13.2: Brownian Dynamics

The Langevin equation for the motion of a Brownian particle can be modified to account for an additional external force, in addition to the drag force and random force. From Newton’s Second Law:

\[ m \ddot{x} = f_d + f_r(t)+ f_{ext}(t) \nonumber \]

where the added force is obtained from the gradient of the potential it experiences:

\[ f_{ext} = -\dfrac{\partial U}{\partial x} \]

With the fluctuation-dissipation relation \( \langle f_r(t)f_r(t')\rangle = 2\zeta k_BT \delta (t-t')\), the Langevin equation becomes

\[ m\ddot{x} + (\partial U/\partial x)+\zeta \dot{x} - \sqrt{2\zeta k_BT } R(t)=0 \]

Here \(R(t)\) refers to a Gaussian distributed sequence of random numbers with \(⟨R(t)⟩ = 0\) and \(⟨R(t) R(t′)⟩ = δ(t ‒ t′)\).

Brownian dynamics simulations are performed using this equation of motion in the diffusion-dominated, or strong friction limit \( |m\ddot{x}|\ll |\zeta \dot{x}|\). Then, we can neglect inertial motion, and set the acceleration of the particle to zero to obtain an expression for the velocity of the particle

\[\dot{x} (t) = \dfrac{\dfrac{\partial U}{\partial x}}{\zeta} -\sqrt{2k_BT/\zeta} R(t) \nonumber \]

We then integrate this equation of motion in the presence of random perturbations to determine the dynamics \(x(t)\).

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