Statistical theories for molecular collisions: A maximum entropy derivation

Statistical theories are particularly appropriate when one can define a strong interaction regime. We consider the distribution of classical trajectories which enter or exit from this regime. That distribution of trajectories which is of maximal entropy subject only to total conservation of flux is shown to lead to the familiar "phase-space" expression for the reaction probability. By including more refined conservation conditions as constraints one obtains improved statistical theories. As an example the "unified" statistical theory of Miller and the Hirschfelder-Wigner expression for the reaction probability are derived by imposing one more conservation constraint. Transition state theory is derived as a special case corresponding to a particular, extreme, numerical value of the constraint. Phase-space theory is obtained when the value of the constraint is at the other extreme (in which case the constraint is not informative). Essentially, exact results for the reaction probability in the collinear H+H₂ reactive collision are obtained using two conservation conditions (beside the conservation of total flux). In general, it is shown that the procedure is variational, i.e., that including additional constraints can only improve the results.