Kinetic Equation for Quantum Fermi Gases and the Analytic Solution of Boundary Value Problems

We construct a kinetic equation describing the behavior of quantum Fermi gases with the molecule collision frequency proportional to the molecule velocity. We obtain an analytic solution of the generalized Smoluchowski problem with the temperature gradient and the mass flow velocity specified away from the surface. We find exact formulas for jumps of the gas temperature, concentration, and chemical potential. Analysis of limit cases demonstrates a transition of the quantum Fermi gas to the classical or degenerate gas.     

Magnetic and drift surfaces in stellarators

In this paper we study the structure of a stellarator field in toroidal geometry. A field line tracing code is developed to explore the structure of magnetic fields on the fine scale of the electron gyroradius p_e, so as to explain anomalous electron transport. The magnetic field is modelled by a simple analytic representation with finite number of parameters, so that we can integrate the field lines to a high accuracy. In a typical Heliac field we find that (i) most of the magnetic surfaces are well behaved on the fine scale, even when there is no two-dimensional symmetry and (ii) the width of an island, formed in the vicinity of a magnetic surface with rational rotational transform i = n/m, decays exponentially with m. Among those numerical studies, we have an example where the island width w is less than the electron gyroradius p_e for m greater than 17. This demonstrates that higher-order islands do not affect the electron transport. Our numerical results indicate that the anomalous electron transport observed in experiments may be due to the presence of an ambipolar electrostatic potential Φ. To reconfirm this proposition we compute the guiding center orbits of electrons and estimate the island widths of the drift surfaces that are swept out. We find that with a small electric potential depending on toroidal and poloidal angles, the drift surface island width w is an order of magnitude larger than that without the electric potential and decays exponentially at a slower rate. Since the drift step size is of the order of the maximum of p_e and w, the electron transport, which scales like the square of the step size, is enhanced when there is an electric potential.     

A Tridimensional Inverse Shaping Problem

We study the following tridimensional magnetostatic inverse shaping problem: can one find a distribution of currents around a levitating liquid metal bubble so that it takes a given shape? It leads to the resolution of an Eilonal equation on the surface of the bubble which has self-contained interest. We answer the question for closed smooth surface which are homeomorphic to a sphere. We give a necessary and sufficient condition on the data for existence and uniqueness of a C¹ solution. When the desired shape is axisymmetric and analytic, the solution is also analytic and the problem can be completely solved. A counterexample proves that not all analytic perturbations of such surface are shapable.     

Oscillations of the inertia moment of a finite Fermi system in the cranking model framework

In the framework of the cranking model with the potential of an anisotropic harmonic oscillator, we rigorously calculate how the moment of inertia of a finite Fermi system depends on the chemical potential at finite temperatures in the adiabatic limit analytically. We show that this dependence involves smooth and oscillating components. We find analytic expressions for these components at arbitrary temperatures and axial deformation frequencies. We show that oscillations of the moment of inertia increase as the spherical limit is approached and decrease exponentially as the temperature increases.     

Exact solution of a boundary value problem describing the uniform cylindrical or spherical piston motion

In this paper, we construct the exact solution for fluid motion caused by the uniform expansion of a cylindrical or spherical piston into still air. Following Lighthill [1], we introduce velocity potential into the analysis and seek a similarity form of the solution. We find both numerical and analytic solutions of the second order nonlinear differential equation, with the boundary conditions at the shock and at the piston. The results obtained from the analytic solutions justify numerical solution and the approximate solution of Lighthill [1]. We find that although the approximate solution of Lighthill [1] gives remarkably good numerical results, the analytic form of that solution is not mathematically satisfactory. We also find that in case of spherical piston motion Lighthill’s [1] solution differs significantly from that of our analytic and numerical solutions. We use Pade′ approximation to extend the radius of convergence of the series solution. We also carry out some local analysis at the boundary to obtain some singular solutions.     

A Fourier method for computing the perturbation to a two‐dimensional electric potential by a conductor of arbitrary shape

Introducing an electric conductor into a region pervaded by an initial electric potential perturbs that potential by inducing a charge distribution on the conductor's surface, necessary to guarantee that the surface is an equipotential of the total potential. Some numerical method is required to compute the perturbation potential, when the conductor's shape does not admit a standard analytic solution. For two‐dimensional situations, a method is proposed for solving for the perturbation potential that involves expansion of the boundary perturbation potential and its normal derivative as truncated Fourier series. This boundary potential is known to within an additive constant from the requirement that its sum with the initial potential must be a constant. The standard representation theorem for the Dirichlet problem gives a consistency relation between the boundary function and its normal derivative, which here becomes a set of linear algebraic relations between Fourier series coefficients, with matrix entries found by appropriate applications of the fast Fourier transform. These are solved for the boundary derivative coefficients; at any point exterior to the conductor, the perturbation potential can then be evaluated from the two sets of Fourier coefficients, using further application of the fast Fourier transform. Examples are shown for two conductor shapes, with several initial potentials. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 673–683, 2001     

Positive semidefinite germs in real analytic surfaces

We find all real analytic surface germs in on which every positive semidefinite function germ is a sum of squares (in fact, of two squares) of analytic function germs. Received: 9 January 2001 / Published online: 24 September 2001     

Stabilization of periodic Stokesian Hele–Shaw flows of ferrofluids

We consider an incompressible ferrofluid in a vertical Hele–Shaw cell and develop a proper analytic framework for the free interface and the velocity potential of the fluid in a periodic geometry. The flow is assumed to obey a non-Newtonian Darcy law. The forces influencing the fluid are gravity, surface tension and the response to a magnetic field induced by a current. In addition, the flow is stabilized at the lower boundary component by an external source b. We prove a well-posedness result for the flow near flat solutions. Moreover, we find conditions on the parameters and on the slope of b for the exponential stability and instability of flat interfaces. Furthermore, we identify values for the current's intensity ι where critical bifurcation of nontrivial finger-shaped solutions from the branch of trivial (flat) solutions takes place.     

Analytic modelling of solar radiation: driven temperature variations of outdoor surfaces

An analytic model is described for the variation of surface temperature T₀(t) ofa freely evaporating or totally dry surface exposed to complicated, non-analytic variations of meteorological conditions expressed in (N + 1)-point time series form. Its basis is a parameterized, analytic representation of T₀, inserted in linearized equations describing the surface energy balance and the concurrent underlying heat conduction. Applied to diurnal variations (with N = 24) over bare soil, the model reproduces observed temperatures with a typical standard deviation less than 1K, and the daily mean within 0.1 K.

The method should find general application in modelling of storage systems subject to complicated, discretely-sampled boundary conditions, where a linear analytic equation links the latter to the storage term.     

Construction of Exact Solutions for Equilibrium Configurations of the Boundary of a Conducting Liquid Deformed By an External Electric Field

In a two-dimensional plane-symmetric formulation, we consider the problem of the equilibrium configurations of the free surface of a conducting capillary liquid placed in an external electric field. We find a one-parameter family of exact solutions of the problem according to which the fluid takes the shape of a blade. Such a configuration provides formally unlimited local field amplification: the field strength is maximum at the edge of the blade and drops to zero at the periphery. For a given potential difference between the liquid and the flat electrode located above it, we find threshold values of the electric field strength at the edge of the liquid blade, the radius of curvature of the edge, and the distance from the edge to the electrode limiting the region of existence of the solutions.     

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.     

On the Spectrum of Non-Selfadjoint Schrödinger Operators with Compact Resolvent

We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schrödinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.     

Analytic surface germs with minimal Pythagoras number

We determine all complete intersection surface germs whose Pythagoras number is 2, and find that they are all embedded in ³ and have the property that every positive semidefinite analytic function germ is a sum of squares of analytic function germs. In addition, we discuss completely these properties for mixed surface germs in ³. Finally, we find in higher embedding dimension three different families with these same properties. Partially supported by DGICYT, BFM2002-04797 and HPRN-CT-2001-00271 Mathematical Subject Classification (2000): 11E25, 14P15.An erratum to this article can be found at     

Two-band superconductivity theory beyond the Migdal theorem

We propose a superconductivity theory of two-band nonadiabatic systems with strong electron correlations in the linear approximation in nonadiabaticity. Assuming a weak electron-phonon interaction, we obtain analytic expressions for the vertex and “intersecting” functions for each of the two bands. With the diagrams involving intersections of two electron-phonon interaction lines taken into account (which means going beyond the Migdal theorem), we determine mass operators of the Green’s functions and use them to derive the basic equations of the superconductivity theory for two-band systems. We find an analytic expression for the superconducting transition temperature T_c that differs from the expression in the case of the standard two-band systems by an essential renormalization of the relevant quantities that results from the nonadiabaticity effects and strong electron correlations. We study the dependence of T_c and of the isotopic coefficient α on the Migdal parameter m = ω₀/ε_F and show that accounting for the overlap of energy bands on the Fermi surface and for the nonadiabaticity effects at small values of the transferred momentum (q ≪ 2p_F) allows obtaining high values of T_c even for the weak electron-phonon interaction. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 111–126, October, 2006. An erratum to this article is available at .     

Analytic solutions of initial-boundary-value problems of transient conduction using symmetries

Lie symmetry method is applied to find analytic solutions of initial-boundary-value problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and closed form solutions is carried out for a physical problem of heat conduction in a semi-infinite solid bar made of AISI 304 stainless steel.     

Magneto-Static Vortices in Two Dimensional Abelian Gauge Theories

We study the existence of vortices of the Klein-Gordon-Maxwell equations in the two dimensional case. In particular we find sufficient conditions for the existence of vortices in the magneto-static case, i.e. when the electric potential f = 0{\phi = 0}. This result, due to the lack of suitable embedding theorems for the vector potential A is achieved with the help of a penalization method.     

Confinement–deconfinement transitions for two-dimensional Dirac particles

We consider a two-dimensional massless Dirac operator coupled to a magnetic field B and an electric potential V growing at infinity. We find a characterization of the spectrum of the resulting operator H in terms of the relation between B and V at infinity. In particular, we give a sharp condition for the discreteness of the spectrum of H beyond which we find dense pure point spectrum.     

Similarity inner solutions for the Pulsar equation

Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.     

Electric field in the Smoluchowski problem in a metal with an arbitrary coefficient of specular reflection

The electric field and the potential in the electron gas temperature jump problem in a metal under the influence of temperature gradient that is normal to the surface are constructed. The electron scattering from the metal surface is assumed to be specular-diffusive, and no restrictions are imposed on the coefficient of specular reflection.     

A numerical model for the trans-membrane voltage of vesicles

The Immersed Interface Method is employed to solve the time-varying electric field equations around a three-dimensional vesicle. To achieve second-order accuracy the implicit jump conditions for the electric potential, up to the second normal derivative, are derived. The trans-membrane potential is determined implicitly as part of the algorithm. The method is compared to an analytic solution based on spherical harmonics and verifies the second-order accuracy of the underlying discretization even in the presence of solution discontinuities. A sample result for an elliptic interface is also presented.