Non-extensive thermodynamics of transition-metal nanoclusters

In recent years, much study has been made by applying the non-extensive statistics (NES) to various non-extensive systems where the entropy and/or energy are not necessarily proportional to the number of their constituent subsystems. The non-extensivity may be realized in many systems such as physical, chemical and biological ones, and also in small-scale nanosystems.After briefly reviewing the recent development in nanomagnetism and nano-thermodynamics, I have discussed, in this article, NES calculations of thermodynamical properties of a nanocluster containing noninteracting M dimers. With bearing in mind a transition-metal nanocluster, each of the dimers is assume to be described by the two-site Hubbard model (a Hubbard dimer). The temperature and magnetic-field dependences of the specific heat, magnetization and susceptibility have been calculated by changing M = 1, 2, 3 and ∞, results for M = ∞ corresponding to those of the conventional Boltzman–Gibbs statistics (BGS). It has been shown that the thermodynamical property of nanoclusters containing a small number of dimers is considerably different from that of macroscopic counterparts calculated within the BGS. The specific heat and susceptibility of spin dimers described by the Heisenberg model have been discussed also by employing the NES.