Projection Operators and Relaxation Equations

the equality sign appears only if A = 0. From condition 1 we see that the norm A = (A, A)112 of the property A, which can be regarded as the length of the property A, is a real quantity. A property whose norm is unity is said to be normalized. Two properties, A and B, are said to be orthogonal if (A, B) = 0. 

Thus all functions At(r) of finite norm, together with the above definition of the scalar product define a Hilbert space. This space is called Liouville space because the Liouvillian generates the motion in this space. 

By analogy with quantum mechanics, C(t) [c.f. Eq. (11.2.7)] can be regarded as an expectation value of eiLt in the state Wa — Poll2A. Let us recall that in quantum mechanics the Hamiltonian operator H is Hermitian and the operator exp [IHtjh] is unitary. Correspondingly here the Liouvillian L is Hermitian and the propagator 

In previous chapters phenomenological relaxation equations were used together with the Onsager regression hypothesis to compute time correlation functions. In this section we present a microscopic derivation of generalized relaxation equations (Zwan-zig, 1961 Berne, Mori, 1965 and 1971). These equations can be used to compute time-correlation functions under circumstances where the usual phenomenological equations do not apply. 

In this section it isshown that arbitrary dynamical properties in complicated systems can be described by equations which are analogous to the Langevin equation of Brownian motion theory [cf. Section (5.9)]. For example, the arbitrary property A is described by the equation 

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