Onsager s regression hypothesis

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. 

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. 

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... 

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 time scale in Eq.7 can be thought of as the time for inelastic scattering events that decrease the net energy of the spins that comprise the magnetic system. According to Onsager s regression hypothesis, the Tg in Eq.7 is identical to the attempt time Tq in Eq.5. 

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

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]. 

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