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The thermodynamics of a free Bose gas with effective temperature scale and hard-sphere Bose gas with the scale are studied. arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction V_osc, which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential V_He between ⁴He atoms. The repulsive part of V_He is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the temperature scale (HSET), and particle mass and density equal to those of ^4He, is investigated, first, by the Bogoliubov–Huang method and next by an improved version of this method, which describes He II in terms of dressed bosons and takes approximate account of those terms of the 2-body repulsion which are linear in the zero-momentum Bose operators a₀, (originally rejected by Bogoliubov). Theoretical heat capacity C_V(T) exhibits good agreement, below 1.9 K, with the experimental heat capacity graph observed in ⁴He at saturated vapour pressure. The phase transition to the He II phase, occurs in the HSET at T_λ = 2.17 K, and is accompanied, in the modified HSET version, by a singularity of C_V(T). The fraction of atoms in the momentum condensate at 0 K equals 8.86% and agrees with other theoretical estimates for He II. The fraction of normal fluid falls to 8.37% at 0 K which exceeds the value 0% found in He II.

A recently developed formalism for Helium II is generalized by introducing a 2-body interaction of spheres with diameter depending on the momentum exchanged between two atoms in an interaction process. A larger class of atomic collisions is also admitted. These modifications allow to account for some details of the interatomic potential V_He(r) between two ⁴He atoms, which were previously disregarded, and to improve the theoretical graphs of Helium II momentum distribution and normal fluid fraction.

The ideas of Mitus et al. are exploited to define the liquid state as a state of matter, in which particles perform locally ordered motion. The presence of the liquid phase is accounted for by a stochastic term in the Hamiltonian, which simulates this property of a liquid. The Bogoliubov–Lee–Huang theory of He II, recently modified by use of effective temperature scale and more stringent reduction procedure (DHSET theory) is extended, by incorporating this term into the ⁴He Hamiltonian. The resulting thermodynamics accounts for effects, which are beyond the scope of other He I and He II theories, e.g., the atomic momentum distribution and excitation spectrum have the form of diffused bands, similarly as in He II; the He I, theoretical heat capacity C_V(T) is a convex function, with a minimum at T_min > T_λ, which qualitatively simulates experimental He I heat capacity. Other thermodynamic functions are similar to those of DHSET theory.

The Bogoliubov-Lee-Huang theory of superfluid ⁴He is modified by introducing an effective temperature scale (which accounts for the deep well of the interatomic potential) and by incorporating into the Hamiltonian a stochastic term V_l, which simulates liquidity of HeI and liquidity of the normal and superfluid component of HeII. V_l depends on two independent random angles α_n, α_s ∈ [0, π], which characterize the locally ordered motion of the two fluids (the normal fluid and superfluid) comprising HeII. The resulting thermodynamics improves the thermodynamic functions and excitation spectrum Ep(α_n, α_s) of the superfluid phase, obtained previously, leaving the heat capacity C_V (T) of the normal phase, with a minimum at T_min > 2.17K, unchanged. The theoretical velocity of sound in HeII, equal to the initial slope of Ep(π, π), agrees with experiment.