The Reciprocal Relations

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... 

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... 

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 ... 

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... 

L. Onsager (Yale) discovery of the reciprocity relations bearing his name, which are fundamental for the thermodynamics of irreversible processes. 

These are the famous Onsager reciprocal relations. Thus there is symmetiy in the ability of a potential to create a flux/, and of the ability of a potential to create a flux The reciprocal relations arc experimentally verifiable connections between effects which superficially might appear to be independent. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The four relations (I.)—(IV.) are usually known as Maxwell s Relations, or the Reciprocal Relations they were deduced by Maxwell by means of an ingenious geometrical method Theory of Heat, Chap. 9). 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

The nonlinear transport matrix satisfies the reciprocal relation... 

The synthesis of ideas described above was first achieved by Maxwell and is described by two equations that summarize the essential experimental observations pertaining to the reciprocally related electromagnetic effects1. In SI units and vector notation these equations, valid at every ordinary point... 

In general, the method is precise to about 2% of the end group concentration. The corresponding uncertainty in M will vary from case to case because of the reciprocal relation between the two. 

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