0) \eqno(5.6) $$ \NI where the initial and boundary conditions (2.9) are replaced by $$ C(x=0,\tau \geq 0)=1-m^2(\tau ) \;\;\; {\rm and} \;\;\; C(x>0,\tau =0) =0 \eqno(5.7) $$ \NI with $$ m(\tau )=1-(1-\mu ){\sl e}^{-\tau } \eqno(5.8) $$ \NI The interface density in the continuum limit is approximated as follows, $$ {\rho \over \sqrt{p}} \simeq \left[- {\D C (x,\tau ) \over \D x } \right]_{x=0} \eqno(5.9) $$ The reader should keep in mind that the continuum limit is an \UN{approximation} valid asymptotically for $0\leq p \ll 1$, $\; t\gg 1$, $\; n\gg 1$. The results must be properly interpreted. For instance, if taken literally, relation (5.9) would imply that the interface density is infinite at $\tau=0$ because the initial conditions for $C(x,\tau )$ are step-like. In fact, the divergence is in the regime where the continuum limiting approximation breaks down; see the next section. The rescaling (5.4)-(5.5) also obscures the $p=0$ case. Indeed, the results must be properly expressed in terms of the variable $x/ \tau^2 = n/t^2$ before taking the limit $p \to 0$. If fact, $p=0$ is reminiscent of the ``critical-point'' limit in which there are no small parameters to rescale $n$ and $t$. Instead, only their ``scaling combination'' enters in the continuum limit. The solution of the equation (5.6) with conditions (5.7) is obtained by the Laplace Transform method. We omit the mathematical details and only quote the final expression, $$ \int\limits_0^\infty {\sl e}^{-\omega \tau } C(x, \tau) d \tau =\left[ {2(1-\mu ) \over \omega +1} - {(1-\mu )^2 \over \omega +2} \right] {\sl e}^{-\sqrt{\omega+2}\, x} \eqno(5.10)$$ \NI which inverse-transforms to $$ {C(x, \tau ) \over 1-\mu } = {\sl e}^{-\tau } \left[ {\sl e}^x {\rm erfc}\! \left({x \over 2 \sqrt{\tau } } + \sqrt{\tau } \right) +{\sl e}^{-x} {\rm erfc}\! \left({x \over 2 \sqrt{\tau } } - \sqrt{\tau } \right) -(1-\mu ) {\sl e}^{-\tau } {\rm erfc}\! \left({x \over 2 \sqrt{\tau } } \right) \right] \eqno(5.11) $$ \NI where $$ {\rm erfc}(\alpha )={2 \over \sqrt{\pi } }\int\limits_\alpha^\infty {\sl e}^{-\beta^2}d\beta \eqno(5.12) $$ \NI is one of the standard error functions, the properties of which are well known. Thus, the expression (5.11) can be used to analyze various properties of connected two-point correlations. Some such results will be presented in the next section. \NP \NI{\bf 6. LENGTH SCALES AND BREAKDOWN OF SCALING} Let us consider the large-$x$ behavior of $C(x,\tau)$ for fixed $\tau > 0$ (and $\mu \neq 1$). All three terms in (5.11) then follow the asymptotic large-argument behavior of the error function. The result turns out to be $$ C(x\to \infty, \tau) \propto {\sl e}^{-2\tau } \left[ {\sqrt{\tau } \over x} \exp \left( - {x^2 \over 4\tau }\right) \right] \eqno(6.1) $$ \NI where we omitted the proportionality constant. The decay-tail length scale, $n_{\rm tail}$, is thus determined by the dependence on the diffusional combination $x^2/\tau =n^2/t$, $$ n_{\rm tail} \sim \sqrt{t} \eqno(6.2) $$ Consider next the moment-definition length scales. We define the $k^{\rm th}$ moment, $$ M_k(\tau )=\int\limits_0^\infty x^k C(x,\tau ) dx \eqno(6.3) $$ \NI and the associated time-dependent length, $n_k (t)$, $$ \sqrt{p}\, n_k = \left( M_k \big / M_0 \right)^{1/k} \eqno(6.4) $$ \NI In the evaluation of $M_k$, the contribution due to the first term in (5.11) can be always used in its large-argument form, while the third term is originally a function of the diffusional combination (times ${\sl e}^{-2 \tau }$). The second term, however, can be written in such a diffusional-scaled form only for $$ x \gg a_1 \sqrt{\tau} + a_2 \tau \eqno(6.5) $$ \NI where from now on the coefficient notation $a_j$ will be defined to stand for ``a slow varying function of $\tau$, of order 1, possibly $k$-dependent (when implied by context).'' The diffusional contribution to the moments is $$ M_k^{\rm (diff)} = a_3 \tau^{(k+1)/2}\, {\sl e}^{-2 \tau } \eqno(6.6) $$ \NI In the range of smaller $x$, not satisfying (6.5), the error function in the second term in (5.11) becomes of order 1. In fact, the fixed-$x$, large-time behavior $$ C(x, \tau\to\infty) \propto {\sl e}^{-\tau -x} \eqno(6.7) $$ \NI is explicit in the Laplace-transformed form (5.10) due to the rightmost pole singularity at $\omega =-1$. The added contribution due to this exponential behavior is of the form $$ M_k^{\rm (exp)} = a_4 {\sl e}^{-\tau } \int\limits_0^{ a_1 \sqrt{\tau} + a_2 \tau} x^k {\sl e}^{-x} dx \eqno(6.8) $$ \NI For small $\tau$, the intergation will yield the same power of $\tau $ as in (6.6). Thus, the moment length scales behave according to $$ n_k \sim \left(\tau^{(k+1)/2} \big / \tau^{1/2}\right)^{1/k}\, p^{-1/2}= \sqrt{t} \;\;\;\;\;\;\;\; (t\ll 1/{p}) \eqno(6.9) $$ \NI However, as $\tau $ increases, the integral in (6.8) saturates at a value of order 1. Since the remaining time dependence, ${\sl e}^{-\tau }$, dominates that of the diffusive contribution (6.6), the length scales saturate at $$ n_k \sim 1/\sqrt{p} \;\;\;\;\;\;\;\; (t\gg 1/{p}) \eqno(6.10) $$ \NI The crossover between the limiting behaviors occurs at $t \sim 1/p$ and is difficult to evaluate in closed form. We next turn to the density of interfaces and the associated length scale. A direct calculation of the right-hand side of (5.9) yields $$ {\rho \over \sqrt{p}} \simeq {1-\mu^2 \over \sqrt{\pi \tau } } {\sl e}^{-2\tau} +2(1-\mu ){\sl e}^{-\tau} {\rm erf} (\sqrt{\tau}) \eqno(6.11) $$ \NI where we kept the approximation sign to emphasize that this result applies only for $t\gg 1$ (as well as $p\ll 1$). Note that ${\rm erf} (\alpha)=1-{\rm erfc} (\alpha)$. The limit $p=0$ is thus correctly reproduced; see (4.3). For small $\tau$, the first term in (6.11) dominates, and the associated length scale behaves according to $$ n_\rho \propto \rho^{-1} \sim \sqrt{t} \;\;\;\;\;\;\;\; (t\ll 1/p) \eqno(6.12) $$ \NI However, for large $\tau$ the second term takes over. Noting that the function ${\rm erf} (\alpha )$ approaches 1 for large $\alpha$, we conclude that $$ n_\rho \sim {\sl e}^{pt} \big / \sqrt{p} \;\;\;\;\;\;\;\; (t\gg 1/p) \eqno(6.13) $$ In the theories of structure-factor scaling,$^{(4)}$ where the structure factor is defined as the spatial Fourier transform of $C(n,t)$, assuming continuous coordinate $n$ and time $t$, the momentum, $q$, dependence is scaled in the form $\hat n (t) q$. In the direct-space notation this amounts to assuming that the coordinate dependence enters via $n/\hat n(t)$. It is tempting to associate $\hat n(t)$ with a typical cluster size measure. In practice, $\hat n$ is determined as the inverse of some momentum scale found at low or fixed $q$-values, corresponding to large or intermediate coordinate values $n$. Our results support this picture \UN{at coexistence}, i.e., at $p=0$. Indeed, due to the critical-point-like scaling expressed by the diffusive scaling combination $n^2/t$, all length scales defined at short or large distances are essentially the same. The identification $\hat n \sim \sqrt{t}$ is unambiguous. However, explicit expressions obtained for $p > 0$ indicate two difficulties with the structure-factor scaling when the growth of the stable phase occurs off coexistence. Firstly, the identification of a unique length scale is no longer possible for large times for which the cluster size distribution deviates significantly from the symmetric case. All three length scales estimated behave differently for large $t$. Secondly, the two-point correlation function no longer has simple scaling properties. In fact, a more general conclusion, alluded to in Section~4, is that in such cases the length scale $n_\rho (t)$ is the appropriate one to use as a typical $+$ cluster size. However, it is characteristic only of the \UN{short-distance} coordinate dependence of the two-point function. In summary, we presented a solvable $1D$ model of cluster growth. Our results indicate that the ideas of structure-factor scaling apply only to cluster coarsening at coexistence. Off coexistence, a typical stable-phase cluster size measure reflects only the short-distance properties of the two-point correlations; the full correlation function no longer obeys scaling. The author acknowledges helpful discussions with M.A.~Burschka, C.R.~Doering, D.A.~Rabson and R.B.~Stinchcombe. 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