Brownian dynamics

In Langevin dynamics, the equation of motion [1] [2] [3] is:

${\displaystyle M{\ddot {X}}=-\nabla U(X)-\zeta {\dot {X}}+{\sqrt {2\zeta k_{B}T}}R(t)}$

where

• ${\displaystyle M}$ is mass
• The dot is a time derivative such that ${\displaystyle {\dot {X}}}$ is the velocity, and ${\displaystyle {\ddot {X}}}$ is the acceleration
• ${\displaystyle U(X)}$ is the particle interaction potential
• ${\displaystyle \nabla }$ is the gradient operator such that ${\displaystyle -\nabla U(X)}$ is the force calculated from the particle interaction potentials
• ${\displaystyle \zeta }$ is the friction constant or tensor in units of mass per time, commonly interpreted as ${\displaystyle \zeta =\gamma M}$ where ${\displaystyle \gamma }$ is the collision frequency with the solvent, a damping constant in units of reciprocal time. For spherical particles of radius r, Stokes' law gives ${\displaystyle \zeta =6\pi \,\eta \,r}$.
• ${\displaystyle k_{B}}$ is Boltzmann's constant
• ${\displaystyle T}$ is the temperature
• ${\displaystyle R(t)}$ is a delta-correlated stationary Gaussian process with zero-mean, satisfying

${\displaystyle \left\langle R(t)\right\rangle =0}$

${\displaystyle \left\langle R(t)R(t')\right\rangle =\delta (t-t').}$

In Brownian dynamics, the ${\displaystyle M{\ddot {X}}(t)}$ term is neglected, and the sum of these terms is zero.[1]

${\displaystyle 0=-\nabla U(X)-\zeta {\dot {X}}+{\sqrt {2\zeta k_{B}T}}R(t)}$

Using the Einstein relation, ${\displaystyle D=k_{B}T/\zeta }$, it is often convenient to write the equation as,

${\displaystyle {\dot {X}}(t)=-{\frac {D}{k_{B}T}}\nabla U(X)+{\sqrt {2D}}R(t).}$

In 1978, Ermack and McCammon suggested an algorithm for efficiently computing Brownian dynamics with hydrodynamic interactions.[2] Hydrodynamic interactions occur when the particles interact indirectly by generating and reacting to local velocities in the solvent. For a system of ${\displaystyle N}$ three-dimensional particle diffusing subject to a force vector F(X), the derived BD scheme becomes:[1]

${\displaystyle X(t+\Delta t)=X(t)+{\frac {\Delta tD}{k_{B}T}}F[X(t)]+R(t)}$

where ${\displaystyle D}$ is a diffusion matrix specifying hydrodynamic interactions in non-diagonal entries and ${\displaystyle R(t)}$ is Gaussian noise vector with zero mean and a covariance matrix satisfying ${\displaystyle <R(t)R(t)^{T}>=2D\Delta (t)}$.

• Angeliki Diane Rigos, research director, business consultant, and former university professor

• Brownian motion
• Immersed boundary method