  This work investigates the evolution of the distribution of charged particles
due to the mechanism of stochastic turbulent acceleration (STA) in presence of
small-scale turbulence with a mean magnetic field. STA is usually modelled as a
biased random walk process in the momentum space of the non-thermal particles.
This results in an advection-diffusion type transport equation for the
non-thermal particle distribution function. Under quasilinear approximation,
and by assuming turbulent spectra with power being available only in the
sub-gyroscale range, we find that the Fokker-Planck diffusion coefficients
$D_{\gamma\gamma}$ and $D_{\mu\mu}$ scale with the Lorentz factor $\gamma$ as:
$D_{\gamma\gamma}\propto\gamma^{-2/3}$ and $D_{\mu\mu}\propto\gamma^{-8/3}$. We
consider Alfv\`{e}n and fast waves in our calculations, and find a universal
trend for the momentum diffusion coefficient irrespective of the properties of
the small-scale turbulence. Such universality has already been reported
regarding the spatial diffusion of the cosmic rays, and, here too, we observe a
universality in the momentum diffusion coefficient. Furthermore, with the
calculated transport coefficients, we numerically solve the advection-diffusion
type transport equation for the non-thermal particles. We demonstrate the
interplay of various mircophysical processes such as STA, synchrotron loss and
particle escape on the particle distribution by systematically varying the
parameters of the problem. We observe that the effect of the small-scale
turbulence is more impactful for the high energy protons as compared to the
electrons and such turbulence is capable of sustaining the energy of the
protons from catastrophic radiative loss processes. Such a finding is novel and
helps us to enhance our understanding about the hadronic emission processes
that are typically considered as a competitor for the leptonic emission.
