| Summary: | The paper presents a study of a non-standard model of fractional statistics. The exponential of the Gibbs factor in the expression for the occupation numbers of ideal bosons is substituted with the Tsallis <i>q</i>-exponential and the parameter <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </semantics> </math> </inline-formula> is considered complex. Such an approach predicts quantum critical phenomena, which might be associated with <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">PT</mi> </semantics> </math> </inline-formula>-symmetry breaking. Thermodynamic functions are calculated for this system. Analysis is made both numerically and analytically. Singularities in the temperature dependence of fugacity and specific heat are revealed. The critical temperature is defined by non-analyticities in the expressions for the occupation numbers. Due to essentially transcendental nature of the respective equations, only numerical estimations are reported for several values of parameters. In the limit of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> some simplifications are obtained in equations defining the temperature dependence of fugacity and relations defining the critical temperature. Applications of the proposed model are expected in physical problems with energy dissipation and inderdisciplinarily in effective description of complex systems to describe phenomena with non-monotonic dependencies.
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