Helium-4 Energy and Specific Heat in Superfluid and Normal Phase

The calculation of the He4 energy and specific heat is carried out in a wide temperature range within the two-time temperature Green functions approach. The approximation improving the random phase approximation is developed providing the correct behaviour of the calculated specific heat at the temperatures $0 < T/T_c 1.32$ where $T_c$ stands for the phase transition point.Comment: 10 pages, 2 figure

Calculation of the Condensate Fraction in Liquid Helium-4

We adduce the results of the condensate fraction calculation for liquid helium-4. The method is derived from the first principles and involves minimum assumptions. The only experimental quantity we need in our calculations is the static structure factor which is easily measurable. We use the approximation in which expressions contain one summation in the wave vector space (or one integration it the radius vector space) in order to demonstrate the validity of our method. The computed values of the condensate fraction lie within $8.8 \div 14$ depending on the model potential for the short-range interaction. This result agrees with recent experimental measurements and numerical estimations.Comment: 8 pages, 1 figur

Representation of the Short-Range Interactions in Liquid Helium via Modified Hard Sphere Potentials

In this paper we propose five different modifications of the hard sphere potential for the modeling a short-range repulsion and the calculation of thermodynamic and transport properties of liquid He4. We calculate the potential energy, the total energy, and the sound velocity at T=0 K. It is shown that three of the proposed potentials give a satisfactory description of these properties.Comment: 7 pages, 4 figure

The Rank-Frequency Analysis for the Functional Style Corpora in the Ukrainian Language

We use the rank-frequency analysis for the estimation of Kernel Vocabulary size within specific corpora of Ukrainian. The extrapolation of high-rank behaviour is utilized for estimation of the total vocabulary size.Comment: 8 page

Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter

For a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameters the emergence of a phase transition is reported and temperature dependences of energy and heat capacity are studied in detail. Estimations towards a possibility to check the obtained jumps in the specific heat are made

The relation between fractional statistics and finite bosonic systems in one-dimensional case

The equivalence is established between the one-dimensional (1D) Bose-system with a finite number of particles and the system obeying the fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. The system of 1D harmonic oscillators is considered providing the model of harmonically trapped Bose-gas. The results are generalized for the system with power energy spectrum

Lviv period for Smoluchowski: Science, teaching, and beyond

A major part of Marian Smoluchowski's achievements in science corresponds to the period of his work at the University of Lviv. Since this part is well described in the literature, in the paper the emphasis is made on some less known activities of this outstanding scientist: his teaching, his organizational efforts, and even his hobbies. The list of publications corresponding to the Lviv period is given.iльшiсть наукових досягнень Марiана Смолуховського припадає на час його роботи у Львiвському унiверситетi. Оскiльки ця сторона дiяльностi вченого добре описана в лiтературi, то у статтi зроблено наголос на менш вiдомих напрямках роботи цього видатного вченого: викладання й орґанiзацiйнi заходи i навiть його хобi. Наведено також перелiк публiкацiй львiвського перiоду

Effective Hamiltonian and excitation spectrum of harmonically trapped bosons

An approach is proposed to obtain an effective Hamiltonian of a harmonically trapped Bose-system. Such a Hamiltonian is quadratic in the creation–annihilation operators and certain approximations allow to simplify higher (three and four operator) products to the required form. After the Hamiltonian diagonalization, the expression for the excitation spectrum is obtained containing in particular temperature-dependent corrections. Numerical calculations are made for a one-dimensional system. Some prospects towards the extension of the suggested approach to study binary bosonic mixtures are briefly discussed

Asymptotic formulas for integer partitions within the approach of microcanonical ensemble

The problem of integer partitions is addressed using the microcanonical approach which is based on the analogy between this problem in the number theory and the calculation of microstates of a many-boson system. For ordinary (one-dimensional) partitions, the correction to the leading asymptotic is obtained. The estimate for the number of two-dimensional (plane) partitions coincides with known asymptotic results.Розглянуто задачу про розбиття цiлих чисел у межах мiкроканонiчного пiдходу, який ґрунтується на аналогiї мiж цiєю задачею з теорiї чисел i обчисленням кiлькостi мiкростанiв багатобозонної системи. Для звичайних (одновимiрних) розбиттiв отримано поправку до головної асимптотики. Оцiнка кiлькостi двовимiрних (плоских) розбиттiв добре узгоджується з вiдомими асимптотичними результатами

Fractional statistics and finite bosonic system: A one-dimensional case

The equivalence is established between the one-dimensional (1D) Bose-system with a finite number of particles and the system obeying the fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. The system of 1D harmonic oscillators is considered providing the model of harmonically trapped Bose-gas. The results are generalized for the system with power energy spectrum.Comment: 10 page