Journal of Thermodynamics

Volume 2017, Article ID 3060348, 12 pages

https://doi.org/10.1155/2017/3060348

## Thermodynamics of Low-Dimensional Trapped Fermi Gases

Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 Ciudad de México, Mexico

Correspondence should be addressed to Francisco J. Sevilla; xm.manu.acisif@allivesjf

Received 8 October 2016; Revised 2 December 2016; Accepted 5 December 2016; Published 26 January 2017

Academic Editor: Felix Sharipov

Copyright © 2017 Francisco J. Sevilla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The effects of low dimensionality on the thermodynamics of a Fermi gas trapped by isotropic power-law potentials are analyzed. Particular attention is given to different characteristic temperatures that emerge, at low dimensionality, in the thermodynamic functions of state and in the thermodynamic susceptibilities (isothermal compressibility and specific heat). An* energy-entropy* argument that physically favors the relevance of one of these characteristic temperatures, namely, the nonvanishing temperature at which the chemical potential reaches the Fermi energy value, is presented. Such an argument allows interpreting the nonmonotonic dependence of the chemical potential on temperature, as an indicator of the appearance of a thermodynamic regime, where the equilibrium states of a trapped Fermi gas are characterized by larger fluctuations in energy and particle density as is revealed in the corresponding thermodynamics susceptibilities.

#### 1. Introduction

The discovery of the quantum statistics that incorporate Pauli’s exclusion principle [1], made independently by Fermi [2] and Dirac [3], allowed the qualitative understanding of several physical phenomena—in a wide range of values of the particle density, from astrophysical scales to subnuclear ones—in terms of the ideal Fermi gas (IFG). The success of the explicative scope of the ideal Fermi gas model relies on Landau’s Fermi liquid theory where fermions interacting repulsively through a short range forces can be described in some degree as an IFG. The situations change dramatically in low dimensions, since Fermi systems are inherently unstable towards any finite interaction [4–6]; thus the IFG in low dimensions becomes an interesting solvable model to study the thermodynamics of possible singular behavior.

On the other hand, the experimental realization of quantum degeneracy in trapped atomic Fermi gases [7–11] triggered a renewed interest, over the last fifteen years, in the study not only of interacting fermion systems [12–15] but also of trapped ideal ones [16–34]. Indeed, the nearly ideal situation has been experimentally realized by taking advantage of the suppression of -wave scattering in spin-polarized fermion gases due to Pauli exclusion principle and of the negligible effects of -wave scattering for the temperature ranges involved. Further, the control achieved on the experimental settings has opened the possibility of directly testing a variety of quantum effects such as Pauli blocking [35] and designing experiments to probe condensed matter models, though much lower temperatures are needed to achieve the phenomena of interest. On this trend, experimentally new techniques are being devised to cool further a cloud of atomic fermions [36–39]. Techniques based on the giving-away of entropy by changing the shape of the trapping potential have become of great importance and, as in many instances, a complete understanding of trapped noninteracting fermionic atoms would become of great value.

In distinction with the ideal Bose gas (IBG), which suffers the so-called Bose-Einstein condensation (BEC) in three dimensions, the IFG shows a smooth thermodynamic behavior as function of the particle density and temperature; this, however, does not preclude interesting behavior as has been pointed out in [28, 40], where it is suggested that the IFG can suffer a condensation-like process at a characteristic temperature . Arguments based on a thermodynamic approach in support of this phenomenon are presented in [40], where the author suggests that the change of sign of the chemical potential, which defines the characteristic temperature , marks the appearance of the condensed phase when the gas is cooled.

Truly, the significance of has motivated the discussion of its meaning and/or importance at different levels and contexts [41–52]. For the widely discussed—textbook—case, namely, the three-dimensional IFG confined by an impenetrable box potential, the chemical potential results to be a monotonic decreasing function of the temperature, diminishing from the* Fermi energy*, , at zero temperature, to the values of the ideal classical gas for temperatures much larger than , where is Boltzmann’s constant, is Planck’s constant divided by , is the mass of the particle, and is the thermal wavelength of de Broglie, where denotes the system’s absolute temperature. A clear, qualitative, physical argument of this behavior is presented by Cook and Dickerson in [41]. In comparison, the chemical potential of the IBG vanishes below a characteristic temperature, called the critical temperature of BEC, , and decreases monotonically for larger temperatures converging asymptotically to the values of the classical ideal gas.

This picture changes dramatically as the dimensionality of the system is lowered. In two dimensions the IBG shows no off-diagonal-long-range order at any finite temperature [53] and therefore the BEC transition does not occur. At this quirky dimension, the chemical potential of both, the Fermi and Bose ideal gases, decreases monotonically with temperature essentially in the same functional way [54], being different only by an additive constant, expressly, the Fermi energy. This results in the same temperature dependence of their respective specific heats at constant volume [54–56]. In general, this last outstanding feature occurs whenever the number of energy levels per energy interval is uniform as in the case of a one-dimensional gas in a harmonic trap [57, 58], or the case where is the exponent of the single-particle energy spectrum of the form , being the particle momentum [59].

In one dimension, the chemical potential of the IBG decreases monotonically with temperature, and as in the two dimensional case, this behavior is related to the impossibility of BEC as shown by Hohenberg [53], at finite, nonzero, temperature. In contrast, the chemical potential of the IFG exhibits a* nonmonotonic* behavior: which starts rising quadratically with above the Fermi energy instead of decreasing from it and returns to its usual monotonic decreasing behavior at temperatures that can be as large as twice the Fermi temperature (see Figure 1; see also Figure in [60]). This* unexpected*, and not well understood behavior, can be exhibited mathematically by the Sommerfeld expansion [60, 61] or by other methods [62–64], though no intuitive physical explanation of it, which predicts its appearance in the more general case, seems to have been given before. (Indeed, the precise argument presented in [41] is only valid for the free IFG in three dimensions.) This forms the basis for the motivation of the present paper.