text: writing the potential energy as depending on a continuous parameter, 
  one has the exact result 
  This can either be directly verified from definitions or seen from the limit of the above Gibbs-Bogoliubov inequalities when 
  we can therefore write
  which is the thermodynamic integration (or TI) result. It can be approximated by dividing the range between states A and B into many values of λ at which the expectation value is estimated, and performing numerical integration.