Symmetric Green's Function Research Articles

Diffusive nonreciprocity and thermal diode

Wave propagation and diffusion in linear materials preserve local reciprocity in terms of a symmetric Green's function. For wave propagations, the relation between the fields entering and leaving a system is more relevant than the detailed information about the fields inside it. In such cases, the global reciprocity of the scattering off a system through several ports is more important, which is defined as the symmetric transmission between the scattering channels. When a two-port system supports nonreciprocal (electromagnetic, acoustic) wave propagation, it is a (optical, phonon) diode directly following the definition. However, to date no concrete definition or discussion has been made on the global reciprocity of diffusive processes through a multiple-port system. It thus remains unclear what are the differences and relations between the three concepts, namely, local nonreciprocity, global nonreciprocity, and diode effect in diffusion. Here, we provide theoretical analysis on the frequency-domain Green's function and define the global reciprocity of heat diffusion through a two-port system, which has a different setup from that of a thermal diode. We further prove the equivalence between a heat transfer system with broken steady-state global reciprocity and a thermal diode, assuming no temperature-dependent heat generation. The validities of some typical mechanisms in breaking the diffusive reciprocity and making a thermal diode have been discussed. Our results set a general background for future studies on symmetric and asymmetric diffusive processes.

Non-reciprocity in nonlinear elastodynamics

Reciprocity is a fundamental property of linear time-invariant (LTI) acoustic waveguides governed by self-adjoint operators with symmetric Green's functions. The break of reciprocity in LTI elastodynamics is only possible through the break of time reversal symmetry on the micro-level, and this can be achieved by imposing external biases, adding nonlinearities or allowing for time-varying system properties. We present a Volterra-series based asymptotic analysis for studying spatial non-reciprocity in a class of one-dimensional (1D), time-invariant elastic systems with weak stiffness nonlinearities. We show that nonlinearity is neither necessary nor sufficient for breaking reciprocity in this class of systems; rather, it depends on the boundary conditions, the symmetries of the governing linear and nonlinear operators, and the choice of the spatial points where the non-reciprocity criterion is tested. Extension of the analysis to higher dimensions and time-varying systems is straightforward from a mathematical point of view (but not in terms of new non-reciprocal physical phenomena), whereas the connection of non-reciprocity and time irreversibility can be studied as well. Finally, we show that suitably defined non-reciprocity measures enable optimization, and can provide physical understanding of the nonlinear effects in the dynamics, enabling one to establish regimes of “maximum nonlinearity.” We highlight the theoretical developments by means of a numerical example.

On Green functions of second-order elliptic operators on Riemannian manifolds: The critical case

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation Pu=0 in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive (super)solution, then such a solution is called a ground state of P in M, and P is said to be critical in M.We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and P⋆. Under some further assumptions, we describe the behavior at infinity of such a Green function. This result extends and sharpens the celebrated result of P. Li and L.-F. Tam concerning the existence of a symmetric Green function for the Laplace–Beltrami operator on a smooth and complete Riemannian manifold M.

Autonomous models on a Cayley tree

.The most general single species autonomous reaction–diffusion model on a Cayley tree with nearest-neighbor interactions is introduced. The stationary solutions of such models, as well as their dynamics, are discussed. To study the dynamics of the system, we obtain a directionally symmetric Green function for the evolution equation of average number density. In some limiting cases the Green function is studied. Some examples are worked out in more detail.

Riemann–Hilbert treatment of Liouville theory on the torus

We apply a perturbative technique to study classical Liouville theory on the torus. After mapping the problem on the cut-plane we give the perturbative treatment for a weak source. When the torus reduces to the square the problem is exactly soluble by means of a quadratic transformation in terms of hypergeometric functions. We give general formulas for the deformation of a torus and apply them to the case of the deformation of the square. One can compute the Heun parameter to first order and express the solution in terms of quadratures. In addition, we give, in terms of quadratures of hypergeometric functions, the exact symmetric Green's function on the square on the background generated by a one-point source of arbitrary strength.

