  Non-thermal pressure in the intracluster gas has been found ubiquitously in
numerical simulations, and observed indirectly. In this paper we develop an
analytical model for intracluster non-thermal pressure in the virial region of
relaxed clusters. We write down and solve a first-order differential equation
describing the evolution of non-thermal velocity dispersion. This equation is
based on insights gained from observations, numerical simulations, and theory
of turbulence. The non-thermal energy is sourced, in a self-similar fashion, by
the mass growth of clusters via mergers and accretion, and dissipates with a
time-scale determined by the turnover time of the largest turbulence eddies.
Our model predicts a radial profile of non-thermal pressure for relaxed
clusters. The non-thermal fraction increases with radius, redshift, and cluster
mass, in agreement with numerical simulations. The radial dependence is due to
a rapid increase of the dissipation time-scale with radii, and the mass and
redshift dependence comes from the mass growth history. Combing our model for
the non-thermal fraction with the Komatsu-Seljak model for the total pressure,
we obtain thermal pressure profiles, and compute the hydrostatic mass bias. We
find typically 10% bias for the hydrostatic mass enclosed within $r_{500}$.
