Initial velocities in a Molecular Dynamics simulation

Question

In classical Molecular Dynamics simulations Newton's equations of motion are solved to get the evolution of particle positions and velocities. To do the integration of the equation of motion, one needs to specify the inital velocities and positions of the particles. In many simulations the initial velocities are given random values which correspond to a certain temperature (say, T) because of the following relation :
$\big<\frac{1}{2}\Sigma_im_i v_i^2\big>=\frac{3}{2}Nk_BT$.
In those simulations, the random initial velocities are chosen such that, they correspond to a certain desired temperature ( say, $T_d$ ). The system of particles which is being simulated is expected to go to equilibrium at this temperature ($T_d$) after running for a couple of time steps.
My questions are: Why the initial velocities are chosen so that they correspond to a certain desired temperature ( $T_d$ ) ? The alternative may be to choose the initial velocities randomly so that they don't correspond to the desired temperature say, $T_d$ but some different temperature say, $T$. If one doesn't choose the initial velocites in this way, can the velocities be relaxed to the desired equilibrium value of temperature ( $T_d$ ) ? Can I choose inital velocities as zero in the simulation ?

Answer 1

Choosing the initial velocities in a way that ensures that the starting initial kinetic energy is correctly related to the temperature is not compulsory. As it has been stated in the comments, it is just a way to reduce the equilibration time. Such equilibration time is usually quite fast and corresponds to two phenomena.

The first is establishing the correct velocity distribution for the simulated ensemble (notice that this is the Maxwell-Boltzmann distribution for canonical and grand-canonical ensembles, but it is different for the microcanonical ensemble, the difference vanishing as $\frac{1}{N}$ for large systems).

The second is the proper transfer of energy between kinetic and potential terms. In general, this second mechanism makes it inefficient to start with zero velocities. However, there are at least two cases where starting with zero velocities and then possibly rescaling velocities to get the target temperature is useful. The first is the case of the simulation of clusters. The problem with free clusters is that one has to carefully monitor the possibility of partial evaporation (atoms or molecules separating from the cluster as an effect of fluctuations of energy). Starting with zero velocity is a way to keep this phenomenon under control better. The second is the case of Car-Parrinello simulations, where the electronic degrees of freedom move according to fictitious classical dynamics. In order to keep them close to the Born-Oppenheimer energy surface, it is customary to start their dynamics with zero velocity. In that case, the classical dynamics of the electronic degrees of freedom is explicitly designed in such a way that the equilibration times are very long.

A final note about your formula connecting kinetic energy and temperature. Temperature is the average kinetic energy in the reference frame where the center of mass is at rest. Therefore the formula is correct only within $\frac{1}{N}$ corrections.