Talk:Onsager reciprocal relations

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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.

Miguel

[edit] Mathematical formulation and derivation

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:

  1. The two subsystems interact only with each other;
  2. The interaction takes place at constant (total) volume; and
  3. The interaction takes place isentropically.

(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:

  1. dU1 + dU2 = 0 (conservation of energy);
  2. dV1 + dV2 = 0; and
  3. dS1 + dS2 = 0.

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.

Also, this simple discrete version generalises into a continuous version, where the differential operator d is interpreted as a flux and the difference operator D is replaced by a gradient.

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)

[edit] Statistical physics

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.

Note that the original work of Onsager only considered exponential relaxation, but the generalized version of the Onsager relations are also fulfilled for non-exponential relaxation.

[edit] Nomenclatura

Is there a reason why you choose small t for temperature and capital T for time? Normally it's the other way round... 80.219.138.98 (talk) 21:51, 17 May 2008 (UTC)