Reciprocity theorem generalized

We can also use the 1-D Green s theorem, equation (15.51), to prove the reciprocity theorem in one dimension, similar to the general reciprocity theorem 27 ... 

In addition to the theoretical advantages offered by the principle of minimum potential energy, our elastic analyses will also be aided by the reciprocal theorem. This theorem is a special example of a more general class of reciprocal theorems and considers two elastic states (u ), or(i)) and where each... 

The preceding sections have been concerned primarily with direct solution techniques for problems in creeping-flow theory. Here, we discuss several general topics that evolve directly from these developments. The first two involve application of the so-called reciprocal theorem of low-Reynolds-number hydrodynamics. 

The reciprocal theorem is derived directly from the general integral formula, (8-111). For this purpose, we identify u and u, as well as T and T, as the solutions of two creeping-flow problems for flow past the same body but with different boundary conditions on the body surface 3 D. 

Problem 8-9. Torque on a Sphere in a General Stokes Flow. Use the reciprocal theorem for Stokes equations to derive the following expression for the torque exerted on a sphere of radius a that is held fixed in the Stokes flow u°°(x) ... 

Note To apply the reciprocal theorem, the shape of the drop for the complementary problem generally would have to be exactly the same as the shape in the problem of interest. However, because the original problem can be reduced by means of domain perturbations to an equivalent problem with the boundary conditions applied at the spherical surface, r = 1, we may also conveniently choose the drop to be spherical for this complementary problem. 

KKTs are tools brought to network theory by the work of Kramers (1926) and Kronig (1929) on X-ray optics. Just as the reciprocity theorem, they are purely mathematical rules of general validity in any passive, linear, reciprocal network of a minimum phase shift type. By minimum-phase networks, we mean ladder networks that do not have poles in the right half plane of the Wessel diagram. A ladder network is of minimum phase type a bridge where signal can come from more than one ladder is not necessarily of the minimum-phase type. The transforms are only possible when the functions are finite-valued at all frequencies. With impedance Z = R- -jX the transforms are ... 

Hurley, J. Garrod, C. (1982). Generalization of Onsager reciprocity theorem. Physical Review Letters, 48 (23), pp. 1575. 

Unlike other branches of physics, thermodynamics in its standard postulation approach [272] does not provide direct numerical predictions. For example, it does not evaluate the specific heat or compressibility of a system, instead, it predicts that apparently unrelated quantities are equal, such as (1 A"XdQ/dP)T = - (dV/dT)P or that two coupled irreversible processes satisfy the Onsager reciprocity theorem (L 2 L2O under a linear optimization [153]. Recent development in both the many-body and field theories towards the interpretation of phase transitions and the general theory of symmetry can provide another plausible attitude applicable to a new conceptual basis of thermodynamics, in the middle of Seventies Cullen suggested that thermodynamics is the study of those properties of macroscopic matter that follows from the symmetry properties of physical laws, mediated through the statistics of large systems [273], It is an expedient happenstance that a conventional simple systems , often exemplified in elementary thermodynamics, have one prototype of each of the three characteristic classes of thermodynamic coordinates, i.e., (i) coordinates conserved by the continuous space-time symmetries (internal energy, U), (ii) coordinates conserved by other symmetry principles (mole number, N) and (iii) non-conserved (so called broken ) symmetry coordinates (volume, V). 

To obtain the group velocity, we need a generalized reciprocity theorem which allows for variations in the frequency w of the implicit time dependence of the fields. We let E and H be the fields of the jth mode at wavelength L These fields satisfy the source-free Maxwell equations, and if we allow for variations in the permeability p, they have the form... 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

In (6.5) the subscript n indicates the band index and fe is a continuous wave vector that is confined to the first Brillouin zone of the reciprocal lattice. The index n appears in the Bloch theorem because for a given k there are many solutions to the Schrodinger equation. Because the eigenvalue problem is set in a fixed finite volume, we generally expect to find an infinite family of solutions with discretely-spaced eigenvalues which we label with the band index n. The wave vector k can always be confined to the first Brillouin zone. The vector k takes on values within the Brillouin zone corresponding to the crystal lattice, and particular directions like r,A,A,Z (see Figures 4.13-4.15). 

In the following we are going to introduce the generalized Onsager constitutive theory for a non-linear system of constitutive (9) satisfying the (18) reciprocity relations, the (13) equilibrium conditions and the (10) second law of thermodynamics. In the following we shall present that the Edelen s decomposition theorem [4] is valid in every class of the thermodynamic forces, which are two times continuously differentiable with respect to fluxes. 

Assume that the entropy production function two times continuously differentiable function with respect to fluxes. Then from the previous first equations and the Young-theorem of the analysis we get the generalized reciprocal relations... 

An important theorem relates the Fourier transform and convolution operations. The Convolution Theorem (8,9) states that the Fourier transform of a convolution is the product of the Fourier transfomis, or F (f g) = F(u)G(u). Applying this to the autocorrelation yields F Kx) f(-x)] = F(u)F(-u). If f(x) is real, F(u)F(-u)=F(u)F (u)= F(u)p. Thus, "the Fourier transform ofthe autocorrelation of a function frx) is the squared modulus of its transform" (Ref. 9, p. 81). Application to scattering replaces frx) with the electron density profile, p(x). We have then the important result that the Fourier transform of the autocorrelation of the electron density profile is exactly equal to the intensity in reciprocal space, F(u). The autocomelation function cf the electron density has a special name it is called the generalized Patterson fimction(8), P(x), given by ... 

This demonstration of generalized minimum production principle has been given by Callen. This theorem is very general and seems to have wide applicability. However, the proof of this theorem is so closely connected with the reciprocity relations that some restriction will appear in applying it to practical problems, whereas the principle of minimum entropy production in the macroscopic description holds even when the reciprocal relations are not used. 

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