0$. The marginal logarithmic vanishing of the rate function is difficult to observe numerically. Our Monte Carlo data were obtained for $k=3,4,5,6$, on lattices of sizes 2000 with periodic boundary conditions. Each data set was averaged over 50 independent runs, all with random initial conditions. A few runs, order 10, were also made for $k=10$. We replace $H$ by $R$ in the mean-field approximation (5) and all relations that derive from it. The Monte Carlo time steps were selected to have the microscopic hopping attempt rate $H=1$. Thus we expect the effective rate values to satisfy $R<1$. Our typical numerical results are illustrated in Figures 3-6. Integration of the mean-field relation yields the function $t(\rho )$ as follows, $$ q 2^k k R \left[ t(\rho ) - t_0 \right] = I(\rho ) - I(\rho_0) + \ln {2- \rho \over 2 - \rho_0 } \;\; , \eqno(12) $$ \NI where $$ I(\rho ) \equiv -\ln \rho + \sum_{j=1}^{k-1} {2^j \over j \rho^j} \;\; , \eqno(13) $$ \NI provided that mean-field theory applies for times $t \geq t_0$, with the corresponding density $\rho_0 = \rho (t_0)$. As exemplified by Figures~3-6, our data eventually reached the large-time behavior not sensitive to the initial density. This observation suggests that the large-time value $R$ only depends on $q$, but not on $\rho(0)$. Thus, we used this large time data to fit the $R$ value from the relation (12) in which the terms which are of order 1 for small densities were discarded, $$ R t \simeq I(\rho )/q 2^k k \;\; . \eqno(14) $$ \NI These curves are shown by dashed lines in Figures 4 and 6, while in Figures 3 and 5 they were too close to other lines (solid lines marked $0.8$) to be shown. On a double-logarithmic plot, variation of the trial $R$ value corresponds to translation of the dashed line along the $\, \log t \, $ axis. Thus, we only obtain the estimates of $\, \log R$. The accuracy of the resulting $R$ values is at best semiquantitative. The values of the large-time asymptotic rate constant, $R$, are summarized in Table 1. The overall trend is as expected from our heuristic discussion of the validity of the mean-field approximation. The effective rates $R/H$ approach $1$ for small $q$, while for $q \simeq 1$ the mean-field relation applies with substantially renormalized values $R3$ counterparts with the same $q$. The dashed lines in Figures 3 and 5, defined by (14), deviate in two ways from similar relations predicted by the simpler rate equation (2) with $\Gamma = 2qkb^{k-1} R$ (which are not shown in the figures). Firstly, the curves differ for short times. A more interesting observation is that for larger times, i.e., for smaller $\rho$ values, the dashed lines in Figures 3 and 5 look nearly straight. However, their slope is somewhat steeper than the prediction of the asymptotic rate equation (2): slope $-1/(k-1)$. Indeed, this deviation is quite small for $k\leq 6$. It becomes more profound as $k$ increases, as was found in our preliminary, limited-statistics runs for $k=10$ as well as in Monte Carlo simulations [13] of related deposition models up to $k=10$. Of course, asymptotically the slope slowly approaches the rate-equation value, for times defined by (6). Statistical noise in our data precluded direct estimation of the rate function (11) because evaluation of the time derivative turns out to be particularly sensitive to statistical fluctuations. However, we used the $R$ values estimated for large times (Table 1) and the initial values $\rho(0)$ to draw mean-field curves (12) corresponding to different initial densities. As expected, the short-time behavior of the data is fitted well only for small $\rho(0)$; the quality of the fit improves as $q\to 0$. These properties are illustrated in Figures 3-6 (solid curves). They were shared by all our data sets listed in Table 1. As discussed earlier, the mean-field theory either fails for short times or applies with the effective rate constant larger than $R$, unless the initial density and $q$ are both sufficiently small. Only in the latter case the fixed-$R$ mean-field approximation extends down to $t=0$. In summary, we analyzed the applicability of a mean-field approximate equation accounting for the hard-core particle dynamics, to chemical reactions in $1d$. Some of our conclusions are generally valid for $d \geq 1$; these include the fact that difficulties with the simplest rate equations for large $k$ are not inherent to mean-field approximations and can be repaired by accounting for the hard-core interactions, although our explicit results were limited to the one-dimensional case. Another well known general feature illustrated by our $1d$ studies, is that mean-field theories break down in those cases when local fluctuations dominate the dynamics of the reaction. Classification of borderline $d$-values at and below which the mean-field theory breaks down for multiparticle-input reactions $k_1 A_1 + k_2 A_2 + k_3 A_3 + \ldots \to {\rm inert}$, etc., by scaling agruments, can be found in [3,14]. However, even in the regimes where the local fluctuations are irrelevant asymptotically, the $1d$ model studies emphasize the fact that mean-field theories can only be applied as ``effective'' asymptotic approximations, with renormalized, ``hydrodynamic'' rate parameters. Our study further suggests that with careful choice of a mean-field equation, one-parameter data fits work quite adequately for large times and in some cases describe the behavior down to $t=0$. The authors wish to thank M.