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A general approach how to analytically describe and understand $SU(3)$lattice thermodynamics in the whole temperature range $[0, \infty)$ isformulated and used. It is based on the effective potential approach forcomposite operators properly extended to non-zero temperature and density. Thismakes it possible to introduce into this general formalism the mass gap, whichis responsible for the large-scale dynamical structure of the QCD ground state.The mass gap dependent gluon plasma pressure adjusted by this approach to thecorresponding lattice data is shown to be a continuously growing function oftemperature being thus differentiable in every point of its domain. At the sametime, the entropy and energy densities have finite jump discontinuities at somecharacteristic temperature $T_c = 266.5 \ \MeV$ with latent heat$\epsilon_{LH}= 1.41$. This is a firm evidence of the first-order phasetransition in $SU(3)$ pure gluon plasma. The heat capacity has a $\delta$-typesingularity (an essential discontinuity) at $T_c$, so that the velocity ofsound squared becomes zero at this point. All the independent thermodynamicquantities are exponentially suppressed below $T_c$ and rather slowly approachtheir respective Stefan-Boltzmann limits at high temperatures. Thosethermodynamic quantities which are the ratios of their independent counterpartssuch as conformity, conformality and the velocity of sound squared approachtheir Stefan-Boltzmann limits rather rapidly and demonstrate a non-trivialdependence on the temperature below $T_c$. We also calculate the trace anomalyrelation (the interaction measure) and closely related to it the gluoncondensate, which are especially sensitive to the non-perturbative effects. Ananalytical description of the dynamical structure of $SU(3)$ gluon plasma isgiven.

Analytic description of SU(3) lattice thermodynamics in the whole temperature range within the mass gap approach