A selfadjoint second order hyperbolic system

We consider a second order hyperbolic system of the type (1) L u = u t t − B u x x = f ( x , t ) , ( x , t ) ∈ T m , where matrix B is a nonsingular constant matrix with positive eigenvalues, ( x , t ) ∈ R 2 and u , f ∈ R n . The set T m is defined to be (2) T m = { ( x , t ) | 0 ⩽ t ⩽ 1 / m , | x | ⩽ 1 − m t } , where m = min { μ k } and μ k 2 is any eigenvalue of the matrix B. We will show that, under the condition u ( x , 0 ) = 0 , | x | ⩽ 1 , a symmetric Green's function G n × n can be constructed [K. Kreith, A selfadjoint problem for the wave equation in higher dimensions, Comput. Math. Appl. 21 (5) (1991) 12–132] so that (3) u ( x , t ) = ∫ ∫ T m G n × n ( x , t ; ξ , τ ) f ( ξ , τ ) d ξ d τ for any function f ∈ L 2 ( T m ) . This will imply that the operator L in (1) over the set L 2 ( T m ) of functions given by Eq. (3) and u ( x , 0 ) = 0 , | x | ⩽ 1 , is selfadjoint. We also note that the same result holds for u in (1), under the condition that u t ( x , 0 ) = 0 , | x | ⩽ 1 . We further note that when B has only one eigenvalue μ 2 , the function u in Eq. (3) satisfies a boundary condition similar to that of Kalmenov [T. Kalmenov, On the spectrum of a selfadjoint problem for the wave equation, Akad. Nauk. Kazakh SSR Vestnik 1 (1983) 63–66] on the characteristic boundaries of T μ .

The Rest-Frame Darwin Potential from the Lienard–Wiechert Solution in the Radiation Gauge

In the semiclassical approximation in which the electric charges of scalar particles are described by Grassmann variables (Q2i=0, QiQj≠0), it is possible to re-express the Lienard–Wiechert potentials and electric fields in the radiation gauge as phase space functions, because the difference among retarded, advanced, and symmetric Green functions is of order Q2i. By working in the rest-frame instant form of dynamics, the elimination of the electromagnetic degrees of freedom by means of suitable second classs contraints leads to the identification of the Lienard–Wiechert reduced phase space containing only N charged particles with mutual action-at-a-distance vector and scalar potentials. A Darboux canonical basis of the reduced phase space is found. This allows one to re-express the potentials for arbitrary N as a unique effective scalar potential containing the Coulomb potential and the complete Darwin one, whose 1/c2 component agrees with the known expression. The effective potential gives the classical analogue of all static and non-static effects of the one-photon exchange Feynman diagram of scalar electrodynamics.

Order parameter suppression in double layer quantum Hall ferromagnets

Double-layer quantum Hall systems at Landau level filling factor ν=1 have a broken symmetry ground state with spontaneous interlayer phase coherence and a gap between symmetric and antisymmetric subbands in the absence of interlayer tunneling. We examine the influence of quantum fluctuations on the spectral function of the symmetric Green's function, probed in optical absorption experiments. We find that as the maximum layer separation at which the ν=1 quantum Hall effect occurs is approached, absorption in the lowest Landau level grows in strength. Detailed line shapes for this absorption are evaluated and related to features in the system's collective excitation spectrum.

Quantum phase transitions in a three-dimensional frustrated S=1/2 Heisenberg antiferromagnet.

The spherically symmetric Green's function approximation is used to study a three-dimensional (3D) S=1/2 antiferromagnet described by isotropic Heisenberg Hamiltonian. Detailed numerical calculations are carried out for the simple cubic lattice with antiferromagnetic nearest- and next-nearest-neighbor exchange interactions (${\mathit{J}}_{1\mathrm{\ensuremath{-}}}$${\mathit{J}}_{2}$ model). We obtained the phase diagram indicating two regions of temperature and frustration parameters p=${\mathit{J}}_{2}$/(${\mathit{J}}_{1}$+${\mathit{J}}_{2}$) where two different ordered states are possible. The absence of the long-range order (LRO) in 3D lattice even at zero temperature at some values of frustration is emphasized. At zero temperature with increasing frustration the system endures the second-order phase transitions from the N\'eel-type LRO state to the stripe-type LRO state through the spin liquid state.

A Method of Solving a Class of CIV Boundary Value Problems

AbstractA method will be introduced to solve problems utt — uss = h(s, t), u(t,t) - u(1+t,1 - t), u(s,0) = g(s), u(1,1) = 0 and for (s, t) in the characteristic triangle R = {(s,t) : t ≤ s ≤ 2 — t, 0 ≤ t ≤ 1}. Here represent the directional derivatives of u in the characteristic directions e1 = (— 1, — 1) and e2 = (1, — 1), respectively. The method produces the symmetric Green's function of Kreith [1] in both cases.

