangle =0$
$\left\langle R(t)R(t') \right\rangle = \delta(t-t').$

In Brownian dynamics, no acceleration is assumed to take place. Thus, the $M\ddot{X}(t)$ term is neglected, and the sum of these terms is zero.

$0 = - \nabla U(X) - \gamma M\dot{X}+ \sqrt{2 \gamma k_B T M} R(t)$

Defining $\zeta = \gamma M$, and using the Einstein relation, $D = k_B T/\zeta$, it is often convenient to write the equation as,

$\dot{X}(t) = - \nabla U(X)/\zeta + \sqrt{2 D} R(t).$