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The Gibbs Free Energy G is a Cartan 0-form

dG is an exact 1-form

The thermodynamic Maxwell Relations MR say that

d^2G = 0

d^2 = 0

That is a vanishing “curl” Cartan 2-form.

Now when G is a multiple-valued scalar 0-form field for topological reasons - the thermodynamic lumped coarse-grained parameter manifold its defined over has topological obstructions with “holes” - non-overlapping homotopy/homology “gauge orbits” (see Hagen Kleinert’s works for rigorous math)

There is induced a new LOCAL GAUGE POTENTIAL 1-form A

i.e. we now have

dG + A


dA Cartan 2-form =/= 0

and that’s your violation of MR.

That’s what you are seeing.

It’s analogous to an electromagnetic field 2-Form F and to the gravity curvature 2-Form R.

It’s a thermodynamic effective gauge field and there is probably a local symmetry group for it.

This is basically the idea in my 1969 PhD in more sophisticated form - I did not know Cartan forms back then and was groping.

In any case, Hagen Kleinert has the details.

Now what the micro-mechanism is - is another story.

From this POV there is no violation of 2nd Law of Thermo - just a deeper understanding of it.

Note that in the EM case, the static Coulomb field is made of coherent Glauber states of longitudinal & timelike virtual photons.

Crystal lattices are made of coherent Glauber states of virtual phonons.

Non-radiating magnetic near fields are coherent states of virtual transverse polarized photons.

curl A =/= 0

Only far field radiation has real transverse (zero rest mass) photons.

Similarly in this thermodynamic case? - New physics here!

On Apr 13, 2012, at 1:19 PM, JACK SARFATTI wrote:

On Apr 13, 2012, at 12:48 PM, This email address is being protected from spambots. You need JavaScript enabled to view it." data-scaytid="26">This email address is being protected from spambots. You need JavaScript enabled to view it. wrote:

Do you agree with the following:
(Assuming temperature constant conditions)
If strain and magnetization are single valued functions of stress and H-field (state functions), and the MR for these variables is violated, then:
1.  The Gibbs free energy is NOT a single valued function of stress and H-field (i.e., not a state function)


2.  The Kelvin Statement (of the 2nd law) is violated.

That’s the question - maybe not. I mean it may tacitly assume G is single-valued. The math was not really known in his time.
I think this is easy to show.
In a message dated 4/12/2012 8:50:17 A.M. Eastern Daylight Time, This email address is being protected from spambots. You need JavaScript enabled to view it. writes:
10-4 however, it’s now more clear to me that you never have to say that violation of Maxwell Relations is tantamount to violation of the Second Law of Thermodynamics - this is not only politically expedient, but I think physically correct. Remember Eddington! ;-)

On Apr 12, 2012, at 1:22 PM, This email address is being protected from spambots. You need JavaScript enabled to view it." data-scaytid="327">This email address is being protected from spambots. You need JavaScript enabled to view it. wrote:

Hi Jack,
Just want to clarify.  We have not yet performed an actual "over-unity" test.  Only actual "over-unity" test so far is UMD experiment.
What we have measured are fairly large MR discrepancies in the magnetomechanical system which (given overall losses can be minimized)      will be capable of "over-unity" operation.
In a message dated 4/12/2012 5:11:16 A.M. Eastern Daylight Time, This email address is being protected from spambots. You need JavaScript enabled to view it. writes:

Pelligrini et-al (North Eastern University, Boston) report such violations in magnetostrictive materials at kilohertz frequencies with an anomalous seemingly over-unity net energy output. Their experiments and allegedly similar ones at a US Navy lab seem to violate the 19th Century Clausius formulation of the Second Law of Thermodynamics. These people are not cranks so what they are seeing needs understanding. Also there are evidently practical energy payoffs if the claims are substantiated.
My basic point is elementary from section 10.2 of Penrose RR.

The Maxwell Relations are generally


This corresponds to a vanishing curl for the gradients of the potentials.

Some specific cases are:




The basic thermodynamic potentials U, H, A, G are Cartan 0-form (scalar fields) of thermodynamic parameters.

Their Cartan 1-form “vector fields,” e.g. dU below where d is the Cartan exterior derivative in thermodynamic parameter space of natural variables - not physical space-time where what we are discussing are “fibers” in sense of mathematics of fiber bundles.



Now we certainly have non-vanishing curls in electrodynamics. This requires a transverse polarized vector potential A in addition to the longitudinal scalar potential. Obviously, something similar is needed to explain Pelligrini’s et-al’s data. Curiously, I introduced such a quantity for superfluid helium turbulence in my 1969  Ph..D. dissertation with Fred W. Cummings (of the Jaynes-Cummings model now ubiquitous in quantum information theory models of strings of qubits). The timelike scalar and longitudinal part of A are constrained together by U1 local gauge invariance.

In terms of Cartan forms d^2 = 0, therefore, the above thermodynamic Cartan 1-forms are not exact 1-forms in Pelligrini’s data. They are not even closed 1-forms because they have local non-vanishing 2-form exterior derivatives.

Now Penrose says that the basic Maxwell relations correspond to C^2 smoothness in the thermodynamic potentials. Obviously, this is violated in Pelligrini’s case assuming that his measurements are good.

Clearly, there is no violation of the 2nd Law of Thermodynamics which must be generalized to include the quasi local gauge field A. A similar trick is used by Frank Wilczek for 2D anyons.

The A field corresponds to topological obstructions and what Hagen Kleinert means by singular “multi-valued fields,” i.e. different homotopy/homology classes. The thermodynamic potential Cartan 0-form scalar fields are no longer single-valued functions in a simply-connected thermodynamic parameter manifold. Indeed, the Bohm-Aharonov and Berry phase effects may be relevant here?

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