  We demonstrate that the reformulation of renormalization group (RG) flow
equations as non-linear heat equations has severe implications on the
understanding of RG flows in general. We demonstrate by explicitly constructing
an entropy function for a zero-dimensional $\mathbb{Z}_2$-symmetric model that
the dissipative character of generic non-linear diffusion equations is also
hard-coded in the functional RG equation. This renders RG flows manifestly
irreversible, revealing the semi-group property of RG transformations on the
level of the flow equation itself. Additionally, we argue that the dissipative
character of RG flows, its irreversibility and the entropy production during
the RG flow may be linked to the existence of a so-called
$\mathcal{C}$-/$\mathcal{A}$-function. In total, this introduces an asymmetry
in the so-called RG time -- in complete analogy to the thermodynamic arrow of
time -- and allows for an interpretation of infrared actions as equilibrium
solutions of dissipative RG flows equations. The impossibility of resolving
microphysics from macrophysics is evident in this framework. Furthermore, we
directly link the irreversibility and the entropy production in RG flows to an
explicit numerical entropy production, which is manifest in diffusive and
non-linear partial differential equations (PDEs) and a standard mathematical
tool for the analysis of PDEs. Using exactly solvable zero-dimensional
$\mathbb{Z}_2$-symmetric models, we explicitly compute the (numerical) entropy
production related to the total variation non-increasing property of the PDE
during RG flows toward the infrared limit. Finally, we discuss generalizations
of our findings and relations to the $\mathcal{C}$-/$\mathcal{A}$-theorem as
well as how our work may help to construct truncations of RG flow equations in
the future, including numerically stable schemes for solving the corresponding
PDEs.
