Due to the finite-size effects, the localization of the phase transition in finite systems and the determination of its order, become an extremely difficult task, even in the simplest known cases. In order to identify and locate the finite-volume transition point $$T_{0}(V)$$T0(V)of the QCD deconfinement phase transition to a colorless QGP, we have developed a new approach using the finite-size cumulant expansion of the order parameter and the $$L_{mn}$$Lmn-method. The first six cumulants $$C_{1,2,3,4,5,6}$$C1,2,3,4,5,6 with the corresponding under-normalized ratios (skewness $$\Sigma $$Σ,kurtosis $$\kappa $$κ,pentosis $$\varPi_{\pm}$$Π±, and hexosis $$\mathcal{H}_{1,2,3}$$H1,2,3)and three unnormalized combinations of them,($$\mathcal {O}={\mathcal{\sigma}^{2}\mathcal{\kappa}}{\mathbf{\Sigma}^{-1}}$$O=σ2κΣ-1,$$\mathcal{U}={\mathcal{\sigma}^{-2}\mathbf{\Sigma}^{-1}}$$U=σ-2Σ-1,$$\mathcal{N}=\mathcal{\sigma}^{2}\mathcal{\kappa}$$N=σ2κ)are calculated and studied as functions of(T,V).A new approach, unifying in a clear and consistent way the definitions of cumulant ratios, is proposed. A numerical FSS analysis of the obtained results has allowed us to locate accurately the finite-volume transition point. The extracted transition temperature value $$T_{0}(V)$$T0(V) agrees with that expected $$T_{0}^{N}(V)$$(formula presented.)(V) from the order parameter and the thermal susceptibility $$\chi_{T}\left(T,V\right)$$χTT,V,according to the standard procedure of localization to within about $$2\,\%$$2%.In addition to this, a very good correlation factor is obtained proving the validity of our cumulants method. The agreement of our results with those obtained by means of other models is remarkable. © 2015, The Author(s).
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