On the linearized system of equations for the condensate–normal fluid interaction at very low temperature

The linearization around one of its equilibrium of a system that describes the correlations between the superfluid component and the normal fluid part of a condensed Bose gas in the approximation of very low temperature and small condensate density is studied. A simple and transparent argument gives a necessary and sufficient condition for the existence of global solutions satisfying the conservation of the total number of particles and energy. The global solutions describe the evolution in time of the density of the thermal cloud and, unlike in previous work, that of the condensate's density. Their convergence to a suitable stationary state is also shown and rates of convergence for the normal fluid and superfluid components are obtained.     

Scattering of Fermions on a Superfluid Bose Condensate

The influence of a fermionic impurity on the spectrum of overcondensate excitations of a superfluid Bose system is investigated. The scattering of neutrons on the Bose condensate of a superfluid ⁴He and a mixture of the liquid ³He and ⁴He is considered. An analytic expression for the spectrum of overcondensate excitations for these two systems is found. Bibliography: 10 titles.     

Splitting superfluid hydrodynamic equations and instability of solutions in the case of degenerate bosons

Properties of solutions to superfluid hydrodynamic equations as applied to the degenerate Bose gas are considered. The equations are split into two independent pairs of equations. One pair is written for the normal component implies the instability of solutions, which manifests itself in the majorant catastrophe with respect to the total density. The case when the thermodynamic functions depend on the difference of the normal and superfluid velocities is also considered. In that case, the system is not split; however, the instability and the majorant catastrophe occur when the initial temperature tends to absolute zero.     

Second sound near lambda transition in presence of quantum vortices

In this paper, temperature waves (also known as second sound) are considered, with their respective coupling with waves in the order parameter describing the transition from normal phase to superfluid phase, and with waves in the vortex length density. We analyze the coupling between these three kinds of waves and explore its relevance in situations not far from the lambda transition. In particular, the expressions for the second sound speed and second sound attenuation are explicitly obtained within some approximations, showing the influence of the order parameter and the vortex length density, which is decisive close to the transition.

     

Molecular hydrodynamics of degenerate weakly nonideal bose gas. I. Generalized equations of two-fluid hydrodynamics

A degenerate weakly nonideal Bose gas is investigated at temperatures near zero. Relations connecting the irreducible Green's functions are used to obtain exact expressions for the two-time temperature-dependent Green's functions. In the case of weak nonideality an expression possessing interpolation properties with respect to the frequency and momentum is obtained for the density—density Green's function. At low frequencies, the results of two-fluid hydrodynamics are reproduced. At wavelengths less than the mean free path the energy of the elementary excitations and the damping obtained in the paper agree with the results of perturbation theory with respect to the coupling constant. An expression for the operator of the superfluid velocity is obtained.In memory of Nikolai Nikolaevich BogolyubovV. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 412–465, December, 1992.     

Existence and uniqueness for a mathematical model in superfluidity

In this paper we propose a model to study superfluidity by considering as state variables the order parameter, describing the concentration of the superfluid phase, the velocity of the superfluid and the absolute temperature. We assume that the order parameter satisfies a Ginzburg–Landau equation and that the velocity is decomposed as the sum of a normal and a superfluid component. The heat equation provides the evolution equation for the temperature. We prove that this model is consistent with the principles of thermodynamics. Well‐posedness of the resulting initial and boundary value problem is shown. Copyright © 2008 John Wiley & Sons, Ltd.     

Superfluid quantum vortex in curved space

The effects of the general theory of relativity in superfluid liquids can be studied with respect to real astrophysical objects, namely, neutron stars. We calculate the energy of a quantum vortex in the neutron superfluid liquid for the Schwarzschild metric.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 474–480, December, 1996.     

Some features of quantum statistical systems with an energy spectrum of the fractional-power type

We derive and investigate the equation of state for a system with a fractional-power spectrum and also derive an expression for the density of states in such systems. We consider a fractional-differential model of electron-phonon interaction.     

Turbulent Superfluid Profiles and Vortex Density Waves in a Counterflow Channel

In this paper we study the two-dimensional profiles of the superfluid component velocity and the quantized vortex-points density in a counterflow channel where the influence of the walls cannot be neglected. The numerical results obtained show the presence of vortex density waves in the channel, as shown in a recent paper by means of the one-fluid model.     

A Weakly Nonideal Bose Gas Model Leading to the Fountain Effect

The ordinary thermodynamics corresponds to the equilibrium ground state of the system. We present a model making it possible to construct a second thermodynamics, an analogue of an equilibrium state of a superfluid, based on which we explain the effect of the appearance of a bottleneck in a thin capillary for a moving superfluid if it is heated to the critical point in the second thermodynamics.     

