On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.      

Ground States in the Spin Boson Model

We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant λ. We show that the ground-state energy is an analytic function of λ and that the corresponding ground state can also be chosen to be an analytic function of λ. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground-state energy can be calculated using regular analytic perturbation theory.     

An anyon model

We construct an infinite-dimensional dynamical Hamiltonian system that can be interpreted as a localized structure (“quasiparticle”) on the plane E ₂. The model is based on the theory of an infinite string in the Minkowski space E _1,3 formulated in terms of the second fundamental forms of the worldsheet. The model phase space H is parameterized by the coordinates, which are interpreted as “internal” (E(2)-invariant) and “external” (elements of T*E ₂) degrees of freedom. The construction is nontrivial because H contains a finite number of constraints entangling these two groups of coordinates. We obtain the expressions for the energy and for the effective mass of the constructed system and the formula relating the proper angular momentum and the energy. We consider a possible interpretation of the proposed construction as an anyon model.     

Well-Posedness and Regularity for a Parabolic-Hyperbolic Penrose-Fife Phase Field System

This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable χ, which is characterized by the presence of an inertial term multiplied by a small positive coefficient μ. This feature is the main consequence of supposing that the response of χ to the generalized force (which is the functional derivative of a free energy potential and arises as a consequence of the tendency of the free energy to decay towards a minimum) is subject to delay. We first obtain well-posedness for the resulting initial-boundary value problem in which the heat flux law contains a special function of the absolute temperature ϑ, i.e. α(ϑ) ∼ ϑ − 1/ϑ. Then we prove convergence of any family of weak solutions of the parabolic-hyperbolic model to a weak solution of the standard Penrose-Fife model as μ ↘ 0. However, the main novelty of this paper consists in proving some regularity results on solutions of the parabolic-hyperbolic system (including also estimates of Moser type) that could be useful for the study of the longterm dynamics.     

Instanton in the Field of a Pointlike Source of a Euclidean Non-Abelian Field

We consider the behavior of an (anti)instanton in the field of a pointlike source of a Euclidean non-Abelian field and investigate (anti)instanton deformations described by variations of its characteristic parameters. We formulate the variational problem of seeking the corresponding “crumpled” topological configurations and solve it algebraically using the Ritz method (of multipole decomposition of deformation fields). We investigate the domain of parameters specific for the instanton liquid model. We propose a simple method of taking contributions of such configurations in the functional integral into account approximately. In the framework of superposition analysis, we obtain an estimate for the mean energy of the source in the instanton liquid, and the estimate increases linearly with distance. In the case of a dipole in the color-singlet state, the energy is a linear function of the distance between the sources with the “strength” coefficient agreeing well with the known model and lattice estimates. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 267–298, February, 2006.     

Convergence of phase-field approximations to the Gibbs–Thomson law

We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van der Waals–Cahn–Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs–Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn–Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta–Kawasaki as a model for micro-phase separation in block-copolymers.     

2D semiclassical model for high harmonic generation from gas

The electron behavior in laser field is described in detail. Based on the ID semiclassical model, a2D semiclassical model is proposed analytically using 3D DC-tunneling ionization theory. Lots of harmonic features are explained by this model, including the analytical demonstration of the maximum electron energy 3.17U _p Finally, some experimental phenomena such as the increase of the cutoff harmonic energy with the decrease of pulse duration and the “anomalous” fluctuations in the cutoff region are explained by this model.     

Exactly solvable two-dimensional complex model with a real spectrum

Using supersymmetric intertwining relations of the second order in derivatives, we construct a two-dimensional quantum model with a complex potential for which all energy levels and the corresponding wave functions are obtained analytically. This model does not admit separation of variables and can be considered a complexified version of the generalized two-dimensional Morse model with an additional sinh ⁻² term. We prove that the energy spectrum of the model is purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. We explicitly find the symmetry operator, describe the biorthogonal basis, and demonstrate the pseudo-Hermiticity of the Hamiltonian of the model. The obtained wave functions are simultaneously eigenfunctions of the symmetry operator. Dedicated to the 80th birthday of Yuri Victorovich Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 102–111, July, 2006.     

Renormalization group in turbulence theory: Exactly solvable Heisenberg model

An exactly solvable Heisenberg model describing the spectral balance conditions for the energy of a turbulent liquid is investigated in the renormalization group (RG) framework. The model has RG symmetry with the exact RG functions (the β-function and the anomalous dimension γ) found in two different renormalization schemes. The solution to the RG equations coincides with the known exact solution of the Heisenberg model and is compared with the results from the ε expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter ε of the RG expansion is introduced by replacing a δ-function-like pumping function in the random force correlator by a powerlike function). The results, which are valid for asymptotically small ε, can be extrapolated to the actual value ε=2, and the few first terms of the ε expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 245–262 May. 1998.     

