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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.

We study a large class of strongly interacting condensate-like materials, which can be characterized by a normalizable complex-valued function. A quantum wave equation with logarithmic nonlinearity is known to describe such systems, at least in a leading-order approximation, wherein the nonlinear coupling is related to temperature. This equation can be mapped onto the flow equations of an inviscid barotropic fluid with intrinsic surface tension and capillarity; the fluid is shown to have a nontrivial phase structure controlled by its temperature. It is demonstrated that in the case of a varying nonlinear coupling an additional force occurs, which is parallel to a gradient of the coupling. The model predicts that the temperature difference creates a direction in space in which quantum liquids can flow, even against the force of gravity. We also present arguments explaining why superfluids, be it superfluid components of liquified cold gases or Cooper pairs inside superconductors, can affect closely positioned acceleration-measuring devices.

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.

A mean-field theory is developed for a Bose liquid enclosed in a large vessel 𝒱. In accord with liquid structure concepts of Mitus et al., the liquid in 𝒱 is assumed to consist of adjacent macroscopic subregions Λ_k. In each subregion the bosons perform a locally ordered motion with prevailing orientation k + x, which varies randomly when passing from one subregion to another. |k| is constant, whereas temperature dependence of |x| is governed by a mean-field theory (MFT). The theory is applied to simulate HeI heat capacity C_V (T) at T > T_λ = 2.17 K and C_V (T) singularity at $T_{\text{λ}}^{+}$. The MFT numerical heat capacity C_n(T) = ΔE/ΔT exhibits behaviour characteristic of a singularity at $T_{\text{λ}}^{+}$: rapid increase with decreasing ΔT. Apart from $T_{\text{λ}}^{+}$, good agreement of C_n(T) with C_V(T) experimental plot is also found above T_λ, at T ∊ (T_λ, 3K].