  Aims: Ultraviolet (UV) lines of neutral carbon observed in absorption in the
local diffuse interstellar medium (ISM) have long revealed that a substantial
fraction of the mass of the gas lies at a thermal pressure one to three orders
of magnitude above that of the bulk of the ISM. In this paper, we propose that
this enigmatic component originates from shocks propagating at intermediate
($V_S > 30$ km s$^{-1}$) and high velocities ($V_S \geqslant 100$ km s$^{-1}$)
in the Warm Neutral Medium (WNM).
  Methods: Shock waves irradiated by the standard interstellar radiation field
(ISRF) are modelled using the Paris-Durham shock code designed to follow the
dynamical, thermal, and chemical evolutions of shocks with velocities up to 500
km s$^{-1}$. Each observed line of sight is decomposed into a high pressure and
a low pressure components. The column density of carbon at high pressure is
confronted to the model predictions to derive the number of shocks along the
line of sight and their total dissipation rate.
  Results: Phase transition shocks spontaneously lead to the presence of high
pressure gas in the diffuse ISM and are found to naturally produce neutral
carbon with excitation conditions and linewidths in remarkable agreement with
the observations. The amounts of neutral carbon at high pressure detected over
a sample of 89 lines of sight imply a dissipation rate of mechanical energy
with a median of $\sim 3x10^{-25}$ erg cm$^{-3}$ s$^{-1}$ and a dispersion of
about a factor of three. This distribution of the dissipation rate weakly
depends on the detailed characteristics of shocks as long as they propagate at
velocities between 30 and 200 kms s$^{-1}$ in a medium with a preshock density
$n_H^0 \ge 0.3$ cm s$^{-3}$ and a transverse magnetic field $B_0 \leqslant 3$
$\mu$G. We not only show that this solution is consistent with a scenario of
shocks driven by supernovae remnants (SNR) but also that this scenario is, in
fact, unavoidable. Any line of sight in the observational sample is bound to
intercept SNRs, mostly distributed in the spiral arms of the Milky Way, and
expanding in the diffuse ionized and neutral phases of the Galaxy.
Surprisingly, the range of dissipation rate derived here, in events that
probably drive turbulence in the WNM, is found to be comparable to the
distribution of the kinetic energy transfer rate of the turbulent cascade
derived from the observations of CO in the Cold Neutral Medium (CNM).
  Conclusions: This work reveals a possible direct tracer of the mechanisms by
which mechanical energy is injected in the ISM. It also suggests that a still
unknown connection exists between the amount of energy dissipated during the
injection process in the WNM and that used to feed interstellar turbulence and
the turbulent cascade observed in the CNM.
