Exact solutions for spectra and Green's functions in random one-dimensional systems

For chains of harmonic oscillators with random masses a set of equations is derived, which determine the spatial Fourier components of the average one-particle Green's function. These equations are valid for complex values of the frequency. A relation between the spectral density and functions introduced by Schmidt is discussed. Exact solutions for this Green's function and the less complicated characteristics function-the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions-are derived for exponential distributions of the masses. For some cases the characteristic function is calculated numerically. For gamma distributions the equations are cast in the form of ordinary, higher order differential equations; these have been solved numerically for determining the characteristic function. For arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the Green's function are discussed.The method is also applied to chains where the spring constants and/or the masses have random values. Also for these systems exact solutions are discussed; for exponential distributions, e.g., of both masses and spring constants the characteristic function is expressed in Bessel functions. The relation with certain random relaxation models is shown. Finally, X-Y Hamiltonians with random exchange constants and/or magnetic fields-or, equivalently, tight-binding electron models with diagonal and/or off-diagonal disorder-are considered. Here the Green's function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin. New solutions for the characteristic function and Green's function are derived for a number of cases, including exponentially distributed magnetic fields and power law distributed exchange constants.     

Wehrl-Lieb's inequality for entropy and the uncertainty relation

Wehrl has defined a new classical entropy by means of coherent states and conjectured that this entropy is greater than one. Lieb proved Wehrl's conjecture. In our note we discuss a certain reformulation of the classical entropy. We show that Wehrl's inequality for reformulated entropy implies the usual version of the Heisenberg uncertainty relation.     

Application of Hamilton's law to forced,damped, stationary systems

The algebraic equations for the forced, damped, periodic, axisymmetric motion of circular plates, solid and annular, are derived directly through the application of Hamilton's law of varying action. The simplicity, for many problems, of direct analytical solutions by means of Hamilton's law has previously been demonstrated. The method is called the Hamilton-Ritz method. In this paper, direct analytical solutions from Hamilton's law are shown to be exactly the same as direct analytical solutions from the ancient and fundamental principle of virtual work. The Hamilton-Ritz formulation is compared to the Galerkin formulation. Results from one- and two-term solutions by direct virtual work (Hamilton-Ritz) are compared to results from the exact solution and to results from the Galerkin method.     

Ersaks's regeneration hypothesis and deviations from the exponential decay law

Recent formal arguments have related deviations from the exponential decay law of an unstable system to the regeneration of the initial unstable state. We investigate this relationship using simple models drawn from quantum optics.     

On the upper critical dimensions of random spin systems

A set of critical exponent inequalities is proved for a large class of classical random spin systems. The inequalities imply rigorous (and probably the optimal) lower bounds for the upper critical dimensions, i.e.,d _u4 for regular and random ferromagnets,d _u6 for spin glasses and random field systems.     

On Wigner's semicircle law for eigenvalues of non-random hamiltonians

It is shown that physical many-body systems with hamiltonians which belong to a large class of non-random matrices of rational Jacobi type have as level density Wigner's famous semicircle law.     

A class of exactly soluble random spin systems

It is shown that a large class of magnetic models with infinite-ranged exchange but local randomnes is exactly soluble.     

A continuity property of the entropy density for spin lattice systems

The entropy density of spin lattice systems is known to be a weak upper semi-continuous functional on the set of the lattice invariant states. (It is even weak discontinuous.) However we prove here that it is continuous with respect to the norm topology on those states.Aspirant van het Belgisch N.F.W.O.     

Noether's theorem and invariants of certain nonlinear systems

The invariants of certain nonlinear systems (having the form of inhomogeneous time-dependent harmonic oscillators) proposed by Ray and Reid are derived from Noether's theorem.     

New constants of motion for systems of free gravitating particles in Newton's and Einstein's theories of gravitation

New quantities have been found which are constants of motion in Newton's gravitational theory. Analogous but different quantities exist in Einstein's theory. The difference between the Newtonian and the relativistic quantities may be used to distinguish experimentally between the theories.     

Upper bounds on the free energy of random spin systems

The sequence of upper bounds on the quenched free energy of a random spin system considered by Goulart Rosa recently is rederived by an alternative method, and a similar sequence is constructed for the annealed free energy. The dependence of the free energy on interactions between spins is discussed.     

Strong decay to equilibrium in one-dimensional random spin systems

We consider a spin system on a lattice with finite-range, possibly unbounded random interactions. We show that for such systems the Glauber dynamics cannot decay to equilibrium exponentially fast inL ₂ even at high temperatures. Additionally, for one-dimensional systems with unbounded random couplings we prove that with probability one the corresponding Glauber dynamics has a fast (subexponential) decay to equilibrium in the uniform norm, provided that the distribution of random couplings satisfies some exponential bound.     

Local realistic photon model compatible with Malus' law for experiments testing Bell's inequalities

A local realistic model is proposed that agrees approximately with quantum mechanics in the predictions about experiments testing Bell's inequalities by measuring polarization correlations of photon pairs emitted in atomic cascades. No free parameters exist, except polarizer efficiencies, which may be measured independently using Malu's law.     

Dynamics of dilute spin systems in random fluctuating magnetic fields

The problem of stochastic evolution of dilute spins in a randomly fluctuating environment is considered within the frameworks of the Schrödinger equation with Gaussian fluctuating magnetic fields. The evolution equations for the averaged correlators are derived and it is shown that the density matrix of a spin system coupled with the thermoequilibrium fluctuations of fields tends asymptotically with time to the thermodynamic density matrix. The general results are illustrated by examples of coupling with magnons, phonons, and relaxation in paramagnetics. The evolution of spins in artificial high-frequency stochastic fields (Simonius' effect) is also considered. The high-temperature limit and the limit of classical spins are considered separately and the applicability of the Bloch equations is discussed. Then the results are generalized to the single-ion anisotropy and movable spins. Finally, it is shown how the results can be generalized to multi-spin systems.     

Evaluation of entanglement measures in spin systems with the random phase approximation

We discuss a general formalism based on the mean field plus random phase approximation (RPA) for the evaluation of entanglement measures in the ground state of spin systems. The method provides a tractable scheme for determining the entanglement entropy as well as the negativity of finite subsystems, which becomes analytic in the case of systems with translational invariance, in one or D dimensions. The approach improves as the spin increases, and also as the interaction range or connectivity increases. Illustrative results for different types of entanglement entropies (single site, block and comb) in the ground state of a small spin lattice with ferromagnetic type XY couplings in a transverse field are shown and compared with the exact numerical result. Effects arising from symmetry breaking at the mean field level are also discussed.