  We investigate the relationship between a galaxy cluster's hydrostatic
equilibrium state, the entropy profile, $K$, of the intracluster gas, and the
system's non-thermal pressure (NTP), within an analytic model of cluster
structures. When NTP is neglected from the cluster's hydrostatic state, we find
that the gas' logarithmic entropy slope, $k\equiv \mathrm{d}\ln K/\mathrm{d}\ln
r$, converges at large halocentric radius, $r$, to a value that is
systematically higher than the value $k\simeq1.1$ that is found in observations
and simulations. By applying a constraint on these `pristine equilibrium'
slopes, $k_\mathrm{eq}$, we are able to predict the required NTP that must be
introduced into the hydrostatic state of the cluster. We solve for the
fraction, $\mathcal{F}\equiv p_\mathrm{nt}/p$, of NTP, $p_\mathrm{nt}$, to
total pressure, $p$, of the cluster, and we find $\mathcal{F}(r)$ to be an
increasing function of halocentric radius, $r$, that can be parameterised by
its value in the cluster's core, $\mathcal{F}_0$, with this prediction able to
be fit to the functional form proposed in numerical simulations. The minimum
NTP fraction, as the solution with zero NTP in the core, $\mathcal{F}_0=0$, we
find to be in excellent agreement with the mean NTP predicted in non-radiative
simulations, beyond halocentric radii of $r\gtrsim0.7r_{500}$, and in tension
with observational constraints derived at similar radii. For this minimum NTP
profile, we predict $\mathcal{F}\simeq0.20$ at $r_{500}$, and
$\mathcal{F}\simeq0.34$ at $2r_{500}$; this amount of NTP leads to a
hydrostatic bias of $b\simeq0.12$ in the cluster mass $M_{500}$ when measured
within $r_{500}$. Our results suggest that the NTP of galaxy clusters
contributes a significant amount to their hydrostatic state near the virial
radius, and must be accounted for when estimating the cluster's halo mass using
hydrostatic equilibrium approaches.
