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arXiv:cond-mat/0408019 · Journal of Physics A Mathematical and General

Finite size effects in canonical Fermi systems

Dragoş-Victor Anghel

Published Apr 19, 2006• 1 citation
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Abstract

I investigate the thermodynamic equivalence between finite systems of constant density of states below the Bose–Einstein condensation temperature. I show that in order to preserve the equivalence, a phenomenon called Fermi condensation has to be taken into account. This phenomenon leads to a correction of the intensive parameters of the Fermi gas and of the Fermi distribution in canonical Fermi systems at low temperatures.

DOI
10.1088/0305-4470/39/18/001

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I investigate the thermodynamic equivalence between finite systems of constant density of states below the Bose–Einstein condensation temperature. I show that in order to preserve the equivalence, a phenomenon called Fermi condensation has to be taken into account. This phenomenon leads to a correction of the intensive parameters of the Fermi gas and of the Fermi distribution in canonical Fermi systems at low temperatures.
The effects of a finite number of particles on the thermodynamic properties of ideal Bose and Fermi gases trapped in any-dimensional harmonic potential are investigated. The orders of relative corrections to the thermodynamic quantities due to the finite number of particles are estimated in different situations. The results obtained for the two trapped quantum gases are compared, and consequently, it is shown that the finite-particle-number effects for the condensed Bose gas (a Bose gas with Bose–Einstein Condensation (BEC) occurring in the system) are much more significant than those for the Fermi gas and normal Bose gas (a Bose gas without BEC).
We show various aspects of finite size effects on Bose-Einstein condensation(BEC). In the first section we introduce very briefly the BEC of harmonically trapped ideal Bose gas. In the second section we theoretically argued that Bose-Einstein(B-E) statistics needs a correction for finite system at ultralow temperatures. As a corrected statistics we introduced a Tsallis type of generalized B-E statistics. The condensate fraction calculated with this generalized B-E statistics, is satisfied well with the experimental result. In the third section we show how to apply the scaling theory in an inhomogeneous system like harmonically trapped Bose condensate at finite temperatures. We calculate the temperature dependence of the critical number of particles by a scaling theory within the Hartree-Fock approximation and find that there is a dramatic increase in the critical number of particles as the condensation point is approached. Our results support the experimental result which was obtained well below the condensation temperature. In the fourth section we concentrate on the thermodynamic Casimir force on the Bose-Einstein condensate. We explored the temperature dependence of the Casimir force.
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called thermodynamically equivalent and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonization technique, of transforming between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In appendix A, I calculate the fluctuation of the ground-state population of a condensed Bose gas in a grand-canonical ensemble and mean field approximation, while in appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single-particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.
The critical temperature and condensate fraction of a trapped interacting Bose gas are investigated when both atom-atom interaction and finite-size effects are taken into account. A canonical ensemble is used to obtain the equations on the condensate fraction for the trapped interacting Bose gas near and below the critical temperature. In our approaches corrections due to atom-atom interaction and finite-size effects are obtained simultaneously for the critical temperature and condensate fraction of the system. An analytical high-order correction to the condensate fraction is given.
The phenomenon of Bose-like condensation, the continuous change of the dimensionality of the particle distribution as a consequence of freezing out of one or more degrees of freedom in the limit of low particle density, is investigated theoretically in the case of closed systems of massive bosons and fermions, described by general single-particle Hamiltonians. This phenomenon is similar for both types of particles and, for some energy spectra, exhibits features specific to multiple-step Bose-Einstein condensation, for instance, the appearance of maxima in the specific heat. In the case of fermions, as the particle density increases, another phenomenon is also observed. For certain types of single particle Hamiltonians, the specific heat is approaching asymptotically a divergent behavior at zero temperature, as the Fermi energy epsilon(F) is converging towards any value from an infinite discrete set of energies epsilon(i)(i>/=1). If epsilon(F)=epsilon(i), for any i, the specific heat is divergent at T=0 just in infinite systems, whereas for any finite system the specific heat approaches zero at low enough temperatures. The results are particularized for particles trapped inside parallelepipedic boxes and harmonic potentials.
We calculate numerically and analytically the grandcanonical fluctuation of the number of particles both, in the fermionic condensate and above it, for ideal Fermi systems of constant density of states. We compare the canonical fluctuations, obtained from the equivalent Bose condensate fluctuation, with the grandcanonical fermionic calculation. The fluctuations of the condensate are almost the same in the two ensembles with a small correction coming from the total particle number fluctuation in the grandcanonical ensemble. On the other hand, well below the condensation temperature, the number of particles above the condensate and its fluctuation are insensitive to the choice of ensemble.