Thermodynamics of Low-Dimensional Trapped Fermi Gases

Francisco J Sevilla

doi:10.1155/2017/3060348

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.

Highlights

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)
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
The situations change dramatically in low dimensions, since Fermi systems are inherently unstable towards any finite interaction [4,5,6]; the IFG in low dimensions becomes an interesting solvable model to study the thermodynamics of possible singular behavior

Summary

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 probability of occupation of the available states in thermal equilibrium must follow the Fermi-Dirac distribution and just a fraction of the excitable fermions are excited into the interval [EF, EF + kBT] (the occupation probability for the states with energy larger than μ is smaller than 1/2) For this case we can certainly apply the argument given by Cook and Dickerson in [41] to infer that when adding adiabatically an extra particle to the system, the internal energy will decrease from EF. In order to quantitatively characterize the incomplete accommodation described above, we consider the ratio R(T) of the number of particles in the energy interval [EF, EF + Δ] to the number of available states in the same energy interval,

The Physical Meaning of T0

Conclusions and Final Remarks