The concept of effective temperature (ET) T * (T 0 , T ) is used in order to approximately "quantize" the thermodynamic functions of the dynamical object which is in the thermal equilibrium with thermal bath being at constant temperature T (T 0 = E 0 /k B , where E 0 is the ground-state energy, k B -Boltzmann constant, is the characteristic "quantum" temperature of the system itself). On these grounds the extensive comparative investigation is carried out for the "standard model" of statistical mechanics -the one-dimensional harmonic oscillator (HO). Three well-known approaches are considered and their thermodynamic consequences thoroughly studied. These are: the exact quantum, or non-classical Planck-Einstein approach, intermediate, or semiclassical Bloch-Wigner approach and, finally, the pure classical, or Maxwell-Boltzmann approach.

The long-standing and highly non-trivial problem of calculating the pressure fluctuations in the Gibbs equilibrium statistical mechanics is fully revised. The previous attempts are critically analyzed and it is shown that the application of the ideas by Bogolyubov gives the full and unambiguous solution of this problem. The crucial role is played by the Bogolyubov's idea of quasiaverages (or rather quasidynamic) quantities -specifically, the pressure P and dynamic compressibility Ψ. The virtual conjugate field which eliminates the translational invariance of the Hamilton function H in the limit ε →0 is the singular potential of the impenetrable walls of the container. The general relations for P and Ψ in terms of the derivatives of H are obtained and some examples are studied -i. e., the cases of the ideal vs. non-ideal as well as of uniform vs. non-and quasi-uniform (in Euler sense) Hamilton function H describing the given system.