As discussed above, an extensive form of entropy is recovered if we divide the volume of phase space, Ω Ω E , ℓ ℓ , by n!. An alternative approach is to argue that the dependence on particle number cannot be trusted on the grounds that changing n also changes the dimensionality of phase space. Such changes in dimensionality lie outside the scope of Hamiltonian mechanics and Liouville's theorem. For that reason it is plausible to allow the arbitrary constant to be a function of n . Defining the function to be, f ( n ) = − − 3 2 n ln ⁡ ⁡ n , we have: