The stochastic Liouville equation and the approach to thermal equilibrium
Some aspects of behaviour of spins in the presence of random molecular motions are discussed. Various theories which deal with the approach of the spin system to thermal equilibrium are reviewed. It is emphasized that two types of description are used. One is the 'p(t) formalism', where the spin behaviour of one randomly moving molecule is considered. The other is the 'p(Q) formalism', where the average behaviour of all the spins which are momentarily in the same environment is described. The conventional relaxation theories make use of the p(t) formalism, whereas the stochastic Liouville method for line shape calculations uses the p(Q) formalism. In the first type the approach to equilibrium has beendealt with for a long time. In the second type the approach to the thermal equilibrium state of p(Q) was formulated only recently in the form of the modified stochastic Liouville equation. It is pointed out that this equation has important implications for both line shape calculations and for relaxation theory. I. SEMI-CLASSICAL THEORY A. The density matrix p(Q) This paper deals with spins which are in random motion. The simplest example of such a type of motion is that of spins which jump back and forth among a number of sites with different chemical or magnetic environment. W e Iabel the si tes wi th the n um bers v = 1, 2, ... , n and define the distribution vector pv(t) as the vector of the probabilities of finding Bpins in the sites v at the time t. Another example is the case of translational motion. The position of a spin is then characterized by the spatial coordinate r and the distribution function is p(r, t). The case of rotatory motion of molecules is very common. The orientation of the molecules is then defined by the Eulerian angles, Q. Weshall treat all these cases in one formalism and choose Q as the notation of the random coordinate. Thus the function p(Q, t) will denote the probability density of finding spins in the environment characterized by a particular Q. We shall sometimes call this parameter 'position' or 'orientation', but this should be considered as a general term which also stands for 'site', 'conformation', etc. Tobegin with, let us assume that the spins can satisfactorily be described in terms of elementary magnets m, which can differ in direction but not in magnitude Im I· The state of the whole spin system is then given by the combined probability density P(Q, m), which we taketobe normalized: