The Continuum Limit

If Qj is sufficiently small such that InIQ. is much larger than interatomic distances in the condensed phase, then the medium can be treated as a continuum, though its nuclear density need be neither constant nor homogeneous. In vacuum, the total energy of the neutron outside the medium is simply its kinetic energy 

In the continuum limit, the effective potential energy in the condensed phase is given by Eq. (3.11) and the total neutron energy is 

The procedure can be expanded in a piecewise fashion [17,18] to obtain the reflection and transmission amplitudes arising from the reflection of neutrons from an arbitrary potential or SLD profile if the potential is divided into a discrete number (j) of rectangular lamellae. The reflection and transmission amplitudes are obtained from a pair of simultaneous equations which, when written in matrix notation, define the transfer matrix  

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In the continuum limit we define the probability of a path The probability is a functional of the path. 

After some manipulations, taking the continuum limit of Eqs. (4) and (S), assuming that... 

We now sketch a simple deterministic lattice gas model of diffusion that becomes exactly Lorentz invariant in the continuum limit. We follow Toffoli ([toff89], [tofiSOb]) and Smith [smithm90]. 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

The second term on the right-hand side of Eq. (146) vanishes in the continuum limit when use is made of electrical neutrality. For the defects in the impure crystal the term is again zero. In the intrinsic case it is not identically zero but is much smaller than the other terms (details can be found in Ref. 4). The final term to be evaluated in Eq. (146) is found, by substituting the asymptotic value of mu, to be... 

In numerical work quoted below the summations in Eq. (166) were treated in the same way as those in Eq. (165) and appreciable differences from the continuum limit value were found.4... 

For simplicity we consider only the continuum limit (i.e. Mayer ionic solution theory). The last equation allows us to calculate the value of p which the association theory should predict in order to be compatible with the true value, which we assume to be given by the Mayer theory in the range considered. It is... 

It can also be seen that the value of p calculated from Eq. (183) will not be identical with the degree of association defined in terms of distribution functions except at infinite dilution. In the continuum limit we have70... 

To obtain the early-time behavior of the mean-square width, we take the continuum limit of the sum over q, note that the integrand is even, convert to dimensionless variables, and integrate by parts (Khare, 1996) ... 

The continuum limit of the Hamiltonian representation is obtained as follows. One notes that if the friction function y(t) appearing in the GLE is a periodic function with period T then Eq. 4 is just the cosine Fourier expansion of the friction function. The frequencies coj are integer multiples of the fundamental frequency and the coefficients Cj are the Fourier expansion coefficients. In practice, the friction function y(t) appearing in the GLE is a decaying function. It may be used to construct the periodic function y(t T) = Y(t TiT)0(t-... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

In an earlier paper (S 044], anomalies in the angular moments of the monomer-monomer distribution function for flexible polymers are established. It is shown here how these anomalies arise from the tetrahedral symmetry of the three-state RIS model and how they disappear in the continuum limit of torsional conformations. It is concluded that the eighth and higher radial moments contain spurious contributions when calculated within the usual three-state model. 

The rotational partition function qimt can be similarly evaluated in the continuum limit as... 

Table 4. Viscoelastic functions of the linear array in the continuum limit...

In this book, it is assumed that the continuum limits exist and coarse-grained functions can be obtained that do not depend significantly on the choice of . 

This defines the local flux Ji(r) as the continuum limit of... 

The free energy ivff] must now be varied with respect to the location f as well as with respect to the transformation coefficients ao, ap j = 1,.. . , N. The details are given in Ref. 107 and have been reviewed in Ref. 49. The final result is that the frequency A and collective coupling parameter C are expressed in the continuum limit as functions of a generalized barrier frequency X One then remains with a minimization problem for the free energy as a function of two variables - the location f and X Details on the numerical minimization may be found in Refs. 68,93. For a parabolic barrier one readily finds that the minimum is such that f = 0 and that X = X. In other words, in the parabolic barrier limit, optimal planar VTST reduces to the well known Kramers-Grote-Hynes expression for the rate. 

Cell to Cell Transport Phenomenology. Consider a mass of cells in an external medium denoted "e" as in Fig. 1. We shall construct a continuum approximation wherein the cells are "smeared out." Such a theory will be accurate when the length scale of interest (i.e., the pattern length) is greater than a cell diameter. An important aspect of the formulation is that the equations derived by taking the continuum limit will relate effective transport rates directly to measurable properties of individual cells or pairs of cells in contact. 

X-ray diffraction is an instructive example of such a nonlocal response. The material is polarizable in proportion to the local density of electrons. It is not polarizable at all points along the sinusoidal wave. The structure factor of x-ray diffraction describes the nonlocal response to a wave that is only weakly absorbed but that is strongly bent by the way its spatial variation couples with that of the sample to which it is exposed. Reradiation from the acceleration of the electrons creates waves that reveal the electron distribution. In no way can the scattering of the original wave be described or formulated in the continuum limit of featureless dielectric response. Because x-ray frequencies are often so high that the material absorbs little energy, it is possible to interpret x-ray scattering to infer molecular structure. 

PROBLEM L2.18 Show that, in the regime of pairwise summability, the continuum limit is violated by terms of the order of (a/z)2 where, just here, a is atomic spacing. 

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