Reciprocal relation

Inserting the definition of G gives the celebrated Onsager reciprocal relations [4, 5]... 

When the reciprocal relations are valid in accord with (A3.2.251 then R is also symmetric. The variational principle in this case may be stated as... 

Once again, these fluxes are not all independent and some care must be taken to rewrite everything so that syimnetry is preserved [12]. Wlien this is done, the Curie principle decouples the vectorial forces from the scalar fluxes and vice versa [9]. Nevertheless, the reaction temis lead to additional reciprocal relations because... 

Onsager L 1931 Reciprocal relations in irreversible processes. I Rhys. Rev. 37 405... 

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

Interestingly, the need for a multiple electronic set, which we connect with the reciprocal relations, was also a keynote of a recent review ([46] and previous publications cited there and in [47]). Though the considerations relevant to this effect are not linked to the complex nature of the states (but rather to the stability of the adiabatic states in the real domain), we have included in Section HI a mention of, and some elaboration on, this topic. 

The question of determination of the phase of a field (classical or quantal, as of a wave function) from the modulus (absolute value) of the field along a real parameter (for which alone experimental determination is possible) is known as the phase problem [28]. (True also in crystallography.) The reciprocal relations derived in Section III represent a formal scheme for the determination of phase given the modulus, and vice versa. The physical basis of these singular integral relations was described in [147] and in several companion articles in that volume a more recent account can be found in [148]. Thus, the reciprocal relations in the time domain provide, under certain conditions of analyticity, solutions to the phase problem. For electromagnetic fields, these were derived in [120,149,150] and reviewed in [28,148]. Matter or Schrodinger waves were... 

The sign alternatives depend on the location of the zeros (or singulai ities) of x i). The above conjugate, or reciprocal, relations aie the main results in this section. When Eqs. (9) and (10) hold, ln x(f) and argx(t) are Hilbert transforms [245,246]. 

We shall now concentrate on several cases where relations equations (18) and (19) simplify. The most favorable case is where lnhalf-plane, (say) in the lower half, so that In <() (t) =0. Then one obtains reciprocal relations between observable amplitude moduli and phases as in Eqs. (9) and (10), with the upper sign holding. Solutions of the Schiddinger equation are expected to be regular in the lower half of the complex t plane (which corresponds to positive temperatures), but singularities of ln4>(f) can still aiise from zeros of <(>( ). We turn now to the location of these zeros. 

Figure 1. Numerical test of the reciprocal relations in Eqs. (9) anti (10) for Cg shown in Eq. (29), The values computeti directly from Eq. (29) are plotted upward and the values from the integral downward (by broken lines) for K/(ii= 16. The two curves are clearly identical, (a) ln C (/) against (//period). The modulus is an even function of /, (b) argC (f) against (//period). The phase is odd in /.

We now present some examples of studied wavepackets for which the reciprocal relations hold (exactly or approximately), but have not been noted. 

By substituting these expressions into Eq. (55), one can see after some algebra that ln,g(x, t) can be identified with lnx (t) + P t) shown in Section III.C.4. Moreover, In (f) = 0. It can be verified, numerically or algebraically, that the log-modulus and phase of In X-(t) obey the reciprocal relations (9) and (10). In more realistic cases (i.e., with several Gaussians), Eq. (56-58) do not hold. It still may be due that the analytical properties of the wavepacket remain valid and so do relations (9) and (10). If so, then these can be thought of as providing numerical checks on the accuracy of approximate wavepackets. 

The following theoretical consequences of the reciprocal relations can be noted ... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

In addition, the common Maxwell equations result from application of the reciprocity relation for exact differentials ... 

Heat-Capacity Relations In Eqs. (4-34) and (4-41) bothdH and dU are exact differentials, and application of the reciprocity relation leads to... 

The reciprocity relation for an exact differential applied to Eq. (4-16) produces not only the Maxwell relation, Eq. (4-28), but also two other usebil equations ... 

Thus, three reciprocal relations must be satisfied for an orthotropic material. Moreover, only 2, V13, and V23 need be further considered because V21, V31, and V32 can be expressed in terms of the first-mentioned group of Poisson s ratios and the Young s moduli. The latter group of Poisson s ratios should not be forgotten, however, because for some tests they are what is actually measured. 

Obviously, if > E2 as is the case for a lamina reinforced with fibers in the 1-direction, then < A2 as we would expect because the lamina is stiffer in the 1-direction than in the 2-direction. However, because of the reciprocal relations, irrespective of the values of E and E2,... 

The stiffness properties should satisfy the reciprocal relations... 

Note that these example material properties are not realistic (i.e., not physically possible) because the reciprocal relation is not satisfied for v,., or Vo, and because Go., must be less than G,2 and G3,. 

Application of the reciprocity relation A F 2 = 2 21) allows the fraction of radiation received by the target (apart from atmospheric attenuation and emissivity) to be expressed as... 

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