This average can be written using the probability density function p ( x ) ; ∫ ∫ ( − − f ′ ( x ) ∂ ∂ V ∂ ∂ x p ( x ) + k B T f ″ ( x ) p ( x ) ) d x = ∫ ∫ ( − − f ′ ( x ) ∂ ∂ V ∂ ∂ x p ( x ) − − k B T f ′ ( x ) p ′ ( x ) ) d x = 0 where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions f , it follows that ∂ ∂ V ∂ ∂ x p ( x ) + k B T p ′ ( x ) = 0 , thus recovering the Boltzmann distribution p ( x ) ∝ ∝ exp ⁡ ⁡ ( − − V ( x ) k B T ) .