  The persistent motion of bacteria produces clusters with a stationary cluster
size distribution (CSD). Here we develop a minimal model for bacteria in a
narrow channel to assess the relative importance of motility diversity (i.e.
polydispersity in motility parameters) and confinement. A mixture of
run-and-tumble particles with a distribution of tumbling rates (denoted
generically by $\alpha$) is considered on a 1D lattice. Particles facing each
other cross at constant rate, rendering the lattice quasi-1D. To isolate the
role of diversity, the global average $\alpha$ stays fixed. For a binary
mixture with no particle crossing, the average cluster size ($L_\text{c}$)
increases with the diversity as lower-$\alpha$ particles trap higher-$\alpha$
ones for longer. At finite crossing rate, particles escape from the clusters
sooner, making $L_\text{c}$ smaller and the diversity less important, even
though crossing can enhance demixing of particle types between the cluster and
gas phases. If the crossing rate is increased further, the clusters become
controlled by particle crossing. We also consider an experiment-based
continuous distribution of tumbling rates, revealing similar physics. Using
parameters fitted from experiments with Escherichia coli bacteria, we predict
that the error in estimating $L_\text{c}$ without accounting for polydispersity
is around $60\%$. We discuss how to find a binary system with the same CSD as
the fully polydisperse mixture. An effective theory is developed and shown to
give accurate expressions for the CSD, the effective $\alpha$, and the average
fraction of mobile particles. We give reasons why our qualitative results are
expected to be valid for other active matter models and discuss the changes
that would result from polydispersity in the active speed rather than in the
tumbling rate.
