  Background: Lower production of H$_2$O$_2$ in water is a hallmark of
ultra-high dose rate (UHDR) compared to the conventional dose rate (CDR).
However, the current computational models based on the predicted yield of
H$_2$O$_2$ are in opposite of the experimental data. Methods: We construct an
analytical model for the rate equation in the production of H$_2$O$_2$ from
\ce{^{.}OH}-radicals and use it as a guide to propose a hypothetical
geometrical inhomogeneity in the configuration of particles in the FLASH-UHDR
beams. We perform a series of Monte Carlo (MC) simulations of the track
structures for a system of charged particles impinging the medium in the form
of clusters and/or bunches. Results: We demonstrate the interplay of diffusion,
reaction rates, and overlaps in track-spacing attribute to a lower yield of
H$_2$O$_2$ at FLASH-UHDR vs. CDR. This trend is reversed if spacing among the
tracks becomes larger than a critical value, with a length scale that is
proportional to the diffusion length of \ce{^{.}OH}-radicals modulated by a
rate of decay due to recombination with other species, available within a
track, and the space among the tracks. The latter is substantial on the
suppressing of the H$_2$O$_2$ population at FLASH-UHDR relative to CDR.
Conclusions: Based on our analysis of the present work, at FLASH-UHDR, the
lower yield in H$_2$O$_2$ can be interpreted as a signature of bunching the
particles in beams of ionizing radiation. The beams enter the medium in closely
packed clusters and form inhomogeneities in the track-structure distribution.
Thus the MC simulations based on the assumption of uniformly distributed tracks
are unable to explain the experimental data.
