  We study the reaction kinetics of end-functionalized polymer chains dispersed
in an unreactive polymer melt. Starting from an infinite hierarchy of coupled
equations for many-chain correlation functions, a closed equation is derived
for the 2nd order rate constant $k$ after postulating simple physical bounds.
Our results generalize previous 2-chain treatments (valid in dilute reactants
limit) by Doi, de Gennes, and Friedman and O'Shaughnessy, to arbitrary initial
reactive group density $n_0$ and local chemical reactivity $Q$. Simple mean
field (MF) kinetics apply at short times, $k \sim Q$. For high $Q$, a
transition occurs to diffusion-controlled (DC) kinetics with $k \approx
x_t^3/t$ (where $x_t$ is rms monomer displacement in time $t$) leading to a
density decay $n_t \approx n_0 - n_0^2 x_t^3$. If $n_0$ exceeds the chain
overlap threshold, this behavior is followed by a regime where $n_t \approx
1/x_t^3$ during which $k$ has the same power law dependence in time, $k \approx
x_t^3/t$, but possibly different numerical coefficient. For unentangled melts
this gives $n_t \sim t^{-3/4}$ while for entangled cases one or more of the
successive regimes $n_t \sim t^{-3/4}$, $t^{-3/8}$ and $t^{-3/4}$ may be
realized depending on the magnitudes of $Q$ and $n_0$. Kinetics at times longer
than the longest polymer relaxation time $\tau$ are always MF. If a DC regime
has developed before $\tau$ then the long time rate constant is $k \approx
R^3/\tau$ where $R$ is the coil radius. We propose measuring the above kinetics
in a model experiment where radical end groups are generated by photolysis.
