Inhomogeneous term

Now suppose the energy density of the electrons in bulk jellium is given by nf(n), so that the density functional f(n) is the energy per electron of a uniform electron gas of density n in the positive background. (The integral of nf(n), plus inhomogeneity terms, is the quantity one minimizes to obtain the density profile.) The pressure in bulk jellium is then n(df/dn)9 so that... 

To the lowest order, we may neglect all correlations and take w) eZpE/fp in Eq. (262) in this case the velocity field term in (273) behaves like exp( —Kr)/r for large distances, while the inhomogeneous term on the l.h.s. 

Equation (13) has a nonoscillatory solution only if its inhomogeneous terms vanish—that is, the dc field must be adjusted to give... 

The group space of 0(3) is doubly connected (i.e., non-simply connected) and can therefore support an Aharonov-Bohm effect (Section V), which is described by a physical inhomogeneous term produced by a rotation in the internal gauge space of 0(3) [24]. The existence of the Aharonov-Bohm effect is therefore clear evidence for an extended electrodynamics such as 0(3) electrodynamics, as argued already. A great deal more evidence is reviewed in this article in favor of 0(3) over U(l). For example, it is shown that the Sagnac effect [25] can be described accurately with 0(3), while U(l) fails completely to describe it. 

It is useful to go through this derivation in detail because it produces the inhomogeneous term responsible for the Aharonov-Bohm effect in 0(3) electrodynamics. The effect of the rotation may be written as... 

The Aharonov-Bohm effect is self-inconsistent in U(l) electrodynamics because [44] the effect depends on the interaction of a vector potential A with an electron, but the magnetic field defined by = V x A is zero at the point of interaction [44]. This argument can always be used in U(l) electrodynamics to counter the view that the classical potential A is physical, and adherents of the received view can always assert in U(l) electrodynamics that the potential must be unphysical by gauge freedom. If, however, the Aharonov-Bohm effect is seen as an effect of 0(3) electrodynamics, or of SU(2) electrodynamics [44], it is easily demonstrated that the effect is due to the physical inhomogeneous term appearing in Eq. (25). This argument is developed further in Section VI. 

If J(t) from Eq. 21-11 or 21-12 is inserted into Eq. 21-4, we get a linear differential equation with a time variable inhomogeneous term but constant rate k. The corresponding solution is given in Box 12.1, Eq. 8. Application of the general solution to the above case is described in Box 21.3. The reader who is not interested in the mathematics can skip the details but should take a moment to digest the message which summarizes our analytical exercise. 

In this section we have only discussed the effect of the temporal variability of the inhomogeneous term (the external force) on the solution of Eq. 21-4. The solution of the FOLIDE given in Box 21.1 includes the effect of a variable rate constant, k(i). A situation in which both terms (Jand k) vary with time is described in Illustrative Example 21.2. 

Thus, we have reduced Eq. 21-30b to the already-familiar case of a FOLIDE with variable inhomogeneous term. Its solution is given by Eq. 8 of Box 12.1. If the initial concentration of B is assumed to be zero (CBo = 0), we get after some algebraic manipulation ... 

Since there is no back reaction from B to A, the resulting differential equations are hierarchical. Furthermore, since after the accident the input of A and B is zero, the inhomogeneous terms JA and JB (see Eq. 21-30) are zero. The initial concentrations are ... 

Here y and Aa are constant coefficients which depend on the matrix K, on the inhomogeneous terms Ja, and, for the case of the ap s, on the initial values of the variables, y . The ka are the negative eigenvalues of the matrix K. As mentioned above, these eigenvalues are negative or zero, so that the ka are positive or zero ... 

J is the vector of the transformed inhomogeneous terms. With the special form of A, Eq. 4 decays into n uncoupled differential equations ... 

Exercise. W has one eigenvalue zero. Hence (4.11) can only be solved when the inhomogeneous term on the left is orthogonal to the left null eigenvector of W. Show that this condition is fulfilled. On the other hand, the solution is not unique show that the requirement (4.9) takes care of that. 

In contrast to (2.2) there is an inhomogeneous term but there are no boundary conditions. (It is not true that nR(x) must be equal to unity at x = R, because when the particle arrives at R it need not exit but may jump back into the interval.) The integral equation is sufficient to determine nR(x) uniquely. In fact, suppose that (7.3) had two different solutions then their difference nR x) obeys... 

I. Linear differential equations in which only the inhomogeneous term is a random function, such as the Langevin equation. Such equations have been called additive and can be solved in principle. 

We will ignore these inhomogenous terms. The polarization vector is going to have contributions from the linear electric susceptibility and the nonlinear electric susceptibility due to the nonlinear response of the atoms ... 

Since this field equation is obtained by the Euler-Lagrange equation the inhomogenous term is the result of... 

For the boundary value problems in this paper, the Lx and L2 are linear Laplacian operators, the R and R2 disappear, the inhomogeneous terms g, (x) and g2(x) also equal to zero. The function f, f2 arising from integrating should be fiX = a0 + axx and... 

Secondly, we note that Eq. (29), having the structure of a KS equation with an additional inhomogeneity term, plus the boundary condition that V iir(r) tends to zero as r —> oo uniquely determines xp a(r). We can prove this statement by assuming that there are two independent solutions V i<7,1(r) and V Tl2(r) of Eq. (29). Then the difference between these two solutions, T a(r) = V io-.tW — iv,2(r)> satisfies the homogeneous KS equation... 

A fiuther interesting aspect of Eq. (2.13) is afforded by the inhomogeneous term [third term on die right-hmd side of Eq. (2.13)]. Since the initial condition of the system in general is not such as to satisfy... 

The roughest approximation consists of replacing the time-ordered exponential appearing in the second term on the right-hand side of Eq. (2.18) by 1. Since we intend to explore cases where the inhomogeneous term should not play any significant role (see Chapter II), Eq. (2.18) becomes... 

For simplicity we shall disregard the inhomogeneous term appearing in Eq. (3.5) by assuming that... 

In Section VII we shall justify our more general neglect of the inhomogeneous term. 

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