arXiv subjects
Publications and source records attributed to H. G. Miller.
Good approximate eigenstates of a Hamiltionian operator which poesses a point as well as a continuous spectrum have beeen obtained using the Lanczos algorithm. Iterating with the bare Hamiltonian operator yields spurious solutions which can easily be identified. The rms radius of the ground state eigenvector, for example, is calculated using the bare operator.
The Maximum Entropy (MaxEnt) technique is applied to the derivation of the Gaussian Dispersion Plume Model as well as to more complex transport phenomena such as the one-dimensional advection equation, the one-dimensional diffusion equation, the one dimensional advection-diffusion equation, and finally to the multi-dimensional advection-diffusion equation. Further application is discussed.
Distribution functions of many static transport equations are found using the Maximum Entropy Principle. The equations of constraint which contain the relevant dynamical information are simply the low-lying moments of the distributions. Systems subject to conservative forces have also been considered.
An algorithm for finding the bound-state eigenvalues and eigenfunctions of a Hermitian Hamiltonian operator using Green's method, developed by Waxman\cite{W98},has been extended to include non-Hermitian Hamiltonian operators.
The nonextensive entropic measure proposed by Tsallis introduces a parameter, q, which is not defined but rather must be determined. The value of q is typically determined from a piece of data and then fixed over the range of interest. On the other hand, from a phenomenological viewpoint, there are instances in which q cannot be treated as a constant. We present two distinct approaches for determining q depending on the form of the equations of constraint for the particular system. In the first case the equations of constraint for an operator O can be written as $Tr[F^{q}O]=C$, where C may be an explicit function of the distribution function, F. In this case one can solve an equivalent MAXENT problem which yields q as a function of the corresponding Lagrange Multiplier. As an illustration the exact solutions to the static Generalized Fokker-Planck Equation (GFP) are obtained from MAXENT. As in the case where C is a constant if q is treated as a variable within the MAXENT framework, the entropic measure is maximized for all values of q trivially. Therefore q must be determined from existing data. In the second case an additional equation of constraint exists which cannot be brought into the above form. In this case the additional equation of constraint may be used to determine the fixed value of q.
The exact solution of a particular form of the stationary state generalized Fokker-Planck equations, which is given under certain conditions by the classical Tsallis distribution, is compared with the solution of the MAXENT equations obtained using the classical Tsallis entropy. The solutions only agree provided the Tsallis parameter, q, is no longer taken to be constant.
The central role of the Rayleigh quotient in many body physics is discussed. Various many body methods can be obtained from either an attempt to evaluate the Rayleigh Quotient directly or through various variational approximations. Rather than dwell on the technical details necessary to obtain the equations of the various many body methods, we concentrate on how they can be obtained from the Rayleigh Quotient, and some of the consequences of the approximations involved in their evaluation.
The $q$-deformed statistics for fermions arising within the non-extensive thermostatistical formalism has been applied to the study of various quantum many-body systems recently. The aim of the present note is to point out some subtle difficulties presented by this approach in connection with the problem of thermodynamic consistency. Different possible ways to apply the $q$-deformed quantum distributions in a thermodynamically consistent way are considered.
Analogous to a model that predicts the linear scaling of the binding energy of a nucleus from the number of nucleons, a simple model was developed to account for the observed linear variation of the quantum-chemically computed total electronic energy of the fully-optimized structures of a homologous series of polymers. This model was tested with both ab-initio DFT and molecular mechanics methods on the ortho-fused spiral-benzenes. Both methods predict linear scaling of total polymer energy with increasing number of repeating units added. Since this is also the case for the linear ortho-fused zigzag-benzenes and other polymers, it is postulated that the model is applicable to polymers in general. It may, therefore, be used to predict physical properties of long-chain polymers.
The generalized non-extensive statistics proposed by Tsallis have been successfully utilized in many systems where long range interactions are present. For high density quark matter an attractive long range interaction arising from single gluon exchange suggests the formation of a diquark condensate. We study the effects on this color superconducting phase for two quark flavors due to a change to Tsallis statistics. By numerically solving the gap equation we obtain a generalization of the universality condition, $\frac{2ϕ_{0}}{T_{C}}\approx 3.52$ and determine the temperature dependence of the gap. For the Tsallis parameter $q\approx 1$ the specific heat is exponential becoming more linear as q increases. This suggests that for larger values of q s-wave color superconductors behave like high $T_c$ superconductors rather than weak superconductors.
The genesis of spurious solutions in finite basis approximations to operators which possess a continuum and a point spectrum is discussed and a simple solution for identifying these solutions is suggested.
Covergent eigensolutions of the Dirac Equation for a relativistic electron in an external Coulomb potential are obtained using the Lanczos Algorithm. A tri-diagonal matrix representation of the Dirac Hamiltonian operator is constructed iteratively and diagonalized after each iteration step to form a sequence of convergent eigenvalue solutions. Any spurious solutions which arise from the presence of continuum states can easily be identified.
Relationships between the coupling constant and the binding energy of threshold bound states are obtained in a simple manner from an iterative algorithm for solving the eigenvalue problem. The absence of threshold bound states in higher dimensions can be easily understood.
The iterative algorithm recently proposed by Waxman for solving eigenvalue problems, which relies on the method of moments, has been modified to improve its convergence considerably without sacrificing its benefits or elegance. The suggested modification is based on methods to calculate low-lying eigenpairs of large bounded hermitian operators or matrices.
In the iterative algorithm recently proposed by Waxman for solving eigenvalue problems, we point out that the convergence rate may be improved. For many non-singular symmetric potentials which vanish asymptotically, a simple analytical relationship between the coupling constant of the potential and the ground state eigenvalue is obtained which can be used to make the algorithm more efficient.
Analytical expressions for Bose-Einstein condensation of an ideal Bose gas analyzed within the strictures of non-extensive, generalized thermostatistics are here obtained.
We discuss a remarkable property of an iterative algorithm for eigenvalue problems recently advanced by Waxman that constitutes a clear advantage over other iterative procedures. In quantum mechanics, as well as in other fields, it is often necessary to deal with operators exhibiting both a continuum and a discrete spectrum. For this kind of operators, the problem of identifying spurious eigenpairs which appear in iterative algorithms like the Lanczos algorithm does not occur in the algorithm proposed by Waxman.
Given the spectrum of a Hamiltonian, a methodology is developed which employs the Landau-Ginsburg method for characterizing phase transitions in infinite systems to identify phase transition remnants in finite fermion systems. As a first application of our appproach we discuss pairing in finite nuclei.