where the power spectral density S x ( ω ω ) is the Fourier transform of the auto-correlation ⟨ ⟨ x ^ ^ ( t ) x ^ ^ ( 0 ) ⟩ ⟩ and n B E ( ω ω ) = ( e β β ℏ ℏ ω ω − − 1 ) − − 1 is the Bose-Einstein distribution function. The same calculation also yields