  Asymptotically safe gravity is one effective approach to quantum gravity. It
is important to differentiate the modified gravity inspired by asymptotically
safe gravity. In this paper, we examine the matter particles dynamics near the
improved version of Schwarzschild black hole. We assume that in the context of
asymptotically safe gravity scenario the ambient matter surrounding the black
hole is of isothermal in nature and investigate the spherical accretion of
matter by deriving solutions at critical points. The analysis for the various
values of the state parameter for isothermal test fluids, viz.,
$k=1,~1/2,~1/3,~1/4$ show the possibility of accretion onto asymptotically safe
black hole. We formulate the accretion problem as Hamiltonian dynamical system
and explain its phase flow in detail which reveals interesting results in
asymptotically safe gravity theory.
