On the superfluidity of classical liquid in nanotubes

In 2001, the author proposed the ultra second quantization method. The ultra second quantization of the Schr\"odinger equation, as well as its ordinary second quantization, is a representation of the N-particle Schr\"odinger equation, and this means that basically the ultra second quantization of the equation is the same as the original N-particle equation: they coincide in 3N-dimensional space. We consider a short action pairwise potential V(x_i -x_j). This means that as the number of particles tends to infinity, $N\to\infty$, interaction is possible for only a finite number of particles. Therefore, the potential depends on N in the following way: $V_N=V((x_i-x_j)N^{1/3})$. If V(y) is finite with support $\Omega_V$, then as $N\to\infty$ the support engulfs a finite number of particles, and this number does not depend on N. As a result, it turns out that the superfluidity occurs for velocities less than $\min(\lambda_{\text{crit}}, \frac{h}{2mR})$, where $\lambda_{\text{crit}}$ is the critical Landau velocity and R is the radius of the nanotube.Comment: Latex, 20p. The text is presented for the International Workshop "Idempotent and tropical mathematics and problems of mathematical physics", Independent University of Moscow, Moscow, August 25--30, 2007 and to be published in the Russian Journal of Mathematical Physics, 2007, vol. 15, #

Probability Theory Compatible with the New Conception of Modern Thermodynamics. Economics and Crisis of Debts

We show that G\"odel's negative results concerning arithmetic, which date back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites paradox") pose the questions of the use of fuzzy sets and of the effect of a measuring device on the experiment. The consideration of these facts led, in thermodynamics, to a new one-parameter family of ideal gases. In turn, this leads to a new approach to probability theory (including the new notion of independent events). As applied to economics, this gives the correction, based on Friedman's rule, to Irving Fisher's "Main Law of Economics" and enables us to consider the theory of debt crisis.Comment: 48p., 14 figs., 82 refs.; more precise mathematical explanations are added. arXiv admin note: significant text overlap with arXiv:1111.610

Побудова критерію довготривалого руйнування втоми для тонкостінних шаруватих оболонок

A model and criterion of long-term fatigue failure for thin-walled layered shells is built, taking into account the influence of the type of stress state. The problem of calculating the number of cycles to failure under combined loading is considered. Solutions are built on the basis of the concept of equivalent stresses. The problem of determining local stresses in composites of random structure is formulated within the framework of the second-order nonlinear theory. The solution of the stochastic boundary value problem on determining the stress concentration in a unidirectional composite with a metal matrix (MMC) was obtained. To build a complete system of equations of the second order, the method of successive approximations is used. The parameters of the stress concentration at the boundary of the components are determined. The given examples show the importance of the influence of nonlinear properties on the redistribution of stresses near the fibers. The possibility of predicting the long-term strength of the material is shown. The necessary information about the material for the formulation of failure criteria is the S-N curves for individual components of the combined stresses. Pages of the article in the issue: 136 - 139 Language of the article: UkrainianПобудовано модель та критерій довготривалого руйнування втоми для тонкостінних шаруватих оболонок із врахуванням впливу виду напруженого стану. Розглянуто задачі розрахунку числа циклів до руйнування при комбінованому навантаженні. Розв'язки будуються на основі концепції еквівалентних напружень. Сформульовано в рамках нелінійної теорії другого порядку задачу визначення локальних напружень у композитах випадкової структури. Отримано розв’язок стохастичної крайової задачі про визначення концентрації напружень в односпрямованому композиті з металевою матрицею (ММК). Для отримання повної системи рівнянь другого порядку використовується метод послідовних наближень. Визначено параметри концентрації напружень на границі компонентів. Наведені приклади показують важливість впливу нелінійних властивостей на перерозподіл напружень біля волокон. Показано можливість прогнозування довготривалої міцності матеріалу, Необхідною інформацією про матеріал для формулювання критеріїв руйнування є криві S-N для окремих компонентів комбінованих напружень

Спадкова повзучість ізотропних композитів випадкової структури при складному напруженому стані

Nonlinear hereditary creep problem of the mechanics of composites is solved within the framework of a second-order theory. The hereditary functionals are used to construct general constitutive relations. A stochastic boundary value problem for determining the stress concentration and its relaxation in metal matrix composite (PMC) is solved in Laplace-Carson image space. Shapery's correspondence principle for quasi-linear viscoelastic media is generalised on the hereditary creep problem and the method of successive approximation is used. The reduced creep functions and the stress concentration parameters are determined. Examples are given showing the importance of the mutual influence of nonlinear elastic and viscous properties of the components on stress redistribution near inclusions with possibility to predicting the long-term strength. Pages of the article in the issue: 77 - 80 Language of the article: UkrainianРозв'язано задачу нелінійної спадкової повзучості композитів випадкової структури в рамках нелінійної теорії в’язкопружності другого порядку. Спадкові функціонали використано для побудови загальних визначальних рівнянь складного напруженого стану. Принцип відповідності Шепері для квазілінійних в'язкопружних середовищ узагальнено щодо проблем спадкової повзучості. Отримано розв'язок стохастичної крайової задачі про визначення концентрації напружень та їх релаксації в композиті з полімерною матрицєю (ПМК). Для виведення повної системи в'язкопружних рівнянь другого порядку використовується метод послідовного наближення. Визначено функції повзучості, локально усереднені за в'язкопружною матрицею та пружними включеннями. Наведені приклади, що показують важливість взаємного впливу нелінійних пружних та в'язких властивостей компонентів на перерозподіл напружень біля включень у багатокомпонентних ПMК. Як практичний результат можна відзначити можливість прогнозування довготривалої міцності матеріалу, коли в'язкопружне поле напружень відомо в результаті комп’ютерного моделювання в околі включень

