Nanoscale physics

Deformed Bose gas models and a kind of duality

Mishchenko Yu. A., Gavrilik A. M., Khelashvili M. V.

Bogolyubov Institute for Theoretical Physics of the NAS of Ukraine. Metrolohichna str., 14-b, Kyiv-03143, Ukraine. 
E-mail: mishchenko@bitp.kiev.ua
The relation of matching (sharing same property) is studied for pairs of deformed Bose gas models from two different classes, namely: a) the one defined using deformed derivative in modified thermodynamic relations [1,2], b) derived through deformed oscillators and distributions [3] in statistical mechanics sector.
For this we also use the term “duality”. Basically, the “physics” in the both models should agree. That is, thermodynamic relations – at least one-particle distribution (and E.O.S.) – must coincide, as encoded in certain relation on the deformation structure functions (DSFs) of the two models.
In particular, between two versions of μ̃,q-Bose gas, see [1,2] and [3], such a duality relation provides the coincidence of one-particle deformed distributions. Moreover, the relation is extended from two partner μ̃,q-deformed Bose gas models, to  general pairs of dual models. This way, distribution  in φ-deformed model defined like in [1,2] (yielding φ-deformed total number of particles N(φ)) is recovered from: N(φ)  = ; on the other hand, distribution , defined by DSF , – is required to agree. For the models to match, DSFs φ and  should be related: . Note that inverse mapping → φ can be applied [4] in the derivation of modified Gross-Pitaevskii equation.
The class of “self-dual” deformed models – such that their DSFs have the same form but with modified (values of) deformation parameters, is established. Say for the μ̃-Bose gas with |q=1 = (1 + μ̃ ) n − μ̃ n2  (1-parameter subclass of μ̃,q-deformed family), the distribution in the dual version is given by “dual” structure function  = (1 + μ̃/2) n – (μ̃/2) n2.
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2.	Gavrilik A.M., Mishchenko Yu.A. Virial coefficients in the (μ̃,q)-deformed Bose gas model related to compositeness of particles and their interaction: Temperature-dependence problem // Phys. Rev. E -2014.- 90, -P. 052147.
3.	Gavrilik A.M., Mishchenko Yu.A. Correlation function intercepts for μ̃,q-deformed Bose gas model implying effective accounting for interaction and compositeness of particles // Nucl. Phys. B.-2015.- 891, -P. 466-481.
4.	Gavrilik A.M., Khelashvili M.V., Nazarenko A.V., Mishchenko Yu.A. Deformed Gross-Pitaevskii equation and its application   (in preparation).