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arXiv:1202.2288 · Physical Review C

Thermodynamical properties of small superconductors with a fixed number of particles

D. Gambacurta, Denis Lacroix A

Published Apr 26, 2012• 18 citations
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Abstract

The variation after projection approach is applied for the first time to the Richardson pairing Hamiltonian to describe the thermodynamics of small superconductors with a fixed number of particles. The minimization of the free energy is made by a direct diagonalization of the entropy. The variation after projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperatures.

DOI
10.1103/physrevc.85.044321

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The variation after projection approach is applied for the first time to the Richardson pairing Hamiltonian to describe the thermodynamics of small superconductors with a fixed number of particles. The minimization of the free energy is made by a direct diagonalization of the entropy. The variation after projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperatures.
A canonical description of the thermodynamical pairing properties of small systems is achieved by using the Variation After Projection approach at finite temperature. The minimization of the free energy is made by a direct evaluation of the energy and full diagonalization of the entropy. We use the Richardson – pairing model whose exact solution allows to study the reliability of different approaches. We show that the Projection After Variation approach, that is usually performed at zero temperature with rather good success, provides a quite poor description at finite temperature. On the contrary, the Variation After Projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperature.
Dedicated to Professor Józef Spa lek on the occasion of his 60th birthdayThe two-level version of the Richardson model presents a unique possibility to calculate numerically exactly thermodynamic properties of the system it describes.The point is that all energies and the degeneracies of the many-body system can be easily calculated.The energies are given by the eigenvalues of the small (of order of N × N , where 2N is a number of electrons in the system) tridiagonal matrices.Here we numerically obtain a complete spectrum of the interacting two-level model and calculate the specific heat and the pairing energy of the small system at finite temperatures.
The thermodynamic properties of a system consisting of a small number of non-interacting indistinguishable particles in the temperature range where the transition from quantum to classical behaviour occurs are discussed. The system is an extrapolation of a perfect monatomic boson gas contained in a box. We calculate the internal energy, specific heat and entropy and discuss their dependence on various factors such as the temperature, the number of particles and the box dimensionality. The results are compared with those obtained for a system consisting of distinguishable particles. The calculations are carried out without approximation starting from first principles, and the results have general validity since reduced units are used.
For a small system at a low temperature, thermal fluctuation and quantum effect play important roles in quantum thermodynamics. Starting from micro-canonical ensemble, we generalize the Boltzmann–Gibbs statistical factor from infinite to finite systems, no matter the interactions between particles are considered or not. This generalized factor, similar to Tsallis’s q-form as a power-law distribution, has the restriction of finite energy spectrum and includes the nonextensivities of the small systems. We derive the exact expression for distribution of average particle numbers in the interacting classical and quantum nonextensive systems within a generalized canonical ensemble. This expression in the almost independent or elementary excitation quantum finite systems is similar to the corresponding ones obtained from the conventional grand-canonical ensemble. In the reconstruction for the statistical theory of the small systems, we present the entropy of the equilibrium systems and equation of total thermal energy. When we investigate the thermodynamics for the interacting nonextensive systems, we obtain the system–bath heat exchange and “uncompensated heat” which are in the thermodynamical level and independent on the detail of the system–bath coupling. For ideal finite systems, with different traps and boundary conditions, we calculate some thermodynamic quantities, such as the specific heat, entropy, and equation of state, etc. Particularly at low temperatures for the small systems, we predict some novel behaviors in the quantum thermodynamics, including internal entropy production, heat exchanges between the system and its surroundings and finite-size effects on the free energy.
Abstract A procedure is developed for obtaining approximate thermodynamic properties of statistical models, based on a well known variational principle for the free energy. The procedure involves the choice of a suitable parametrized trial Hamiltonian which contains some elementary interactions. The free trial Hamiltonian leads to the usual mean‐field approximation. When the trial Hamiltonian includes a certain number of pair interactions only, the results are regained of the constant‐coupling approximation for the Ising model if the number of interacting pairs is conveniently chosen by a comparison with the first few terms of the exact high‐temperature expansion. This procedure is simpler than existing approximations and can be easily generalized. Besides treating the Ising model, some applications are presented to the spin‐1/2 antiferromagnetic Heisenberg model, and to the spin‐1 Heisenberg model with uniaxial anisotropy.