It is usually admitted in many textbooks that probability for a bimolecular (or biatomic) collision is computed as

Prob = [molecule1] [molecule2]

This is valid for gas-phase systems and for high-dilution situations only.

The best way to see this is from classical statistical mechanics. The basic classical equation for the colliding evolution of a molecule 1 is

partial rho

_{1} / partial t = Tr

_{2} L

_{V} rho

_{12}rho

_{1} is the classical distribution function for the molecule. Tr denotes the integration in the phase space (Q, P) of the dynamics. The integration is done over the classical degrees of freedom of the molecule 2 (i.e. coordinates Q and impulses P). L

_{V} is the interaction Liouvillian, which is simply the Poisson bracket for the intermolecular potential. The most interesting entity is rho

_{12}, which denotes s the classical distribution function for the collision pair.

For gaseous systems at very high dilution,

rho

_{12} =aprox= rho

_{1} rho

_{2}.

This is the famous Boltzmann’s assumption of molecular chaos (called by him Stosszahlansatz). It is an approximation valid for diluted gases only and the equation derived from the introduction of the assumption in above classical equation named Boltzmann equation. Of course, the Boltzmann equation is also valid for gases in high dilution. For most condensed phase systems the Boltzmann equation is just wrong.

In general the probability for a collision is not rho

_{1} rho

_{2} because that approximation is assuming statistical independence, that is, absence of correlations between the molecules; this is only valid for high-diluted systems with short range potentials.

A lot of stuff could be said about computing probabilities for collisions. I have no time for a serious review of all has been published in the topic since is a very sophisticated topic. However, I will introduce some basic ideas.

Boltzmann like equations (e.g. chemical bimolecular reaction rates) are local in time and space and assume absence of correlations before the collision.

In ionised gases (i.e. plasmas), a basic equation is Balescu one. Therein the distribution function for the colliding pair appears in the denominator instead of in the numerator. This feature is characteristic of systems with collective effects, that is, systems where molecules (in this case ions) are not randomly distributed but acting as a whole. In collective phenomena. Usual methods based in two-body or three-body processes are not valid in collective phenomena. Precisely this is one of technical difficulties are founding in the developing of tokamak reactors.

The more general expression for a collision may include non-local effects in space and time. The introduction of temporal non-localities is done via non-Markovian equations (Resibois-Prigogine kinetic equation is a characteristic example). Prigogine was Nobel Prize for Chemistry in 1977. A problem of his non-Markovian kinetic equation is that is very difficult to be solved even with most advanced mathematical techniques.

Boltzmann equation and hidrodynamics are valid in certain length regimes. Hydrodynamics correspond to situations when mean free path for molecules (or atoms) is much more larger than range of interactions, whereas the range of inhomogeneities (gradients in concentration for instance) is much more larger than mean free path.

However, the mean free path is much more larger than the range of inhomogeneities in rarified gases. In that case, Boltzmann-like equations (or hydrodynamics) are again not valid. It is really interesting to compare the large mean free path collision term with the Boltzmann collision term. The former (assuming molecular chaos) is non-local in space

Collision= rho

_{alpha} Int {rho

_{j}(r)} dr.

That is, molecule alpha can collide with the molecule j sited at some arbitrary distance. In the own Prigogine words [1]:

Molecule alpha interacts, so to speak, with an “average molecule” coming from an arbitrary distance

Some of those collisions at-a-distance are valid even with short-range forces due to large correlations. Those phenomena are often called “dynamical bubbles”.