  Observations have suggested substantial departures from pressure equilibrium
in the interstellar medium (ISM) in the plane of the Galaxy, even on scales
under 50 pc. Nevertheless, multi-phase models of the ISM assume at least
locally isobaric gas. The pressure then determines the density reached by gas
cooling to stable thermal equilibrium. We use two different sets of numerical
models of the ISM to examine the consequences of supernova driving for
interstellar pressures. The first set of models is hydrodynamical, and uses
adaptive mesh refinement to allow computation of a 1 x 1 x 20 kpc section of a
stratified galactic disk. The second set of models is magnetohydrodynamical,
using an independent code framework, and examines a 200 pc cubed periodic
domain threaded by magnetic fields. Both of these models show broad pressure
distributions with roughly log-normal functional forms produced by both shocks
and rarefaction waves, rather than the power-law distributions predicted by
previous work, with rather sharp thermal pressure gradients. The width of the
distribution of the logs of pressure in gas with log T < 3.9 is proportional to
the rms Mach number in that gas, while the distribution in hotter gas is
broader, but not so broad as would be predicted by the Mach numbers in that
gas. Individual parcels of gas reach widely varying points on the thermal
equilibrium curve: no unique set of phases is found, but rather a
dynamically-determined continuum of densities and temperatures. Furthermore, a
substantial fraction of the gas remains entirely out of thermal equilibrium.
Our results appear consistent with observations of interstellar pressures, and
suggest that the pressures observed in molecular clouds may be due to ram
pressure rather than gravitational confinement.
