Nonextensive Thermodynamics of a Cluster Consisting of M

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 nonex-tensive statistics (NES). We have calculated temperature and magnetic-field dependences of the specific heat and susceptibility for M = 1, 2, 3 and ∞, assuming the relation between M and the entropic index q given by q = 1 + 1/M , which was previously derived by several methods. For relating the physical temperature T to the Lagrange multiplier β, two methods have been adopted: T = 1/kBβ in the method A [Tsallis et al. denotes the Boltzman constant, cq = i p q i , and pi the probability distribution of the ith state. A comparison between the results calculated by the two methods suggests that the method B may be more appropriate than the method A for small-scale systems. In the last several years, much study has been made with the use of nonextensive statistics (NES) which was initiated by Tsallis. 1)–4) Before discussing the NES, let's recall the basic feature of the Boltzman-Gibbs statistics (BGS) for a system with internal energy E and entropy S, which is immersed in a large reservoir with energy E 0 and entropy S 0. The temperature of the small system T is the same as that of the reservoir T 0 where T = δE/δS and T 0 = δE 0 /δS 0. If we consider the number of possible microscopic states of Ω(E 0) in the reservoir, its entropy is given by S 0 = k B lnΩ(E 0) where k B denotes the Boltzman constant. The probability of finding the small system with the energy E is given by p(E) = Ω(E 0 − E)/Ω(E) ∼ exp(−E/k B T) with E ≪ E 0. When the physical quantity Q of a system containing N particles is expressed by Q ∝ N γ , it is classified into two groups in the BGS: intensive (γ = 0) or extensive one (γ = 1). The temperature and energy are typical intensive and extensive quantities, respectively. This is not the case in the NES, as will be shown below. In the NES, on the contrary, the temperature of a nanosystem which is in contact with the reservoir, is expected to fluctuate around the temperature of the reservoir T 0 because of the smallness of nanosystems and …