Adjusting density and temperature

Since the number of particles is usually fixes and conserved, the way to achieve the desired density os to adjust the volume. This is done scaling all coordinates by a suitable factor.

In Molecular Dynamics, the temperature is a measurable quantity. Kinetic theory tells us that

$$T=m\langle \vert\mathbf{v}\vert^{2}\rangle /3k,$$

where $k$ is the Boltzmann's constant, and $\langle \vert\mathbf{v}\vert^{2}\rangle$ is the averaged square of the velocity. In a simulation, we have to average this quantity over a number of MD steps to get a measurement of $T$. If we want to simulate a particular temperature, we rescale each component of the velocity vector of every particle by $\sqrt{T_{desired}/T_{actual}}$. Obviously $T$ is a fluctuating quantity and can therefore be adjusted only approximately.