  A system of particles is studied in which the stochastic processes are
one-particle type-change (or one-particle diffusion) and multi-particle
annihilation. It is shown that, if the annihilation rate tends to zero but the
initial values of the average number of the particles tends to infinity, so
that the annihilation rate times a certain power of the initial values of the
average number of the particles remain constant (the double scaling) then if
the initial state of the system is a multi-Poisson distribution, the system
always remains in a state of multi-Poisson distribution, but with evolving
parameters. The large time behavior of the system is also investigated. The
system exhibits a dynamical phase transition. It is seen that for a k-particle
annihilation, if k is larger than a critical value k_c, which is determined by
the type-change rates, then annihilation does not enter the relaxation exponent
of the system; while for k < k_c, it is the annihilation (in fact k itself)
which determines the relaxation exponent.
