Kinetic energy of Bose systems and variation of statistical averages

TL;DR

This paper investigates the proper way to define and compute the average kinetic energy in Bose systems, clarifying the conditions for existing formulas and deriving new general expressions, with applications to low-temperature Bose gases.

Contribution

It derives a general formula for the mass derivative of the internal energy in Bose systems without adiabatic approximation, clarifying the validity of kinetic energy formulas.

Findings

Incorrect use of kinetic energy formulas leads to paradoxes.

Derived a new equation for the parametric derivative of the average of any operator.

Applied results to a low-temperature dilute Bose gas.

Abstract

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.