  We present a theoretical framework for 'collective transition quenching', a
quantum many-body dissipative phenomenon that occurs in systems with multiple
collective decay channels. Despite the competition, interactions suppress all
but the dominant decay transition, leading to a 'winner takes all' dynamic
where the system primarily settles into the dominant ground state. We prove
that, in the presence of permutation symmetry, this problem is exactly solvable
for any number of competing channels. Additionally, we develop an approximate
model for the dynamics by mapping the evolution into a continuity equation for
a fluid, and show analytically that the dominant transition ratio converges to
unity with increasing system size as a power-law, for any branching ratio. This
near-deterministic preparation of the dominant ground state has broad
applicability. As an example we discuss a protocol for molecular
photoassociation where collective dynamics effectively acts as a catalyst,
amplifying the yield in a particular final state. Our results open new avenues
for many-body strategies in the preparation and control of quantum systems.