Expansion Around the Mean-Field Solution of the Bak-Sneppen Model

We study a recently proposed equation for the avalanche distribution in the Bak-Sneppen model. We demonstrate that this equation indirectly relates $\tau$,the exponent for the power law distribution of avalanche sizes, to $D$, the fractal dimension of an avalanche cluster.We compute this relation numerically and approximate it analytically up to the second order of expansion around the mean field exponents. Our results are consistent with Monte Carlo simulations of Bak-Sneppen model in one and two dimensions.Comment: 5 pages, 2 ps-figures iclude

Peculiarities of dynamics of Dirac fermions associated with zero-mass lines

Zero-mass lines result in appearance of linear dispersion modes for Dirac fermions. These modes play an important role in various physical systems. However, a Dirac fermion may not precisely follow a single zero-mass line, due to either tunneling between different lines or centrifugal forces. Being shifted from a zero-mass line the Dirac fermion acquires mass which can substantially influence its expected "massless" behavior. In the paper we calculate the energy gap caused by the tunneling between two zero-mass lines and show that its opening leads to the delocalization of linear dispersion modes. The adiabatic bending of a zero-mass line gives rise to geometric phases. These are the Berry phase, locally associated with a curvature, and a new phase resulting from the mass square asymmetry in the vicinity of a zero-mass line.Comment: 6 pages, 4 figures. In the second version some references were added and minor changes were made in the introductio

Mathematical Conception of "Phenomenological" Equilibrium Thermodynamics

In the paper, the principal aspects of the mathematical theory of equilibrium thermodynamics are distinguished. It is proved that the points of degeneration of a Bose gas of fractal dimension in the momentum space coincide with critical points or real gases, whereas the jumps of critical indices and the Maxwell rule are related to the tunnel generalization of thermodynamics. Semiclassical methods are considered for the tunnel generalization of thermodynamics and also for the second and ultrasecond quantization (operators of creation and annihilation of pairs). To every pure gas there corresponds a new critical point of the limit negative pressure below which the liquid passes to a dispersed state (a foam). Relations for critical points of a homogeneous mixture of pure gases are given in dependence on the concentration of gases.Comment: 37 pages, 9 figure, more precise explanations, more references. arXiv admin note: substantial text overlap with arXiv:1202.525

Semiclassical Estimates of Electromagnetic Casimir Self-Energies of Spherical and Cylindrical Metallic Shells

The leading semiclassical estimates of the electromagnetic Casimir stresses on a spherical and a cylindrical metallic shell are within 1% of the field theoretical values. The electromagnetic Casimir energy for both geometries is given by two decoupled massless scalars that satisfy conformally covariant boundary conditions. Surface contributions vanish for smooth metallic boundaries and the finite electromagnetic Casimir energy in leading semiclassical approximation is due to quadratic fluctuations about periodic rays in the interior of the cavity only. Semiclassically the non-vanishing Casimir energy of a metallic cylindrical shell is almost entirely due to Fresnel diffraction.Comment: 12 pages, 2 figure

Critical exponents of the anisotropic Bak-Sneppen model

We analyze the behavior of spatially anisotropic Bak-Sneppen model. We demonstrate that a nontrivial relation between critical exponents tau and mu=d/D, recently derived for the isotropic Bak-Sneppen model, holds for its anisotropic version as well. For one-dimensional anisotropic Bak-Sneppen model we derive a novel exact equation for the distribution of avalanche spatial sizes, and extract the value gamma=2 for one of the critical exponents of the model. Other critical exponents are then determined from previously known exponent relations. Our results are in excellent agreement with Monte Carlo simulations of the model as well as with direct numerical integration of the new equation.Comment: 8 pages, three figures included with psfig, some rewriting, + extra figure and table of exponent

Comments on the Sign and Other Aspects of Semiclassical Casimir Energies

The Casimir energy of a massless scalar field is semiclassically given by contributions due to classical periodic rays. The required subtractions in the spectral density are determined explicitly. The so defined semiclassical Casimir energy coincides with that obtained using zeta function regularization in the cases studied. Poles in the analytic continuation of zeta function regularization are related to non-universal subtractions in the spectral density. The sign of the Casimir energy of a scalar field on a smooth manifold is estimated by the sign of the contribution due to the shortest periodic rays only. Demanding continuity of the Casimir energy under small deformations of the manifold, the method is extended to integrable systems. The Casimir energy of a massless scalar field on a manifold with boundaries includes contributions due to periodic rays that lie entirely within the boundaries. These contributions in general depend on the boundary conditions. Although the Casimir energy due to a massless scalar field may be sensitive to the physical dimensions of manifolds with boundary, its sign can in favorable cases be inferred without explicit calculation of the Casimir energy.Comment: 39 pages, no figures, references added, some correction