Non-Markovian differential theory

The theory of Section 1.8 is sometimes qualified as non-Markovian since it accounts for non-exponential angular momentum relaxation, unlike impact theory which is Markovian in this sense. However, it is not a unique non-Markovian generalization of impact theory. Not less known is a differential version of the theory 

It is often obtained from Eq. (1.71) when the kernel is assumed to relax much quicker than the solution to be found. Then it is nothing more than a low-density gas approximation to Eq. (1.71), valid when conditions (1.83) or (1.88) are met. For these conditions the differential theory is expected to be binary in collisions, and 

Although non-Markovian, the differential theory surely has Markovian asymptotics at sufficiently long times  

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

This is exactly an integral of Eq. (1.103) with (r) defined in Eq. (1.104). Hence the integral of differential equation (1.105) 

---

NMR see nuclear magnetic resonance non-Markovian binary theory 138-40 non-Markovian differential theory 38-45, 65... 

In non-Markovian theory, the off-diagonal elements of G cannot be looked upon as transfer rates between two or more discrete eigenfrequen-cies of the system as they are functions of the continuous variable to. The transfer rates concept is only acceptable in the Markovian limit of the theory (t tc) when co-dependence is eliminated. To obtain this limit, we must first pass to differential formulation of non-Markovian theory and after that let t -> oo. In the literature there is complete unanimity on how the transition from integral to differential formalism can be carried out correctly. According to Eq. (4.28) the integrand in Eq. (4.26) may be written as... 

The time-dependent (non-Markovian) rate constant k(t) determines the rate of energy quenching in the differential kinetic equation that constitutes the basis of this theory ... 

---