The Keilson-Storer kernel

Here the averaging denoted by an overbar is performed with a two-dimensional probability density (1.5), i.e. 

The change of average z projection of the angular momentum per unit time is its change for a single collision A/ = (D — l)Jo(t) multiplied by the collision frequency Tq 1  

After integrating this equation and substituting the result into (1.8), we find 

Because non-adiabatic collisions induce transitions between rotational levels, these levels do not participate in the relaxation process independently as in (1.11), but are correlated with each other. The degree of correlation is determined by the kernel of Eq. (1.3). A one-parameter model for such a kernel adopted in Eq. (1.6) meets the requirement formulated in (1.2). Mathematically it is suitable to solve integral equation (1.2) in a general way. The form of the kernel in Eq. (1.6) was first proposed by Keilson and Storer to describe the relaxation of the translational velocity [10]. Later it was employed in a number of other problems [24, 25], including the one under discussion [26, 27]. 

Owing to space isotropy of collision in the coordinate frame strongly coupled with a molecule s frame and hereafter referred to as the molecular system (MS), the kernel may be represented as 

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The width of this Lorentzian line is half as large as that found in (3.37). This, however, is not a surprise because the perturbation theory equation (3.23) predicted exactly this difference in the width of the line narrowed by strong and weak collisions. This is the maximal difference expected within the framework of impact theory when the Keilson-Storer kernel is used and 0 < y < 1. 

It is necessary to get insight into the kernel of the integral equation (3.26). Since frequency exchange is initiated by non-adiabatic collisions, it is reasonable to use the Keilson-Storer model. However, before employing kernel (1.16) it should be integrated over the angle... 

In order to make this system more concrete, let us employ the Keilson-Storer model for definition of its kernel... 

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