Account for gains and losses in velocity space

Between encounters a particle streams through space and is accelerated by any external force. Along that trajectory the distribution changes only through the collision term. A force term differentiates with respect to velocity and includes acceleration, force divided by mass, when the velocity measure is used. This is a common convention trap when translating a formula originally written in momentum coordinates. The same physical equation must preserve its units after that change of variables.

At a given position, binary collisions couple four velocities: two incoming and two outgoing. Elasticity preserves their total momentum and kinetic energy but generally preserves neither particle’s velocity. The interaction range and duration must be small compared with macroscopic lengths and the mean time between collisions. Dense liquids, strongly correlated recollisions, and long-range cumulative deflections can require different kinetic descriptions; calling every interaction a Boltzmann collision does not establish those approximations.

For a particle at velocity v, partners with velocity v₁ approach at relative speed g. Multiplying g by the differential cross section gives the collision kernel for scattering into a small solid angle. Its product with the partner velocity density and velocity volume gives a rate. Integrate over partner velocities and scattering directions to count losses from the selected cell. Reversed elastic encounters supply the corresponding gain term through the product of the associated primed distributions.

The symmetric notation uses microscopic reversibility and the collision-map Jacobian to express gain and loss with the same nonnegative kernel. It should not be read as a claim that the primed and unprimed particle populations are always equal. Only detailed balance at equilibrium makes each paired product equal. There is no extra one-half in the tagged-particle loss frequency; a half appears when counting total unordered collision events, because summing tagged encounters counts each pair twice.

For equal masses, introduce center-of-mass velocity and relative velocity. An elastic collision retains the center velocity and rotates the relative vector without changing its magnitude. This construction gives an immediate momentum and energy audit for any proposed numerical collision. Merely preserving speed for each particle separately is too restrictive, while independently drawing two outgoing velocities usually violates the shared conservation laws. A valid collision sampler must enforce the correct pair constraints.

The vector lab isolates one part of this geometry. Reset and set angle to 0° and magnitude to 2; predict components (2,0). Reset and set angle to 90° and magnitude to 2; predict (0,2). A fixed-length relative vector can rotate without changing relative kinetic energy. The instrument contains one vector only, so it does not reconstruct the two outgoing particles, use a differential cross section, or sample collision events.