The probability distribution functions ( PDF ) of density of the interstellar medium ( ISM ) in galactic disks and global star formation rate are discussed .
Three-dimensional hydrodynamic simulations show that the PDFs in globally stable , inhomogeneous ISM in galactic disks are well fitted by a single log-normal function over a wide density range .
The dispersion of the log-normal PDF ( LN-PDF ) is larger for more gas-rich systems , and whereas the characteristic density of LN-PDF , for which the volume fraction becomes the maximum , does not significantly depend on the initial conditions .
Supposing the galactic ISM is characterized by the LN-PDF , we give a global star formation rate ( SFR ) as a function of average gas density , a critical local density for star formation , and star formation efficiency .
Although the present model is more appropriate for massive and geometrically thin disks ( \sim 10 pc ) in inner galactic regions ( < a few kpc ) , we can make a comparison between our model and observations in terms of SFR , provided that the log-normal nature of the density field is also the case in the real galactic disk with a large scale height ( \sim 100 pc ) .
We find that the observed SFR is well-fitted by the theoretical SFR in a wide range of the global gas density ( 10 - 10 ^ { 4 } M _ { \odot } pc ^ { -2 } ) .
Star formation efficiency ( SFE ) for high density gas ( n > 10 ^ { 3 } cm ^ { -3 } ) is SFE = 0.001 - 0.01 for normal spiral galaxies , and SFE = 0.01 - 0.1 for starburst galaxies .
The LN-PDF and SFR proposed here could be applicable for modeling star formation on a kpc-scale in galaxies or numerical simulations of galaxy formation , in which the numerical resolution is not fine enough to describe the local star formation .