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The application of the fractional exclusion statistics to the BCS theory--a redefinition of the quasiparticle energies

The effective energy of a superconductor $E_{eff}(T)$ at temperature $T$ is defined as the difference between the total energy at temperature $T$ and the total energy at 0~K. We call the energy of the condensate, ${\mathcal E}_c$, the difference between $E_{eff}$ and the sum of the quasiparticle energies $E_{qp}$. ${\mathcal E}_c$, $E_{qp}$, as well as the BCS quasiparticle energy $ε$ are positive and depend on the gap energy $Δ$, which, in turn, depends on the populations of the quasiparticle states (equivalently, they depend on $T$). So from the energy point of view the superconductor is a Fermi liquid of non-ideal quasiparticles. We show that the choice of quasiparticles is not unique, but there is an infinite range of possibilities. Some of these possibilities have been explored in the context of the fractional exclusion statistics (FES), which is a general method of describing interacting particle systems as ideal gases. We apply FES here and transform the Fermi liquid of BCS excitations into an ideal gas by redefining the quasiparticle energies. The new FES quasiparticles exhibit the same energy gap as the BCS quasiparticles, but a different DOS, which is finite at any quasiparticle energy. We also discuss the effect of the remnant electron-electron interaction (electron-electron interaction beyond the BCS pairing model) and show that this can stabilize the BCS condensate, increasing the critical temperature.