We simulate the multi-phase interstellar medium randomly heated and stirred
by supernovae, with gravity, differential rotation and other parameters of the
solar neighbourhood. Here we describe in detail both numerical and physical
aspects of the model, including injection of thermal and kinetic energy by SN
explosions, radiative cooling, photoelectric heating and various transport
processes. With 3D domain extending 1 kpc^2 horizontally and 2 kpc vertically,
the model routinely spans gas number densities 10^-5 - 10^2 cm^-3, temperatures
10-10^8 K, local velocities up to 10^3 km s^-1 (with Mach number up to 25).
The thermal structure of the modelled ISM is classified by inspection of the
joint probability density of the gas number density and temperature. We confirm
that most of the complexity can be captured in terms of just three phases,
separated by temperature borderlines at about 10^3 K and 5x10^5 K. The
probability distribution of gas density within each phase is approximately
lognormal. We clarify the connection between the fractional volume of a phase
and its various proxies, and derive an exact relation between the fractional
volume and the filling factors defined in terms of the volume and probabilistic
averages. These results are discussed in both observational and computational
contexts. The correlation scale of the random flows is calculated from the
velocity autocorrelation function; it is of order 100 pc and tends to grow with
distance from the mid-plane. We use two distinct parameterizations of radiative
cooling to show that the multi-phase structure of the gas is robust, as it does
not depend significantly on this choice.Comment: 28 pages, 22 figures and 8 table
