Pwfen

Fluctuation-dissipation theorem

The fluctuation-dissipation theorem article in Wikipedia seems to have a long history of inaccurate and plain wrong statements in it. I recall many years ago reading that linear response was required for theorem validity (it isn't). The fluctuation-dissipation theorem is subtle, and the intervening years have only seen moderate improvements in the entry. There are still problematic statements in the article.

(1) The first sentence was a problem until 25-January-2015 (I fixed it). The theorem's validity depends on detailed balance holding true. The previous version had no statement of this clear, concise and absolutely vital point. The old version mentioned non-equilibrium. That isn't the point and is misleading! The FDT connects fluctuations and dissipations, just like the name says. The issue of non-equilibrium here is auxilary, and potentially overly-inclusive. There exist non-equilibrium system where the FDT does /not/ hold. A better first sentence is what I changed it to: "The fluctuation-dissipation theorem (FDT) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance."

(2) "The fluctuation-dissipation theorem was originally formulated by Harry Nyquist in 1928" seems misleading in three ways. The first is that Nyquist's beautiful paper isn't a general formulation, and thus should probably not be called the "fluctuation dissipation theorem". The (general) FDT proves a relationship between admittance and fluctuations of thermodynamic variables. Nyquist proved a special case for EMFs. The second misleading point is that Nyquist's paper would (at least to a physicist) count as a proof. The implication of the text is that Callen and Welton did the first proof. The third misleading way is that if we're going to talk about specific examples of flucuation-dissipation relationships instead of the general theorem, Einstein's 1905 Brownian motion paper is a better candidate for the "first".

(3) I believe the sentence, "The fluctuation-dissipation theorem relies on the assumption that the response of a system in thermodynamic equilibrium to a small applied force is the same as its response to a spontaneous fluctuation", is catastrophically wrong. The FDT doesn't /assume/ this! It is a mathematical /proof/ of it for systems that obey detailed balance. When the FDT is violated, then the microscopic degrees of freedom corresponding to thermodynamic variables are known to violate detailed balance. The rest of the paragraph is not pertinent, and potentially misleading. The comment about exponential decay is wholly out of place. The "small" part of the sentence is irrelevant, so long as detailed balance holds true.

(4) It is probably worth mention R Kubo as the second discoverer of the FDT.

(5) "The probability distribution function W(x,0) is an equilibrium distribution and hence given by the Boltzmann distribution for the Hamiltonian" just shouldn't be in the FDT entry given that the introduction goes out of its way to mention that quantum mechanical systems that obey detailed balance are also covered by the FDT (slightly adapting the actual claim from the intro to better accord with reality).

I'm assuming that more work and thought would uncover more problems with the page given that at least 3 of 5 above are science problems.

What is the stomach of the 39 other people watching this page for a wholesale revision of the page? I'm not in a hurry to do this quickly (given that it has been a decade that the entry has had bad problems), but hopefully sometime in the next few months I and others will have time to fix the bad problems with the fluctuation-dissipation theorem entry.

Pwfen (talk) 00:06, 4 January 2015 (UTC)

It sounds like you're well qualified to improve and rewrite it, and I hope you do so. --Steve (talk) 13:47, 26 January 2015 (UTC)

Your submission at Articles for creation: Theodore A. Welton has been accepted

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The Herald (Benison) (talk) 15:43, 10 November 2018 (UTC)

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