Kinetic energy of Bose systems and variation of statistical averages
  The problem of defining the average kinetic energy of statistical systems is addressed. The conditions of applicability for the formula, relating the average kinetic energy with the mass derivative of the internal energy, are analysed. It is shown that incorrectly using this formula, outside its region of validity, leads to paradoxes. An equation is found for a parametric derivative of the average for an arbitrary operator. A special attention is paid to the mass derivative of the internal energy, for which a general formula is derived, without invoking the adiabatic approximation and taking into account the mass dependence of the potential-energy operator. The results are illustrated by the case of a low-temperature dilute Bose gas. 
V.I. Yukalov