Microscopic justification of the stochastic f-model of critical dynamics

We use a microscopic approach to derive the stochastic Langevin equations for a quantum Bose liquid near the phase transition to the superfluid state. These equations can be obtained in local form if either the finiteness of the system dimensions or the boundedness of the time interval under study is taken into account. We give the microscopic expressions for random forces in the stochastic equations and show that the white noise approximation holds only in the critical region. We prove that the model for properly describing the critical fluctuations is precisely the F-model.     

Classifying magnetic and superfluid equilibrium states in magnets with the spin <Emphasis Type="Italic">s</Emphasis> = 1

Based on the method of quasiaverages, we classify magnetic and superfluid equilibrium states in magnets with the spin s = 1. Under certain simplifications, assumptions about the residual symmetry of degenerate states and the transformation properties of order parameter operators under transformations generated by additive integrals of motions lead to linear algebraic equations for a classification of the equilibrium means of the order parameters. We consider different cases of the magnetic SO(3) or SU(3) symmetry breaking and obtain solutions for the vector and tensor order parameters for particular forms of the parameters of the residual symmetry generators. We study the equilibriums of magnets with simultaneously broken phase and magnetic symmetries. We find solutions of the classification equations for superfluid equilibrium states and establish relations between the parameters of the residual symmetry generator that allow the thermodynamic coexistence of nonzero equilibrium means of the order parameters.     

Averaging the operators for a large number of clusters: Phase transitions

We develop the theory of averaging the operators in a Fock space, introduced in our previous papers. We find the algebra of mean operators. We introduce the quantum entropy and quantum free energy using the function f(z)=zlog(z) of the mean unit operator (the “measure” of mean operators). Such a “quantum thermodynamics” determines the temperature dependence of the critical speed (“the Landau criterion”) and the temperature distribution at which the speed of a superfluid system is nonzero even at zero temperature. We generalize the consideration to the case where sparsely distributed bosons form clusters. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 297–314, November, 2000.     

Singlet-triplet fermion coupling theory

At zero temperature, the superfluid Fermi liquid self-consistency equations that describe superposition of states with singlet and triplet fermion coupling are obtained and analyzed. The conditions for creating such states are found. Equations for the critical temperature of the phase transition from a one-gap (singlet or triplet) superfluid state to a singlet-triplet superfluid state and the temperature dependence of the order parameters near the transition temperature are derived. A mechanism for creating new states is proposed, and the thermodynamic stability of these states is studied. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115, No. 3, pp. 459–476, June, 1998.     

Computation of localization degree in the sense of the Anderson criterion for a one-dimensional diagonally disordered system

For a one-dimensional diagonally disordered half-infinite chain, we consider the problem of finding the limit value as t → ∞ of the average excitation density D at the edge site of the chain under the condition that the excitation is localized at this site at t = 0. For a binary disordered chain, we obtain an expression for D that is exact in the small defect concentration limit for an arbitrary defect energy. In this case, the density D depends nonanalytically on the energy. We obtain an expression for D in the case of an arbitrary small diagonal disorder. We also calculate the relative contribution to D from states with a given energy. All the obtained results agree well with the computer simulation data.     

Quasiaverages and Degenerate Quantum Equilibriums of Magnetic Systems with SU(3) Symmetry of the Exchange Interaction

We consider magnetic systems with the SU(3) symmetry of the exchange interaction. For degenerate equilibriums with broken magnetic and phase symmetries, we formulate classification equations for the order parameter using the concept of residual symmetry. Based on them, we obtain an explicit form of the equilibrium values of the order parameters of a spin nematic and an antiferromagnet in the general form. We clarify the existence conditions for six types of superfluid equilibriums for the order parameter describing the Bose pair condensate. We study inhomogeneous equilibriums and obtain the explicit coordinate dependence of the magnetic order parameters.     

Quasithermodynamic correction to the Stefan-Boltzmann law

We provide a correction to the Stefan-Boltzmann law and discuss the problem of a phase transition from the superfluid state into the normal state. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 207–208, January, 2008.     

On the expected discounted penalty function for risk process with tax

In this paper we consider the generalized Cramér-Lundberg risk model including tax payments. We investigate how tax payments affect the behavior of a Cramér-Lundberg surplus process by defining an expected discounted penalty function at ruin. We derive an explicit expression for this function by solving a differential equation. Consequently, the explicit formulas for the discounted probability density function of the surplus immediately before ruin and the discounted joint probability density function of the surplus immediately before ruin and the deficit at ruin are obtained. We also give explicit expressions for the function for exponential claims.     

Pricing and hedging of Asian options: quasi-explicit solutions via Malliavin calculus

We use Malliavin calculus and the Clark–Ocone formula to derive the hedging strategy of an arithmetic Asian Call option in general terms. Furthermore we derive an expression for the density of the integral over time of a geometric Brownian motion, which allows us to express hedging strategy and price of the Asian option as an analytic expression. Numerical computations which are based on this expression are provided.