~Barma for helpful comments and suggestions. This research was partially supported by the Science and Engineering Research Council (UK) under grant number GR/G02741. One of the authors (V.P.) also wishes to acknowledge the award of a Guest Research Fellowship at Oxford from the Royal Society. \NP \centerline{\bf REFERENCES} {\frenchspacing \item{1.} M. Bramson and D. Griffeath, Ann. Prob. {\bf 8}, 183 (1980). \item{2.} D.C. Torney and H.M. McConnell, J. Phys. Chem. {\bf 87}, 1941 (1983). \item{3.} K. Kang, P. Meakin, J.H. Oh and S. Redner, J. Phys. A{\bf 17}, L665 (1984). \item{4.} T. Liggett, {\sl Interacting Particle Systems\/} (Springer-Verlag, New York, 1985). \item{5.} Z. Racz, Phys. Rev. Lett. {\bf 55}, 1707 (1985). \item{6.} A.A. Lushnikov, Phys. Lett. A{\bf 120}, 135 (1987). \item{7.} M. Bramson and J.L. Lebowitz, Phys. Rev. Lett. {\bf 61}, 2397 (1988). \item{8.} Review: V. Kuzovkov and E. Kotomin, Rep. Prog. Phys. {\bf 51}, 1479 (1988). \item{9.} D. ben--Avraham, M.A. Burschka and C.R. Doering, J. Stat. Phys. {\bf 60}, 695 (1990). \item{10.} V. Privman and M. Barma, Oxford preprint {\sl OUTP--92--24S\/} (1992). \item{11.} L. Braunstein, H.O. Martin, M.D. Grynberg and H.E. Roman, J. Phys. A{\bf 25}, L255 (1992). \item{12.} V. Privman and P. Nielaba, Europhys. Lett. {\bf 18}, 673 (1992). \item{13.} P. Nielaba and V. Privman, Modern Phys. Lett. B (1992), in print. \item{14.} S. Cornell, M. Droz and B. Chopard, Physica A (1992), in print. } \NP \centerline{\bf TABLE} \hphantom{AA} \NI\hang {\bf Table~1.}~$\;$Large time estimates of the phenomenological mean-field rate constant $R \leq H$, based on Monte Carlo data up to $t=10^6H^{-1}$. Due to statistical noise in the data, the values $R/H$ shown are semiquantitative; no reliable error limits can be offered. (The $q=0.001$ Monte Carlo run was only for $k=3$. Limited-statistics results for $k=10$, up to times $tH=10^7$, were also obtained, but no reliable $R$ estimates can be offered.) \vskip 0.20 in $$\vbox{ \settabs\+ &AAAAAAAAA&AAAAAAAA&AAAAAAAA&AAAAAAAA&AAAAAAAA\cr \+ & & $k=3$ & $k=4$ & $k=5$ & $k=6$ \cr \+ & $q=1$ & $0.05$ & $0.13$ & $0.18$ & $0.2$ \cr \+ & $q=0.1$ & $0.3$ & $0.5$ & $0.6$ & $0.6$ \cr \+ & $q=0.01$ & $0.7$ & $0.9$ & $0.9$ & $0.9$ \cr \+ & $q=0.001$ & $0.96$ &\NI{------} &\NI{------} &\NI{------} \cr }$$ \NP \centerline{\bf FIGURE CAPTIONS} \NI\hang {\bf Fig.~1.}~$\;$The active particle {\bf a} attempts to hop to the right. The upper panel shows a configuration in which the attempt is successful: the particle will move to the empty site {\bf e}. The lower panel shows a blocked configurations. In both cases, the $k-1$ lattice sites in the direction of the hopping attempt are occupied ($k=5$ here). Thus, all $k$ particles shown will annihilate with probability $q$. \NI\hang {\bf Fig.~2.}~$\;$The function defined by the right-hand side of the inequality (10), shown for $k=2,3,4,5$. \NI\hang {\bf Fig.~3.}~$\;$Numerical data for $k=6$ and $q=1$, with the initial densities $\rho(0)=0.8$ ($\circ$), $0.5$ ($\triangle$), 0.2 ($\square$). The solid lines, labeled by the $\rho (0)$ values, represent the mean-field relation (12) with the large-time $R$ value, forced to obey the initial condition $\rho = \rho(0)$ at $t=0$. \NI\hang {\bf Fig.~4.}~$\;$Numerical data for $k=6$ and $q=0.01$, with the initial densities $\rho(0)=0.8$ ($\circ$), $0.5$ ($\triangle$), 0.2 ($\square$). The solid lines, labeled by the $\rho (0)$ values, represent the mean-field relation (12) with the large-time $R$ value, forced to obey the initial condition $\rho = \rho(0)$ at $t=0$. The dashed line indicates the large-time asymptotic expression (14) used to fit the $R$ value. \NI\hang {\bf Fig.~5.}~$\;$Numerical data for $k=3$ and $q=0.1$, with the initial densities $\rho(0)=0.8$ ($\circ$), $0.5$ ($\triangle$), 0.2 ($\square$). The solid lines, labeled by the $\rho (0)$ values, represent the mean-field relation (12) with the large-time $R$ value, forced to obey the initial condition $\rho = \rho(0)$ at $t=0$. \NI\hang {\bf Fig.~6.}~$\;$Numerical data for $k=3$ and $q=0.001$, with the initial densities $\rho(0)=0.8$ ($\circ$), $0.5$ ($\triangle$), 0.2 ($\square$). The solid lines, labeled by the $\rho (0)$ values, represent the mean-field relation (12) with the large-time $R$ value, forced to obey the initial condition $\rho = \rho(0)$ at $t=0$. The dashed line indicates the large-time asymptotic expression (14) used to fit the $R$ value. \hphantom{A} \NI {\bf To get the preprint with the figures write to \NL PRIVMAN@CRAFT.CAMP.CLARKSON.EDU} \bye