Brownian dynamics (BD) can be used to describe the motion of molecules for example in molecular simulations or in reality. It is a simplified version of Langevin dynamics and corresponds to the limit where no average acceleration takes place. This approximation can also be described as 'overdamped' Langevin dynamics, or as Langevin dynamics without inertia.
In Langevin dynamics, the equation of motion is
where is the particle interaction potential; is the gradient operator such that is the force calculated from the particle interaction potentials; the dot is a time derivative such that is the velocity and is the acceleration; T is the temperature, kB is Boltzmann's constant; and is a delta-correlated stationary Gaussian process with zero-mean, satisfying
In Brownian dynamics, no acceleration is assumed to take place. Thus, the term is neglected, and the sum of these terms is zero.
Defining , and using the Einstein relation, , it is often convenient to write the equation as,
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