 Introducing these two expressions into the expression of the structure function evaluated at the mean value of the energy leads to
The denominator is exactly Stirling's approximation for formula_33, and if the structure function retains the same functional dependency for all values of the energy, the canonical probability density,
will belong to the family of exponential distributions known as gamma densities.Consequently, the canonical probability density falls under the jurisdiction of the local law of large numbers which asserts that a sequence of independent and identically distributed random variables tends to the normal law as the sequence increases without limit.The expressions given below are for systems that are close to equilibrium and have negligible quantum effects.