Onsager regression hypothesis

The Onsager regression hypothesis, stated mathematically for the chemically reacting system just described, is given in the classical limit by... 

The most likely terminal position was given as Eq. (24), where it was mentioned that if the coefficient could be shown to scale linearly with time, then the Onsager regression hypothesis would emerge as a theorem. Hence the small-x behavior of... 

On the left side of (11.15) we have the time evolution of a prepared deviation from equilibrium of the dynamical variable B. On the right side we have a time correlation function of spontaneous equilibrium fluctuations involving the dynamical variables A, which defined the perturbation, and B. The fact that the two time evolutions are the same has been known as the Onsager regression hypothesis. (The hypothesis was made before the formal proof above was known.)... 

The use of macroscopic transport equations for the determination of time-correlation functions of fluctuating quantities is equivalent to the Onsager regression hypothesis outlined in Section 10.2 and discussed in Chapter 11. 

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. 

Thus there is no correlation between A(0) and the random force F(x). This is a very important formal conclusion. It is precisely this lack of correlation between F(x) and A( 0) that is at the foundation of the Onsager regression hypothesis. 

This equation was derived without appealing to the Onsager regression hypothesis. 

A somewhat simpler approach has been taken by Li and Wilson i who used the Onsager regression hypothesis to model aspects of the A + BC reaction in rare gas solution. The Onsager regression hypothesis states that the... 

Three general approaches lead to values for gp q,r). First, scatterers can be treated microscopically as objects having hydrodynamic and direct, e.g., excluded volume, interactions, and dg]p q, r)/dr can be calculated. Second, because ag t) = exp[iq (rj(f)] is the q spatial Fourier component of the scatterer density, semicontinuum hydrodynamics and the Onsager regression hypothesis predict the average temporal evolution of fluctuations. Third, scatterers could be assumed to perform simple Brownian motion. 

The Onsager regression hypothesis(20, 21) proposes that the relaxation of a macroscopically imposed concentration gradient and the relaxation of a transient concentration fluctuation should be governed by the same transport coefficient D. The application of this hypothesis to light scattering is straightforward. QELSS observes fluctuations in the th spatial Fourier component a (t) of the scat-terer concentration, which as a macroscopically nonuniform concentration can be written... 

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