Nonextensive thermodynamic functions in the Schrödinger-Gibbs ensemble

Schrödinger suggested that thermodynamical functions cannot be based on the gratuitous allegation that quantum-mechanical levels (typically the orthogonal eigenstates of the Hamiltonian operator) are the only allowed states for a quantum system [E. Schrödinger, Statistical Thermodynamics (Courier Dover, Mineola, 1967)]. Different authors have interpreted this statement by introducing density distributions on the space of quantum pure states with weights obtained as functions of the expectation value of the Hamiltonian of the system. In this work we focus on one of the best known of these distributions and prove that, when considered in composite quantum systems, it defines partition functions that do not factorize as products of partition functions of the noninteracting subsystems, even in the thermodynamical regime. This implies that it is not possible to define extensive thermodynamical magnitudes such as the free energy, the internal energy, or the thermodynamic entropy by using these models. Therefore, we conclude that this distribution inspired by Schrödinger's idea cannot be used to construct an appropriate quantum equilibrium thermodynamics.

Figure 1

Plot of the functions $S_{N=1}^{\text{th}}$ (dotted line), $S_{N=2}^{\text{th}}$ (thin line), and $S_{N=3}^{\text{th}}$ (thick line) versus $T$ for $k_B=8.617\times {10}^{-5}$ and $\Delta =0.001$ for the SG ensemble. We can easily verify the nonadditivity property of the function $S^{\text{th}}$ and the violation of the third law, since the entropy does not go to zero in the limit $T\to 0$.

Figure 2

Plots of the entropy $S^{c,\text{th}}$ of the canonical ensemble vs $T$ with $N=1$ (dotted line), $N=2$ (thin line), and $N=3$ (thick line) for $k_B=8.617\times {10}^{-5}$ and $\Delta =0.001$. We see how additivity is clearly preserved and that the limit $T\to 0$ is equal to zero.

Figure 3

Plot of the specific heat of the SG ensemble vs $T$ with $N=1$ (dotted line), $N=2$ (thin line), and $N=3$ (thick line) for $k_B=8.617\times {10}^{-5}$ and $\Delta =0.001$. We can see that the limit when $T\to 0$ increases with $N$, while for a consistent thermodynamic description it must go to zero.

Figure 4

Plot of von Neumann's entropy of the SG ensemble, for $N=1$ (dotted line), $N=2$ (thin line), and $N=3$ (thick line), vs $T$, with $k_B=8.617\times {10}^{-5}$ and $\Delta =0.001$. We can notice that the three levels are not equally spaced.

Figure 5

Representation of the integration region $\Gamma$ in the complex plane. The poles are represented on the real axis.