The default is to specify an initial expression for each component of the velocity, v0. In the case of the Newtonian, first-order Newtonian, and Lagrangian formulations the following condition is implemented:

In this case the initial velocity of every particle is defined using the same expression. The particles may have different initial velocity if v0 is a function of initial position.

If the initial kinetic energy E0 (SI unit: J) and initial particle direction L0 (dimensionless) are specified, for the Newtonian, first-order Newtonian, and Lagrangian formulations the following condition is implemented:

where mp is the mass of the particle. If either Newtonian formulation is used and a relativistic correction is enabled, instead the following condition is implemented:

where mr (SI unit: kg) is the rest mass of the particle and c = 2.99792458 × 108 m/s is the speed of light in a vacuum.

When a Maxwellian initial velocity distribution is selected, an array of Nv initial velocity values is created for each particle in each velocity direction. For a given initial temperature, in each velocity direction, the distribution function is given by:

The probability a particle has an initial velocity vi is given by:

This can clearly generate a large number of particles because the total number of initial velocities are Nvnsdim. This option is particularly useful in plasma modeling, AC/DC modeling, and in Monte Carlo simulations for molecular dynamics applications.

Sometimes it is desirable to release Nvel particles with the same speed isotropically in velocity space. Defining the speed as c, the following expressions generate such a velocity distribution in 2D according to:

where θ goes from 0 to 2π in Nvel steps.

The azimuthal angle ϕ is uniformly distributed from 0 to 2π. The polar angle θ is sampled from the interval [0, π] with probability density proportional to sin θ. The polar angle is arbitrarily chosen as the angle that the initial velocity makes with the positive z-axis, but any direction could be chosen because the sphere is isotropic.

The Constant speed, hemispherical option is the same as the Constant speed, spherical option, except that in 2D θ goes from 0 to π and in 3D θ goes from 0 to π/2. The angle θ is measured from the direction given by the Hemisphere axis setting.

The Constant speed, cone option is the same as the Constant speed, spherical option, except that θ goes from 0 to α. The angle is measured from the direction given by the Cone axis setting.

The Constant speed, Lambertian option releases particles within a hemisphere in 3D velocity space, but the probability distribution function is different from that of the Constant speed, hemisphere option. Recall that for an isotropic hemispherical distribution the polar angle θ has a probability density proportional to sin θ; for the Lambertian distribution the probability density is instead proportional to sin θ cos θ. Because of this extra cosine term, distributions following this probability density are said to follow Lambert’s cosine law or (in molecular dynamics) Knudsen’s cosine law.