High temperature regime for a multidimensional Sherrington–Kirkpatrick model of spin glass

Comets and Neveu have initiated in [5] a method to prove convergence of the partition function of disordered systems to a log-normal random variable in the high temperature regime by means of stochastic calculus. We generalize their approach to a multidimensional Sherrington-Kirkpatrick model with an application to the Heisenberg model of uniform spins on a sphere of ℝ ^d , see [9]. The main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical energy path. Received: 30 October 1996 / In revised form: 13 October 1997     

Exactly solvable quantum mechanical models with Stückelberg divergences

We consider an exactly solvable quantum mechanical model with an infinite number of degrees of freedom that is an analogue of the model of N scalar fields (λ/N)(ϕ^{a a})² in the leading order in 1/N. The model involves vacuum and S-matrix divergences and also the Stückelberg divergences, which are absent in other known renormalizable quantum mechanical models with, divergences (such as the particle in a δ-shape potential or the Lee model). To eliminate divergences, we renormalize the vacuum energy and charge and transform the Hamiltonian by a unitary transformation with a singular dependence on the regularization parameter. We construct the Hilbert space with a positive-definite metric, a self-adjoint Hamiltonian operator, and a representation for the operators of physical quantities. Neglecting the terms that lead to the vacuum divergences fails to improve and, on the contrary, worsens the renormalizability properties of the model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 91–106, October, 2000.     

Copolymer–homopolymer blends: global energy minimisation and global energy bounds

We study a variational model for a diblock copolymer–homopolymer blend. The energy functional is a sharp-interface limit of a generalisation of the Ohta–Kawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Furthermore we characterise the conditions under which the minimisers may be non-unique. In higher dimensions we construct lower and upper bounds on the energy of minimisers, and explicitly compute the energy of spherically symmetric configurations.     

The null energy condition and cosmology

Field theories that violate the null energy condition (NEC) are of interest both for the solution of the cosmological singularity problem and for models of cosmological dark energy with the equation of state parameter w < −1. We consider two recently proposed models that violate the NEC. The ghost condensate model requires higher-derivative terms in the action, and this leads to a heavy ghost field and energy unbounded from below. We estimate the rates of particle decay and discuss possible mass limitations to protect the stability of matter in the ghost condensate model. The nonlocal stringy model that arises from a cubic string field theory and exhibits a phantom behavior also leads to energy unbounded from below. In this case, the energy spectrum is continuous, and there are no particle-like excitations. This model admits a natural UV completion because it comes from superstring theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 3–12, April, 2008.     

Viscoelastic Rayleigh waves in a layered structure with a weak lateral inhomogeneity

Rayleigh waves in an almost layered viscoelastic medium are studied by using the “surface” ray method based on real rays. Viscoelasticity is described in terms of the Maxwell-Boltzmann-Volterra model and, for high frequencies, is treated as perturbed perfect elasticity. In addition to the leading term of the ray asymptotics, which corresponds to the balance of energy along rays, the correction term describing anomalous displacements of the Love type is discussed. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 7–13.     

Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈ℝ² where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of Navier Stokes equation coupled on the interface to the dynamic system of elasticity. The characteristic feature of this coupled model is that the resolvent is not compact and the energy function characterizing balance of the total energy is weakly degenerated. These combined with the lack of mechanical dissipation and intrinsic nonlinearity of the dynamics render the problem of asymptotic stability rather delicate. Indeed, the only source of dissipation is the viscosity effect propagated from the fluid via interface. It will be shown that under suitable geometric conditions imposed on the geometry of the interface, finite energy function associated with weak solutions converges to zero when the time t converges to infinity. The required geometric conditions result from the presence of the pressure acting upon the solid.     

Coherentization of the energy of heat fluctuations by a two-channel bilinear control system

We propose and investigate a mathematical model of an open bilinear control system for the conversion of heat energy in a coherent form. We show that the use of a combinational parametric resonance formed by the control system in a one-temperature ensemble of weakly dissipative elastic-gyroscopic subsystems enables one to obtain a positive energy output without using any cooling device apart from the control system. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1557–1573, November, 2007.     

A Monotonicity Property for Weighted Delaunay Triangulations

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa’s result.     

λφ4model and Higgs mass in standard model calculated by Gaussian effective potential approach with a new regularization-renormalization method

Based on a new regularization-renormalization method, the λφ⁴ model used in standard model (SM) is studied both perturbatively and nonperturbatively by Gaussian effective potential (GEP). The invariant property of two mass scales is stressed and the existence of a (Landau) pole is emphasized. Then after coupling with theSU(2) ×U(1) gauge fields, the Higgs mass in standard model (SM) can be calculated to bem _H≈138 GeV. The critical temperature (T _c ) for restoration of symmetry of Higgs field, the critical energy scale (μ_max, the maximum energy scale under which the lower excitation sector of the GEP is valid) and the maximum energy scale (μ_max, at which the symmetry of the Higgs field is restored) in the SM areT _c ≈476 GeV, μ_c≈0.547 × 10¹⁵ and μ_max≈0.873 × 10¹⁵, respectively. Project supported in part by the National Natural Science Foundation of China.     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

Spectral Analysis of an Effective Hamiltonian in Nonrelativistic Quantum Electrodynamics

We investigate spectral properties of an effective Hamiltonian which is obtained as a scaling limit of the Pauli–Fierz model in nonrelativistic quantum electrodynamics. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian.