Regression hypothesis

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

If it can be shown that the prefactor is the identity matrix plus a matrix linear in x, then this is, in essence, Onsager s regression hypothesis [10] and the basis for linear transport theory. 

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... 

These indicate that the system returns to equilibrium at a rate proportional to the displacement, which is Onsager s famous regression hypothesis [10]. 

This confirms Onsager s regression hypothesis, namely, that the flux following a fluctuation in an isolated system is the same as if that departure from equilibrium were induced by an externally applied force. 

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

This result confirms Onsager s regression hypothesis. The most likely velocity in an isolated system following a fluctuation from equilibrium, Eq. (229), is equal to the most likely velocity due to an externally imposed force, Eq. (237), when the internal force is equal to the external force, Ts i =T. ... 

The relationship between fluctuation and dissipation is reminiscent of the reciprocal Onsager relations that link affinity to flux. The two relationships become identical under Onsager s regression hypothesis which states that the decay of a spontaneous fluctuation in an equilibrium system is indistinguishable from the approach of an undisturbed non-equilibrium system to equilibrium. The conclusion important for statistics, is that the relaxation of macroscopic non-equilibrium disturbances is governed by the same (linear) laws as the regression of spontaneous microscopic fluctuations of an equilibrium system. In the specific example discussed above, the energy fluctuations of a system in contact with a heat bath at temperature T,... 

The major advantage of the reactive flux method is that it enables one to initiate trajectories at the barrier top. instead of at reactants or products. Computer time is not wasted by waiting for the particle to escape from the well to the barrier. The method is based on the validity of Onsager s regression hypothesis/ which assures that fluctuations about the equilibrium state decay on the average with the same rate as macroscopic deviations from equilibrimn. It is sufficient to know the decay rate of equilibrimn correlation fimctions. There isn t any need to determine the decay rate of the macroscopic population as in the previous subsection. 

Onsager assumed that the variables and the rate laws were the same on the macroscopic and the microscopic level this is the so-called regression hypothesis. Also using the assumption of microscopic reversibility, he proved the reciprocal relations ... 

It is thus sufficient that the relaxation of the nonequilibrium population is proportional to the decay of equilibrium fluctuations. While the above expression (also known as Onsager s regression hypothesis) follows for weak perturbations, a linear dependence of the system s response on the perturbation... 

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.)... 

Onsager s treatment of nonequilibrium fluctuations rests on his fluctuation-regression hypothesis [1], To explain this hypothesis, we first note the following. 

The fluctuation-regression hypothesis, rephrased in modern language may now be stated as follows. To describe the dynamical fluctuations just mentioned, it is sufficient to use Onsager s purely macroscopic eqs. (A.38) and (A.40) modified to account for microscopic effects solely by the inclusion of random forces of the standard Brownian motion type namely, zero mean white noise Gaussian forces that obey fluctuation-dissipation relations that ensure recovery of Eq. (A.45) as r -> oc [2]. 

Next we review the more refined theories. The connection between the hydrodynamics of film motion and the light scattering experiments from thermal fluctuations is based on Onsager s regression hypothesis, namely, the relaxations of the surface elevations derived from macroscopic theories also pertain to the relaxation of thermally excited fluctuations. 

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... 

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