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Onsager's 1931 regression hypothesis asserts that “…the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process". (Here is the links to Onsager's original paper WikipedeA's article (about the related Onsager's reciprocal principle).)

I will be very thankful and happy for a self-contained explanation of what this hypothesis says and also of some demonstration, perhaps with some models mathematicians hang around with.

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Onsager's regression hypothesis

“…the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process"
comes vividly to life when experimentalists observe the Brownian motion $q(t)$ of a damped oscillator (as nowadays they commonly do). Setting

$\qquad q(t)= x(t) \cos(\omega_0 t) - y(t) \sin(\omega_0 t)$

for $\omega_0$ the resonant frequency of the oscillator and $x(t),\,y(t)$ the (slowly varying) in-phase and quadrature amplitudes, these amplitudes are observed to satisfy

$\displaystyle\qquad \langle x(t) x(t+\tau)\rangle = \langle y(t) y(t+\tau)\rangle = \left[\frac{k_\text{B}T}{m \omega_0^2}\right]\,e^{-\omega_0|\tau|/(2 Q)}$

where $m$ is the mass of the oscillator and $Q$ is its mechanical quality. This example illustrates Onsager's regression principle as follows

“…the average regression of fluctuations (in the above oscillator example, the autocorrelation $\langle x(t) x(t+\tau)\rangle$) will obey the same laws (in the example, exponential decay of fluctuations with rate constant $\Gamma = \omega_0/(2 Q)$) as the corresponding macroscopic irreversible process (in the example, macroscopic damping of the oscillator motion with the same rate constant $\Gamma$)"
It is common experimental practice to deduce $Q$ not from observations of macroscopic damping, but rather by statistical analysis of the observed regression of Brownian motion fluctuations. Thus, in this practical sense, Onsager's regression hypothesis nowadays is universally accepted.

By a similar analysis of coupled fluctuations in larger-dimension dynamical systems, Onsager deduced certain reciprocity relations that bear his name (and for which he received the Nobel Prize in Chemistry in 1968). Accessible discussions of the Onsager relations in textbooks include Charles Kittel's Elementary statistical physics (see Ch. 33, "Thermodynamics of Irreversible Processes and the Onsager Reciprocal Relations") and Landau and Lifshitz' Statistical Physics: Part 1 (see Ch. 122, "The Symmetry of the Kinetic Coefficients").

In the context of separative transport (where these relations find common application) Onsager's principle demonstrates from general thermodynamic that if an imposed current $j_\text{A}$ of conserved quantity $\text{A}$ induces a current $j_\text{B}$ of conserved quantity $\text{B}$ via $j_\text{B} = L_\text{BA}\,j_\text{A}$, then a reciprocal flow induction occurs with $j_\text{A} = L_\text{AB}\,j_\text{B}$ and $L_\text{AB}=L_\text{BA}$. As Kittel and Landau/Lifshitz both discuss, this principle follows by considering the temporal decay of microscopic fluctuations (assuming local thermodynamic equilibrium).

Physically speaking, if a flow of $A$ linearly induces a flow of $B$, then the reciprocal induction occurs too, with equal constant of proportionality. This relation apples in a great many physical systems, including for example (and non-obviously) the coupled transport of electrolytes and nutrients across cell membranes.

active cross-membrane transport

Whether Onsager's dynamical assumptions hold in a given instance has to be carefully analyzed on a case-by-case basis. That is why Kittel's text cautions, prior to working through an example involving thermoelectric coupling (Chapters 33 and 34):

It is rarely a trivial problem to find the correct choice of (generalized) forces and fluxes applicable to the Onsager relation.
In consequence of this necessary admixture of physical reasoning in applying the Onsager relations in particular cases, it sometimes happens that practical applications of Onsager's formalism are accompanied by lively theoretical and/or experimental controversies, which are associated not to the Onsager formalism itself, but to the applicability (or not) of various microscopic dynamical models that justify its use.

We thus see that the Onsager relations are not rigorous constraints in the sense of the First and Second Laws, but rather describe simplifying symmetries that emerge in a broad range of idealized (chiefly, linearized & spatially localized) descriptions of dynamical behavior; with these symmetries providing a vital key to the general description of a large set of transport processes that have great practical importance.

Perhaps I should mention, that I would myself be very interested in any references that generalize Onsager's relation to the coupled quantum dynamical flow of symbol-function measures; this is associated to the practical challenge of generating quantum spin hyperpolarization via separative transport processes.

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Many thanks, John. (For me (given my own lack of background) an even more elementary/sef-contained/mathematical answer could be perhaps even more accesible/helpful.) – Gil Kalai Oct 5 '11 at 10:59
Gil, I have just returned from a visit to IBM Almaden where "an even more elementary/self-contained/mathematical answer" is what they want too. Kittel's discussion is witty and Landau/Lifshitz' is short, but my experience was that their statistical arguments are too abstract to be grasped intuitively by mainly experimental audiences. So I am now seeking a microscopic model whose dynamics everyone already understands intuitively, that illuminates the statistical arguments. Onsager's macro/micro relations are subtle classically (and become yet more subtle when quantum nonlocality enters). – John Sidles Oct 5 '11 at 11:41
Gil, I have now amended the answer to provide a discussion of the Brownian motion of oscillators as a concrete example of the Onsager regression hypothesis. However, to date I have not found a concrete example of an Onsager reciprocity relation that is notably more illuminating than (e.g.) the thermoelectric example that is worked in Kittel. And so: (1) I hope the added discussion is helpful, (2) there's still plenty of work to do at the quantum level, and (3) thank you for contributing a wonderful question to this exciting new forum. – John Sidles Oct 5 '11 at 13:03
Dear John, Many thanks!! – Gil Kalai Oct 6 '11 at 10:04
Thank you, Gil. As it happens, I am presently writing-up a quantum dynamical analysis of Onsager-style polarization transport (in fact, racing to complete the theory before several experimental groups can observe the effect) and so your (wonderful!) question was timely. It is characteristic of transport theory (both classical and quantum) that often-times it is more straightforward to solve the microscopic dynamical equations---for particular cases and by more-or-less brute force---than it is figure out what is going-on in any general/natural sense. Here Onsager's methods are very helpful! – John Sidles Oct 6 '11 at 12:53

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