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arXiv:0904.0053 · Physical Review A

Comparative study of the finite-temperature thermodynamics of a unitary Fermi gas

Fang Qin, Jisheng Chen

Published Apr 23, 2009• 22 citations
arXivHTMLPDF

Abstract

The finite-temperature thermodynamics of a unitary Fermi gas is studied in detail. The chemical potential, energy density, and entropy are given analytically with the quasilinear approximation. The ground-state energy agrees with previous theoretical and experimental results. Recently, the generalized exclusion statistics is applied to the discussion of the finite-temperature unitary Fermi-gas thermodynamics. A concrete comparison between the two different approaches is performed. Emphasis is made on the behavior of the entropy per particle. In physics, the slope of entropy gives the information on the effective fermion mass ${m}^{\ensuremath{\ast}}/m$ in the low-temperature strongly degenerate region. Compared with ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}0.70<1$ given in terms of the generalized exclusion statistics, our quasilinear approximation determines that ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1.11>1$.

DOI
10.1103/physreva.79.043625

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The finite-temperature thermodynamics of a unitary Fermi gas is studied in detail. The chemical potential, energy density, and entropy are given analytically with the quasilinear approximation. The ground-state energy agrees with previous theoretical and experimental results. Recently, the generalized exclusion statistics is applied to the discussion of the finite-temperature unitary Fermi-gas thermodynamics. A concrete comparison between the two different approaches is performed. Emphasis is made on the behavior of the entropy per particle. In physics, the slope of entropy gives the information on the effective fermion mass m\ensuremath∗/m{m}^{\ensuremath{\ast}}/mm\ensuremath∗/m in the low-temperature strongly degenerate region. Compared with m\ensuremath∗/m\ensuremath≈0.70<1{m}^{\ensuremath{\ast}}/m\ensuremath{\approx}0.70<1m\ensuremath∗/m\ensuremath≈0.70<1 given in terms of the generalized exclusion statistics, our quasilinear approximation determines that m\ensuremath∗/m\ensuremath≈1.11>1{m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1.11>1m\ensuremath∗/m\ensuremath≈1.11>1.
We utilize a fractional exclusion statistics of Haldane and Wu hypothesis to study the thermodynamics of a unitary Fermi gas trapped in a harmonic oscillator potential at ultra-low finite temperature. The entropy per particle as a function of the energy per particle and energy per particle versus rescaled temperature are numerically compared with the experimental data. The study shows that, except the chemical potential behavior, there exists a reasonable consistency between the experimental measurement and theoretical attempt for the entropy and energy per particle. In the fractional exclusion statistics formalism, the behavior of the isochore heat capacity for a trapped unitary Fermi gas is also analyzed.
At zero-temperature and finite-temperature, the thermodynamic properties of finite unitary Fermi gas in a three-dimensional harmonic trap are investigated by using fractional exclusion statistics, and the results are compared with those of the system which satisfies the thermodynamic limit. At zero-temperature, Fermi energy and average energy of per particle increase with the increase of the number of particles for finite unitary Fermi gas, and their limits are the corresponding parameters of the system which satisfy thermodynamic limits. Fermi energy and average energy of per particle each have a maximum value changing with the boundary of the potential well. For the finite-temperature trapped unitary Fermi system, when the number of particles is certain the average energy of per particle, average entropy of per particle, average heat capacity of per particle each have a characteristic temperature, respectively, when the temperature is equal to the characteristic temperature of the physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the temperature is lower (or higher) than the characteristic temperature of parameter, the physical parameter of the finite system will be greater (or less) than the corresponding parameter of the thermodynamic limit system. The characteristic temperature has particle number effect and boundary effect. When the temperature is determined, the average energy of per particle, average entropy of per particle and average heat capacity of per particle each have a characteristic number of particles, respectively, when the number of particles is equal to the characteristic number of particles for physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the number of particles is less (or more) than the characteristic number of particles for corresponding parameter, the corresponding parameter of the finite system will be less (or larger) than the thermodynamic limit of system.
We investigate the low-temperature thermodynamics of the unitary Fermi gas by introducing a model based on the zero-temperature spectra of both bosonic collective modes and fermonic single-particle excitations. We calculate the Helmholtz free energy and from it we obtain the entropy, the internal energy, and the chemical potential as a function of the temperature. By using these quantities and the Landau's expression for the superfluid density we determine analytically the superfluid fraction, the critical temperature, the first sound velocity, and the second sound velocity. We compare our analytical results with other theoretical predictions and experimental data of ultracold atoms and dilute neutron matter.
Thermodynamic properties of matter generally depend on the details of interactions between its constituent parts. However, in a unitary Fermi gas where the scattering length diverges, thermodynamics is determined through universal functions that depend only on the particle density and temperature. By using only the general form of the equation of state and the equation of force balance, we measured the local internal energy of the trapped gas as a function of these parameters. Other universal functions, such as those corresponding to the Helmholtz free energy, chemical potential, and entropy, were calculated through general thermodynamic relations. The critical parameters were also determined at the superfluid transition temperature. These results apply to all strongly interacting fermionic systems, including neutron stars and nuclear matter.
Thermodynamic properties of matter generally depend on the details of interactions between its constituent parts. However, in a unitary Fermi gas where the scattering length diverges, thermodynamics is determined through universal functions that depend only on the particle density and temperature. By using only the general form of the equation of state and the equation of force balance, we measured the local internal energy of the trapped gas as a function of these parameters. Other universal functions, such as those corresponding to the Helmholtz free energy, chemical potential, and entropy, were calculated through general thermodynamic relations. The critical parameters were also determined at the superfluid transition temperature. These results apply to all strongly interacting fermionic systems, including neutron stars and nuclear matter.
Strongly interacting, dilute Fermi gases exhibit a scale-invariant, universal thermodynamic behavior. This is notoriously difficult to understand theoretically because of the absence of a small interaction parameter. Here, we present a systematic comparison of theoretical predictions from different quantum many-body theories with recent experimental data of Nascimbne et al (2010 Nature 463 1057). Our comparisons have no adjustable parameters, either theoretically or experimentally. All the model approximations seem to fluctuate rather than converge on the experimental data. It turns out that a simple Gaussian pair fluctuation theory gives the best quantitative agreement, except at the critical superfluid transition region. In the normal state, we also calculate the equation of state by using a quantum cluster expansion theory and explore in detail its applicability to low temperatures. Using the accurate experimental result for the thermodynamic function S ( T ), we determine the temperature T of a trapped Fermi gas at unitarity as a function of a non-interacting temperature T i , which can be obtained by an adiabatic sweep to the free gas limit. By analyzing the recent experimental data, we find a characteristic temperature ( T / T F ) 0 =0.19±0.02 or ( T i / T F ) 0 =0.16±0.02 in a harmonic trap, below which there are deviations from normal Fermi-liquid-like behavior that may be attributed to pairing effects. Here, T F is the Fermi temperature for a trapped ideal, non-interacting Fermi gas. Our thorough comparison may shed light on the further theoretical development of strongly interacting fermions.