Continuum limit

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... 

It is easy to see that this expression has two minima within the Brillouin zone. One minimum is at fc = 0 and gives the correct continuum limit. The other, however, is at k = 7t/a and carries an infinite momentum as the lattice spacing a 0. In other words, discretizing the fermion field leads to the unphysical problem of species doubling. (In fact, since there is a doubling for each space-time dimension, this scheme actually results in 2 = 16 times the expected number of fermions.)... 

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) ... 

Rettori and Villain (1988), and Langon and Villain (1990) have written down equations of motion for the one-dimensional groove profile in both discrete and continuous forms. In the discrete form, the variables are the average positions x , t) of step n in the step train leading from the top to bottom of the groove (or vice versa). In the continuous form, the surface profile is specified by a height function h(x, t). The equation for h(x,t) can be obtained from the equations for x (0 by taking a suitable continuum limit. 

For small particles, subject to noncontinuum effects but not to compressibility, Re is very low see Eq. (10-52). In this case, nonradiative heat transfer occurs purely by conduction. This situation has been examined theoretically in the near-free-molecule limit (SI4) and in the near-continuum limit (T8). The following equation interpolates between these limits for a sphere in a motionless gas ... 

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 translational partition function trans is evaluated in the high-7" continuum limit as... 

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

In the classical limit, the equation for ktt takes on the relatively simple form shown in equation (29). In equation (29), both intramolecular (A ) and solvent (A0) contributions are included. 2 was defined in equation (22) and A0 in equation (23) in the dielectric continuum limit. 

The expression for ket in equation (29) is still not a complete expression for the total electron transfer rate constant. Both the electronic coupling term V and A0 are dependent upon the interreactant separation distance r, and, therefore, so is ktt in equation (29). The dependence of /.0 on r is shown in equation (23) in the dielectric continuum limit. The magnitude of V depends upon the extent of donor-acceptor electronic orbital overlap (equation 17) and the electronic wave-functions fall off exponentially from the centers of the reactants. In order to make comparisons between ktt and experimental values of electron transfer rate constants, it is necessary to include the dependence of ktt on r as discussed in a later section. 

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... 

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