# Quasi-particles, thermodynamic consistency and the gap equation

## TL;DR

This paper addresses thermodynamic inconsistencies in the quasi-particle approach to superconductivity by deriving corrected potentials via the Bogoliubov-Valatin formalism, ensuring the validity of thermodynamic relations and deriving the gap equation.

## Contribution

It demonstrates how the Bogoliubov-Valatin transformation ensures thermodynamic consistency and provides corrected thermodynamic potentials in the superconducting state.

## Key findings

- Thermodynamic consistency is recovered using the Bogoliubov-Valatin formalism.
- Corrected quasi-particle potentials are derived, including the vacuum energy and chemical potential dependence.
- The gap equation is shown to coincide with the stationarity condition of the grand potential.

## Abstract

The thermodynamic properties of superconducting electrons are usually studied by means of the quasi-particles distribution; but in this approach, the ground state energy and the dependence of the chemical potential on the electron density cannot be determined. In order to solve these problems, the thermodynamic potentials are derived by means of the Bogoliubov-Valatin formalism. The thermodynamic potentials can be obtained by computing the free energy of a gas of quasi-particles, whose energy spectrum is conditional on the gap function. However, the nontrivial dependence of the gap on the temperature jeopardises the validity of the standard thermodynamic relations. In this article it is shown how the thermodynamic consistency (i.e. the validity of the Maxwell relations) is recovered, and the correction terms to the quasi-particles potentials are computed. It is shown that the Bogoliubov-Valatin transformation avoids the problem of the thermodynamic consistency of the quasi-particle approach; in facts, the correct identification of the variables, which are associated with the quasi-particles, leads to a precise calculation of the quasi-particles vacuum energy and of the dependence of the chemical potential on the electron density. The stationarity condition for the grand potential coincides with the gap equation, which guarantees the thermodynamic consistency. The expressions of various thermodynamic potentials, as functions of the (T,V,N) variables, are produced in the low temperature limit; as a final check, a rederivation of the condensation energy is presented.