No obvious physical meaning comes to me. Matters are quite different for the time integral of the product of two fluxes of intertwinned magnetic fields (e.g. the toroidal and the poloidal magnetic field in a torus): this is the amount of magnetic helicity destroyed by dissipation during the time interval of interest. remarkably, it is relativistically invariant.

Thanks Andrea,

My interest is in the extrapolation of eletric device called memristor.

By the way do you have any reference on what you indicate (some paper easily understandable by non-physicits of course).

There are a lot of them; unfortunately, all of them are for specialists' eyes only (i.e. plasma physicists, astrophysicists).

Try M. A. Berger, 'Introduction to magnetic helicity' Plasma Phys. Controlled Fus. 41 (1999) B167-B175 as for a general introduction -in particular, Stokes' theorem shows that the first term of the R.H.S. equation (15) in this paper is exactly the product of two fluxes I referred to. See also equation (5) for the definition of magnetic helicity (but Wikipedia too is helpful to this purpose, of course).

As for a review of physical problems involved, I have greatly enjoyed H.K.Moffatt's excellent book, 'Magnetic field generation in electrically conducting fluids', Cambridge Univ. Press 1978; there also many amusing laboratory applications.

As for the mathematical details, alas rather involved, tro refences may be helpful: The elementary one is L.S.Solovev, V.D.Shafranov, 'Number of loops for two closed curves' Appencidx III of Review of Plasma Physics V, Consultants Bureau (1970) edited by M.A.Leontovich. The hard stuff (to me, at least) is G.E.Marsh, Phys. Rev. E 47, 5 (1992) , 3607-3611; above all, equation (10) and Fig.2 of this paper have triggered in my head my previous answer to you.

Finally, as for the relativistic invariance of the helicity dissipation rate see L.D.Landau, E.M.Lifshits, Theory of fields, §25 of Chapter III, where it is shown that one invariant of the electromagnetic field is the dot product of electric and magnetic field, i.e. the density of the amount of helicity dissipated per unit time.

Dear René,

you write "..My interest is in the extrapolation of electric device called memristor..".

Memristor is defined in [1] as an element providing a unique link between the charge and flux (flux-linkage in [1]). Flux is the time integral of the voltage. This quantity is not necessarily the "magnetic" flux. Therefore, I think that in this case it is not necessary to examine the physical meaning of the integral of the "magnetic" flux, but it is useful to think about the meaning of the integral of flux (or the second integral of the voltage).

The integral of the flux is one of the constitutive variables of the memory inductor - meminductor, see figure [2] from [3].

Ref. [4] is a link to some interesting information on the integration of common variables.

[1] Chua, L.O. Memristor – The Missing Circuit Element. IEEE Transactions on Circuit Theory 1971; CT-18(5):507–519

[2] http://www.thinkartlab.com/Memristics/Power%20of%20Four/HTMLFiles/Power%20of%20Four_34.gif

[3] Biolek, D., Biolek, Z., Biolková, V. SPICE Modeling of Memristive, Memcapacitative and Meminductive Systems. In Proceedings of the European Conferenceon Circuits Theory and Design 2009 (ECCTD '09), Antalya, Turkey, 2009; 249–252

[4] http://wearcam.org/absement/Derivatives_of_displacement.htm

Dear Zdenek, Thanks a lot for your help. I agree with your remark on magnetic flux.

In fact my background is mostly math, hence I do not master the physical aspect of memristor.

BW

René