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arXiv:1510.07109

Thermodynamics of extensive but nonadditive systems: modified Gibbs-Duhem equation in the dipolar gas

Takashi Mori

Published Oct 24, 2015• 0 citations

Abstract

Thermodynamic properties of extensive but nonadditive systems are investigated. The precise definitions of additivity and extensivity are presented, and we will see that additivity derives several important properties including the shape-independence of the thermodynamic functions, the concavity of the entropy, and the equivalence of ensembles. In nonadditive systems, some of the above properties can be violated. It is pointed out that the shape-dependence of the entropy density in a nonadditive system results in the violation of the Gibbs-Duhem equation. As an example, the dipolar gas is numerically studied and the violation of the Gibbs-Duhem equation is confirmed. The predicted violation of the Gibbs-Duhem equation should be observable in cold-atom experiment with polarized dipolar gas.

DOI: 10.48550/arxiv.1510.07109

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Thermodynamic properties of extensive but nonadditive systems are investigated. The precise definitions of additivity and extensivity are presented, and we will see that additivity derives several important properties including the shape-independence of the thermodynamic functions, the concavity of the entropy, and the equivalence of ensembles. In nonadditive systems, some of the above properties can be violated. It is pointed out that the shape-dependence of the entropy density in a nonadditive system results in the violation of the Gibbs-Duhem equation. As an example, the dipolar gas is numerically studied and the violation of the Gibbs-Duhem equation is confirmed. The predicted violation of the Gibbs-Duhem equation should be observable in cold-atom experiment with polarized dipolar gas.

We present a thermodynamic description of ultracold gases with dipolar interactions which properly accounts for the long-range nature and broken rotation invariance of the interactions. It involves an additional thermodynamic field conjugate to the linear extension of the gas along the direction of the dipoles. The associated uniaxial pressure shows up as a deviation from the Gibbs–Duhem relation in the density profile of a trapped gas. It has to vanish in self-bound droplets, a condition which determines the observed dependence of the aspect ratio on particle number. A tensorial generalization of the virial theorem and a number of further exact thermodynamic relations are derived. Finally, extending a model due to Nozières, a simple criterion for the freezing transition to a superfluid mass density wave is given.

The Boltzmann-Gibbs-von Neumann entropy of a large part (of linear size L ) of some (much larger) d -dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to Ld-1. Here we show, for d=1,2 , that the (nonadditive) entropy Sq satisfies, for a special value of q not equal to 1, the classical thermodynamical prescription for the entropy to be extensive, i.e., Sq proportional variant Ld. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited in mathematical models with scale-invariant correlations [C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sci. U.S.A.102 15377 (2005)]. Finally, we find that the system critical features are marked by a maximum of the special entropic index q.

It is proposed that some of the formulae derived from the Gibbs-Duhem relation may serve as a simple test of consistency for approximation employed in solving many-body problems. When two of the intensive parameters are partly developed in terms of normalized extensive parameters, such as density, simple connections among the expansion coefficients follow directly from the Gibbs-Duhem relation. As an example chemical potential and external magnetic field are chosen as the two intensive parameters, to apply the relation to the Hubbard model. It is shown that the Hartree-Fock approximation surely satisfies the relation, while Hubbard's first approximation does not. Thus the latter approximation is thermodynamically inconsistent, and its validity should be seriously questioned.

Based on the semiclassical theory, we investigate the thermodynamic properties of a dipolar Fermi gas. Through a self-consistent procedure, we numerically obtain the phase-space distribution function at finite temperature. We show that the deformations in both momentum and real space become smaller and smaller as the temperature is increased. For the homogeneous case, we also calculate pressure, entropy, and heat capacity. In particular, at the low-temperature limit and in the weak interaction regime, we obtain an analytic expression for the entropy which agrees qualitatively with our numerical result. The stability of a trapped gas at finite temperature is also explored.