Nonextensive thermodynamics of a cluster consisting of $M$ Hubbard dimers ($M=1,2,3$ and $\infty$)

TL;DR

This paper investigates the thermodynamic properties of small clusters of Hubbard dimers using nonextensive statistics, comparing two methods for relating physical temperature to Lagrange multipliers, and analyzing their effectiveness.

Contribution

It introduces a nonextensive statistical approach to small Hubbard dimer clusters and compares two temperature relation methods, suggesting method B is more suitable for small systems.

Findings

Method B better describes small-scale systems.

Temperature dependence of thermodynamic quantities calculated.

Comparison shows differences between the two methods.

Abstract

The thermodynamical property of a small cluster including $M$ Hubbard dimers, each of which is described by the two-site Hubbard model, has been discussed within the nonextensive statistics (NES). We have calculated the temperature dependence of the energy, entropy, specific heat and susceptibility for $M = 1, 2, 3$ and $\infty$, assuming the relation between the entropic index $q$ and the cluster size $N$ given by $q=1+2/N$ ($N = 2 M$ for $M$ dimers), which was previously derived by several methods. For relating the physical temperature $T$ to the Lagrange multiplier $\beta$, two methods have been adopted: $T=1/k_B \beta$ in the method A [Tsallis {\it et al.} Physica A {\bf 261} (1998) 534], and $T=c_q/k_B \beta$ in the method B [Abe {\it et al.} Phys. Lett. A {\bf 281} (2001) 126], where $k_B$ denotes the Boltzman constant, $c_q= \sum_i p_i^q$, and $p_i$ the probability distribution of the $i$th state. The susceptibility and specific heat of spin dimers described by the Heisenberg model have been discussed also by using the NES with the methods A and B. A comparison between results calculated by the two methods suggests that the method B may be better than the method A for small-scale systems.