  The interplay between stochastic chemical reactions and diffusion can
generate rich spatiotemporal patterns. While the timescale for individual
reaction or diffusion events may be very fast, the timescales for organization
can be much longer. That separation of timescales makes it particularly
challenging to anticipate how the rapid microscopic dynamics gives rise to
macroscopic rates in the non-equilibrium dynamics of many reacting and
diffusing chemical species. Within the regime of stochastic fluctuations, the
standard approach is to employ Monte Carlo sampling to simulate realizations of
random trajectories. Here, we present an alternative numerically tractable
approach to extract macroscopic rates from the full ensemble evolution of
many-body reaction diffusion problems. The approach leverages the Doi-Peliti
second-quantized representation of reaction-diffusion master equations along
with compression and evolution algorithms from tensor networks. By focusing on
a Schl\"{o}gl model with one-dimensional diffusion between $L$ otherwise
well-mixed sites, we illustrate the potential of the tensor network approach to
compute rates from many-body systems, here with approximately $3 \times
10^{15}$ microstates. Specifically, we compute the rate for switching between
metastable macrostates, with the expense for computing those rates growing
subexponentially in $L$. Because we directly work with ensemble evolutions, we
crucially bypass many of the difficulties encountered by rare event sampling
techniques$\unicode{x2013}$detailed balance and reaction coordinates are not
needed.
