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I don't remember the derivation being this simple, and I know it fundamentally involves statistical mechanics. Moreover, the hypothesis of isentropic variation is false in the situations we are talking about: thermodynamic flows of this kind increase entropy.
Start with the basic thermodynamic equation dU = pdV + TdS. Here, U is the energy of the system, p is the pressure, V is the volume, T is the temperature, and S is the entropy. Actually, this equation applies to two different parts of the systems (call them subsystem 1 and subsystem 2), so we really get two equations: one for subsystem 1 (indicated by a subscript 1 on all variables) and the other for subsystem 2 (indicated with a subscript 2).
The fact that p1, p2, T1, and T2 are well-defined values means that each subsystem is in local equilibrium, but the two subsystems may not be in equilibrium with each other; thus p1 - p2 and T1 - T2 will not be zero. We will refer to these differences in shorthand as Dp := p1 - p2 and DT := T1 - T2.
For the simplest version of the Onsager relations, assume that:
(These assumptions generally will not be valid in the long term, but they may be valid in the short term.) Then we get the following relations:
We now have the equation Dp dV + DT dS = 0, where dV and dS refer to subsystem 1 by arbitrary convention.
Note that the differential d is relevant, in this application, to the change of that variable through time (t), so we may regard it as the derivative with respect to time. Then these equation will be about the rate with which the locally defined thermodynamic variables for each subsystem change with time.
To get the result stated in the introduction, we must relate volume change to density flow, and entropy change to heat flow. For volume, we have V = m/r, where m is the mass (of subsystem 1, by our convention) and r is the density. Since m is constant, we have dV = -(m/r2)dr. Then for entropy, we have dQ = TdS, where dQ is the rate of heat flow (into subsystem 1, for our convention). Thus dS = (1/T)dQ. Now the equation becomes Dp (m/r2) dr = DT (1/T) dQ.
This can be transformed to dQ/Dp = (mT/r2)(dr/DT), which is the promised proportionality.
I'm glad that you got a precise version, so that I didn't have to move this here unsatisfied, replacing it with nothing.
I noticed a discrepancy between energy and energy density, since U doesn't have a partial time derivative but u (the density) does. This can potentially make things even simpler, by removing the unexplained feature of the flux densities -- JU is just uv, right (where v is local velocity). -- Toby Bartels 00:48, 15 Feb 2004 (UTC)
There is a velocity of matter flow, but there is also heat conduction. The flux of internal energy has a contribution from the energy carried by matter and from heat. You can use the equation JU = uv as a definition of velocity of energy transfer if you want, but it's not necessary. -- Miguel Sat Feb 14 22:31:10 PST 2004
Ah, so the two velocities are not the same. (Which I should have realised. Duh me!) So no purpose to changing this. -- Toby Bartels 07:01, 15 Feb 2004 (UTC)
I don't think this is correct: "both coefficients are measured in the same units of temperature times mass density" -- Preceding unsigned comment added by Daviesk24 (talk o contribs) 02:53, 9 July 2012 (UTC)
It seems that the word Onsager relation is used to describe several different things. In statistical physics the term Onsager relation is often used to describe a of the Maxwell relations to linear response. These generalizations follow from the fluctuation dissipation theorem, and they do not require any assumption local equilibrium. Actually they don't even have to involve any diffussion.
As an example there is an Onsager relation for the Poisson ratio of a viscoelastic material. For an elastic solid the Poisson ratio is a ratio between two strains (The radial stress and the longitunal strain of a rod with constant radial stress). But it also denotes a ratio between two stresses (The ratio between radial stress and the longitunal stress for constant length). Therefore the Poison ratio generalizes to two different linear response experiments, but according to an Onsager relations the resulting linear response functions are the same.
Huh, okay. Landau's is a proof of reciprocity while the original content was a generalised formulation of what is meant by (extensive) thermodynamic flows and (intensive) forces. Maybe I'll reintroduce the old material at some point and unify the notation. Miguel (talk) 08:14, 3 July 2010 (UTC)
I can't believe Clifford Truesdell is not mentioned in the article. Unfortunately, right now I don't feel competent enough to add the criticism section, hope I'll be able someday. So I just merely ask if anyone here is aware of the criticism of the relations concerned. Those who are interested I refer to the book "Rational Thermodynamics" by Truesdell (Lecture 7). Maybe citations of this book will help to figure out the current state of the polemics. Yrogirg (talk) 19:42, 23 December 2010 (UTC)
The article states "since the chemical potential is monotonically increasing with density at a fixed temperature" in the section "The phenomenological equations". I believe that this is only true if the density is small and interactions between the particles can be neglected. Maybe a sentence should be added that this is a crucial requirement for the validity of the diffusion equation anyway. -- Preceding unsigned comment added by 22.214.171.124 (talk) 11:49, 21 June 2013 (UTC)
Many refer to the relations as "Onsager-Casimir". I'm not exactly sure why, but it seems the reason is that Onsager did not fully elaborate on his principle in 1931, and Casimir's 1945 article finished the job. Anyway I find it surprising that the name Casimir appears absolutely nowhere in this article. Any good reason for that?
Perhaps related: Another very strange omission in the article is the how the Onsager relations work in nonzero magnetic field or nonzero Coriolis forces. The transport coefficients matrix is no longer symmetric, but, its transpose is equal to the coefficient matrix for the time-reversed system.
In this section of the article it is said (in the first line) that Onsager demonstrated that "L is positive semi-definite". This is not true. It was well known long before Onsager's work appeared.
The entropy production may be written in terms of the thermodynamic fluxes J and the thermodynamic forces X as
The use of the inequality is due to the fact that internal entropy generation within the system is always positive except for when it is in equilibrium. In equilibrium it is zero. This is one of the main assumptions of the thermodynamics of irreversible systems.
In the near-equilibrium case the fluxes are linear combinations of the forces
We have one such equation for each flux.
When these are substituted into the entropy production we get the quadratic expression
This expression had been in used for many years, back to the nineteenth century, before Onsager's 1931 work.
Starting at this last equation we can ask what conditions its positive definite nature requires of the array Lkj. The answer to this question is due to J. J. Sylvester. See the following two references:
J. J. Sylvester, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 4, Vol. 4, No. 23, p. 138-142, 1852.
J. J. Sylvester, On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure, Philosophical Transactions of the Royal Society of London, Vol. 143, p. 407-548, 1853.
When you boil it all down a necessary and sufficient condition for the quadratic form to be positive definite is
There are other such conditions that can be obtained by row-column permutations.
For all these relations there is still nothing to show that L is symmetric, i.e., L12 = L21, etc.
To summarize: The positive definite nature of the L comes from the principle that internal entropy production is greater than or equal to zero and the assumption that the thermodynamic fluxes can be expressed as linear combinations of the thermodynamic forces. Sylvester's work give a necessary and sufficient for the positive definiteness. This was well known before 1931. -- Preceding unsigned comment added by Zorpoid (talk o contribs) 19:39, 21 September 2016 (UTC)
What is meant by ? The two quantities have different units. The typical range of validity of the linear phenomenological equations is that the Taylor expansion of the flux in powers of the affinities can be truncated at the first term (see Callen, Ch. 14). -- Preceding unsigned comment added by 126.96.36.199 (talk) 01:20, 14 December 2021 (UTC)