We study the roles of the dynamical high order perturbation and statistically non-linear infrared fluctuation/correlation in the virial equation of state for the Fermi gas in the unitary limit. Incorporating the quantum level crossing rearrangement effects, the spontaneously generated entropy departing from the mean-field theory formalism leads to concise thermodynamical expressions. The dimensionless virial coefficients with complex non-local correlations are calculated up to the fourth order for the first time. The virial coefficients of unitary Fermi gas are found to be proportional to those of the ideal quantum gas with integer ratios through a general term formula. Counterintuitively, contrary to those of the ideal bosons ($a^{(0)}_2=-\frac{1}{4 \sqrt{2}}$) or fermions($a^{(0)}_2=\frac{1}{4 \sqrt{2}}$), the second virial coefficient $a_2$ of Fermi gas at unitarity is found to be equal to zero. With the vanishing leading order quantum correction, the BCS-BEC crossover thermodynamics manifests the famous pure classical Boyle's law in the Boltzmann regime. The non-Gaussian correlation phenomena can be validated by studying the Joule-Thomson effect.
Deep Dive into The virial equation of state for unitary fermion thermodynamics with non-Gaussian correlations.
We study the roles of the dynamical high order perturbation and statistically non-linear infrared fluctuation/correlation in the virial equation of state for the Fermi gas in the unitary limit. Incorporating the quantum level crossing rearrangement effects, the spontaneously generated entropy departing from the mean-field theory formalism leads to concise thermodynamical expressions. The dimensionless virial coefficients with complex non-local correlations are calculated up to the fourth order for the first time. The virial coefficients of unitary Fermi gas are found to be proportional to those of the ideal quantum gas with integer ratios through a general term formula. Counterintuitively, contrary to those of the ideal bosons ($a^{(0)}_2=-\frac{1}{4 \sqrt{2}}$) or fermions($a^{(0)}_2=\frac{1}{4 \sqrt{2}}$), the second virial coefficient $a_2$ of Fermi gas at unitarity is found to be equal to zero. With the vanishing leading order quantum correction, the BCS-BEC crossover thermodynamic
Unconventional unitary fermion physics is associated with a variety of strongly interacting topics. This theme can test the many-body theories, from neutron stars and nuclear matter to quark-gluon plasmas etc.
In recent years, the study for the interacting fermion matter properties has attracted much attention in quantum many-body community [1]. This is attributed to the rapid progress of atomic Fermi gas experiments. Controlling the S-wave scattering length between two different spin components allows one to control the interaction strength by using a magnetically tuned Feshbach resonance. With the magnetic tuning technique, increasing the interaction strength of atomic fermions with scattering length a from -∞ to +∞ resulting in bound boson systems exhibits the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCS-BEC crossover). The two regimes with positive and negative S-wave scattering length meet in the strongly interacting limit with the divergent scattering length. At the resonant point, the scattering cross-section will be saturated as σ = 4π/k 2 (with k being the relative wave-vector magnitude of the colliding particles) due to the fundamental unitary property limit. The strongly interacting BCS-BEC crossover topic is literally called the unitary Fermi gas thermodynamics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].
Generally, the thermodynamical properties of the low energy dilute Fermi system are determined by the Swave scattering length a, the particle number density n as well as the temperature T . In the resonant regime with a zero-energy bound state, the divergent scattering length will certainly drop out in the thermodynamical quantities; i.e., the thermodynamical properties are universal [1][2][3]. The divergent scattering length poses an intractable many-body problem.
In addressing the BCS-BEC crossover thermodynamics, the fundamental issue is the ground state energy. On the basis of the dimensional analysis, the dimensionless coefficient ξ relates the energy per particle E/N = ξ 3 5 ǫ f with the Fermi kinetic energy ǫ f = k 2 f /(2m). Here, m is the bare fermion mass while k f is the Fermi momentum. The fundamental universal coefficient ξ has prompted many theoretical or experimental efforts in recent years [1]. Furthermore, the finite temperature thermodynamic properties of unitary Fermi gas are as intriguing as the zero-temperature ground state energy and many experimental/theoretical efforts have been made [5,6] [7,8].
Compared with the zero-temperature ground state energy, the additional energy scale, i.e, the reciprocal thermodynamical de Broglie wavelength λ -1 = mT /(2π) complicates the theme and makes the universal property analysis more profound [3]. In the weak degenerate Boltzmann regime, the universal properties are characterized by the virial coefficients [3,4,9,15]. For example, the virial equation of state is related with the neutrinosphere physics in supernovae for the dilute nuclear matter matter. It is believed that the virial equation of state will influence the detailed information of the neutrino response of low-density neutron matter [9]. How to calculate the virial coefficient in the unitary limit regime remains an important many-body topic. Like in addressing the zero-temperature ground state energy problem, the central task is to understand the novel non-linear fluctuation/correlation physics.
The virial expansion is the basic tool for use in discussing the thermodynamical properties and should be model independent. In thermodynamics, the non-linear virial expansion is the infinite series of the pressure according to the particle number density. Even as a fundamental theme, this question is very challenging and by no means resolved yet. To derive the high order virial coefficients of strongly correlated fermions in the unitary limit, the involved quantum statistical fluctuation/correlation and detailed dynamical effects must be clarified, which is the novel systematic requirement for the sound theoretical efforts. With the aim of calibrating the universal virial coefficients, we strive to examine the spontaneously quantum level shift contribution on the entropy (counting the microscopic states) in a dynamically and thermodynamically self-consistent way.
In this work, the dynamical and statistical correlation analysis explicitly demonstrates that the dense and hot thermodynamics at unitarity obey the virial theorem for the ideal non-interacting gas, i.e., P = 2/3E/V [3,5]. Meanwhile, the calculated virial coefficients at unitarity are found to be proportional to those for the ideal Fermi gas with integer ratios through an universal general term formula.
In the strongly interacting system, the dynamical effects compete with the non-linear fluctuation/correlations. The second virial coefficient is found to be vanishing due to the complicated correlations. From the viewpoint of bulk properties, is the dilute unitary Fermi gas behavior much
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