From these two distributions it is possible to calculate properties such as the internal energy per unit volume u , the number of particles N , specific heat capacity c , and thermal conductivity k . The relationships between these properties and the product of the density of states and the probability distribution, denoting the density of states by g ( E ) instead of D ( E ) , are given by u = ∫ ∫ E f ( E ) g ( E ) d E N = V ∫ ∫ f ( E ) g ( E ) d E c = ∂ ∂ ∂ ∂ T ∫ ∫ E f ( E ) g ( E ) d E k = 1 d ∂ ∂ ∂ ∂ T ∫ ∫ E f ( E ) g ( E ) ν ν ( E ) Λ Λ ( E ) d E