Geodesic
Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type
Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 1, pp. 135-148 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter $\alpha$, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of $\alpha$. We consider Bose–Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for $1<\alpha<2$ is greater in the case under consideration than in the case of an ideal system, where $\alpha=2$.
Keywords: fractional-order derivative, nonquadratic spectrum, Green's function, van der Waals equation, Bose–Einstein condensation. 
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Z. Z. Alisultanov; R. P. Meylanov. Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type. Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 1, pp. 135-148. http://geodesic.mathdoc.fr/item/TMF_2012_173_1_a7/

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