Reorientation in the impact approximation

Markovian theory of orientational relaxation implies that it is exponential from the very beginning but actually Eq. (2.26) holds for t zj only. If any non-Markovian equations, either (2.24) or (2.25), are used instead, then the exponential asymptotic behaviour is preceded by a short dynamic stage which accounts for the inertial effects (at t zj) and collisions (at t Tc). 

Let us start from the simplest model, which is impact relaxation of angular momentum. According to Eq. (1.21) it proceeds exponentially with relaxation time (1.22)  

With this kernel the exact solution of Eq. (2.25) is similar to that of Eq. (1.71) with a kernel Eq. (1.77)  

For one-dimensional rotation (r = 1), orientational correlation functions were rigorously calculated in the impact theory for both strong and weak collisions [98, 99]. It turns out in the case of weak collisions that the exact solution, which holds for any happens to coincide with what is obtained in Eq. (2.50). Consequently, the accuracy of the perturbation theory is characterized by the difference between Eq. (2.49) and Eq. (2.50), at least in this particular case. The degree of agreement between approximate and exact solutions is readily determined by representing them as a time expansion 

By keeping higher powers of in these polynomials, one exceeds the 

---