The Mechanics of Molecular Dynamics

To understand how molecular dynamics works, consider a simple analogy: a ball rolling across hills and valleys. At any moment, the ball has a defined position \(r\), velocity \(v\), and mass \(m\). The shape of the terrain—our potential energy surface \(U(r)\)—determines how the ball will move. From this potential energy surface, we can calculate two crucial quantities: first, the force acting on the ball (\(F = -\frac{\mathrm{d}U}{\mathrm{d}r}\)), and second, using Newton’s second law (\(F = ma\)), the ball’s acceleration \(a\).

The ball’s motion is described by two fundamental differential equations:

(1)\[\begin{equation} \frac{\mathrm{d}r}{\mathrm{d}t} = v \end{equation}\]

(2)\[\begin{equation} \frac{\mathrm{d}v}{\mathrm{d}t} = a \end{equation}\]

By integrating these equations of motion, we can derive expressions for the changes in position and velocity over a time interval \(\delta t\):

(3)\[\begin{equation} \frac{dv}{dt} = a \rightarrow v(t + \delta t) = v(t) + a(t)\delta t \end{equation}\]

(4)\[\begin{equation} \frac{dr}{dt} = v \rightarrow r(t + \delta t) = r(t) + v(t)\delta t + \frac{1}{2}a(t)\delta t^2 \end{equation}\]

These equations are exact as \(\delta t\) approaches zero. For practical calculations, however, we need to work with finite time intervals. We introduce a “timestep” \(\Delta t\) and use these equations to predict how our system evolves:

From the current position \(r(t)\), we calculate the force \(F(r) = -\frac{\mathrm{d}U(r)}{\mathrm{d}r}\).

From this force and the ball’s mass, we determine the acceleration \(a(t) = F(r)/m\).

Using \(r(t)\), \(v(t)\), and \(a(t)\), we can approximately calculate \(r(t+\Delta t)\) and \(v(t+\Delta t)\).

By repeating this process—using each new position and velocity to calculate the next—we can trace the ball’s entire trajectory through time.