Uniform zeros for vibrating strings

In this paper we show that it is possible for a finite uniform flexible string which is initially at equilibrium to assume this position again at T = 1. The string will be subject to the linear force –pu, where u is the displacement, and external force ∞ Our technique is a generalization of symmetric Green's function method of Kreith2 and Haws1

Symmetric Green's functions for certain hyperbolic problems

The problems associated with a hyperbolic partial differential operator are usually posed as initial value problems. However, it is possible to view a certain class of problems as boundary value problems, and thus make use of the techniques generally reserved for problems associated with elliptic operators. In the process, some additional “boundary conditions” along the characteristics will be introduced, in order to guarantee a unique solution. In this paper, the Conjugate Point Problem and the Focal Point Problem are studied on a characteristics triangle domain. The criteria for the existence of a Green's function are established, and some additional conditions are derived. The Green's functions for several examples are then constructed by following the prescribed method, and the method is analyzed to determine the most general type of problem to which it can be extended. The method is then generalized to problems with a different domain.

Symmetric Green's Function for a Class of CIV Boundary Value Problems

AbstractGeneralized boundary value problems are considered for hyperbolic equations of the form utt — uss + λp(s, t)u = 0. By constructing symmetric Green's functions appropriate to such problems the existence of eigenvalues is established.

Symmetric Green's Functions on Complete Manifolds

On demontre d'une facon constructive que l'operateur de Laplace admet une fonction de Green symetrique G(x,y) sur une variete de Riemann complete

Quantum field theory and the antipodal identification of black-holes

The antipodal points ( U, V, θ, ϕ) and (− U, − V, π − θ, ϕ + π) of the Schwarzchild-Kruskal manifold, usually interpreted as two different events (in two different worlds) are considered here as physically identified (to give one single world). This has fundamental consequences for the QFT formulated on this manifold. The antipodal symmetric fields have (globally) zero norm. The usual particle-antiparticle Fock space definition breaks down. There is no quantum operator (unitary, antiunitary or projection) giving antipodal symmetric states from the usual Kruskal ones. The antipodal symmetric Green functions have the same periodicity β = 8 π M in imaginary (Schwarzschild) time as the usual (non-symmetric) ones. (Identification with “conical singularity” would give a period 1 2 β ). In any case, no usual thermal interpretation is possible for T = β −1 (nor for T 0 = 2 β or any other value) in the theory. In the light of these results we discuss “old” ideas and more recent ones on identification.

A generalized approach to the solution of variable systems subjected to arbitrary source functions and boundary conditions

Under study are layered inhomogeneous systems with or without forcing functions as represented by sources and boundary conditions. Under a forced or unforced state, such systems pervade the fields of sound and vibration. A common formulation to the constraints and physical laws that apply is given by using the Fredholm integral equation. A unified and versatile solution is developed analytically to treat these problems under forced or unforced conditions. Then the approach is dissected to implement the results on the computer or to treat, at the outset, the lumped approximation of the distributed system. This development modifies, extends, and generalizes previous studies that have been applied to a restricted class of systems. Here the treated systems need not be limited to those describable by a strictly bilinear or symmetric Green function. Also, the source function and boundary conditions are not limited to those of special type or form. Throughout, examples are given to introduce, motivate and illustrate the analysis.

The formulation of spontaneous breakdown in the path integral method

It is proposed as to how spontaneous breakdown of symmetry should be dealt with in the framework of the path-integral. The Goldstone theorem is derived from the path-integral method and the relation between symmetric Green functions and broken ones is discussed.

Reliable Perturbative Results for Strong Interactions?

An explicit calculation shows perturbation theory to be arbitrarily good for the deep Euclidean Green's functions of any Yang-Mills theory and of many Yang-Mills theories with fermions. Under the hypothesis that spontaneous symmetry breakdown is of dynamical origin, these symmetric Green's functions are the asymptotic forms of the physically significant spontaneously broken solution, whose coupling could be strong.

Calculation of an Enclosed Cylindrically Symmetric Green's Function and Its Comparison to Field Measurements

An analytic representation for the field from an arbitrary current distribution (either expandable in a Fourier series or for a current distribution that varies as sin2kz over one period) with cylindrical symmetry and enclosed within conducting inner and outer walls is derived. The Green's function applicable to the dc field from a single loop lying between two perfectly conducting cylinders is first calculated. The inversion from measured magnetic field to current distribution, which can be obtained in principle from the Green's function, is done in detail only for the previously mentioned current distributions. For most experimental parameters, the Green's function is so sharply peaked, one can assume a periodic current distribution for long current layers (length-to-diameter ratio > 2). The predicted magnetic field amplitude and field shapes were experimentally verified on the Astron machine.