Let x 1 , x 2 , … … , x n denote fluctuations from equilibrium values in several thermodynamic quantities, and let S ( x 1 , x 2 , … … , x n ) be the entropy. Then, Boltzmann's entropy formula gives for the probability distribution function w = A exp ⁡ ⁡ ( S / k ) , where A is a constant, since the probability of a given set of fluctuations x 1 , x 2 , … … , x n is proportional to the number of microstates with that fluctuation. Assuming the fluctuations are small, the probability distribution function can be expressed through the second differential of the entropy w = A ~ ~ e − − 1 2 β β i k x i x k ; β β i k = β β k i = − − 1 k ∂ ∂ 2 S ∂ ∂ x i ∂ ∂ x k , where we are using Einstein summation convention and β β i k is a positive definite symmetric matrix.