Finite Number Of Coordinates Research Articles

Hubbard model on the Bethe lattice via variational uniform tree states: Metal-insulator transition and a Fermi liquid

We numerically solve the Hubbard model on the Bethe lattice with finite coordination number $z=3$, and determine its zero-temperature phase diagram. For this purpose, we introduce and develop the `variational uniform tree state' (VUTS) algorithm, a tensor network algorithm which generalizes the variational uniform matrix product state algorithm to tree tensor networks. Our results reveal an antiferromagnetic insulating phase and a paramagnetic metallic phase, separated by a first-order doping-driven metal-insulator transition. We show that the metallic state is a Fermi liquid with coherent quasiparticle excitations for all values of the interaction strength $U$, and we obtain the finite quasiparticle weight $Z$ from the single-particle occupation function of a generalized "momentum" variable. We find that $Z$ decreases with increasing $U$, ultimately saturating to a non-zero, doping-dependent value. Our work demonstrates that tensor-network calculations on tree lattices, and the VUTS algorithm in particular, are a platform for obtaining controlled results for phenomena absent in one dimension, such as Fermi liquids, while avoiding computational difficulties associated with tensor networks in two dimensions. We envision that future studies could observe non-Fermi liquids, interaction-driven metal-insulator transitions, and doped spin liquids using this platform.

An efficient estimator for Gibbs random fields

An efficient estimator for the expectation R f dP is constructed, where P is a Gibbs random field, and f is a local statistic, i. e. a functional depending on a finite number of coordinates. The estimator coincides with the empirical estimator under the conditions stated in Greenwood and Wefelmeyer [6], and covers the known special cases, namely the von Mises statistic for the i.i.d. underlying fields and the case of one-dimensional Markov chains.

A crossover random lattice fluid model for hydrocarbons and carbon dioxide

A random lattice fluid model with finite coordination number is a versatile molecular-based lattice fluid equation of state, but this model fails to reproduce the non-analytical, singular behavior of fluids in the critical region. In this work, a method of obtaining the classical critical properties is presented in the random lattice fluid model. This model is combined with the crossover theory to obtain the crossover random lattice fluid model and to calculate the thermodynamic properties of hydrocarbons and carbon dioxide. This crossover random lattice fluid model presents much better agreement with experimental data near to and far from the critical region than the classical random lattice fluid model.

An Order Model for Infinite Classical States

In 2002 Coecke and Martin (Research Report PRG-RR-02-07, Oxford University Computing Laboratory, 2002) created a model for the finite classical and quantum states in physics. This model is based on a type of ordered set which is standard in the study of information systems. It allows the information content of its elements to be compared and measured. Their work is extended to a model for the infinite classical states. These are the states which result when an observable is applied to a quantum system. When this extended order is restricted to a finite number of coordinates, the model of Coecke and Martin is obtained. The infinite model retains many desirable aspects of the finite model, such as pure states as maximal elements and expected behavior of thermodynamic entropy. But it looses some of the important domain theoretic aspects, such as having a least element and exactness. Shannon entropy is no longer defined over the entire model and both it and thermodynamic entropy cease to be a measurements in the sense of Martin.

A dynamical proof for the convergence of Gibbs measures at temperature zero

We give a dynamical proof of a result due to Brémont (2003 Nonlinearity 16 419–26). It concerns the problem of maximizing measures for some given observable ϕ: for a subshift of finite type, and when ϕ only depends on a finite number of coordinates, it was proved in Brémont (2003) that the unique equilibrium state associated with βϕ converges to some measure when β goes to +∞. This measure has maximal entropy among the maximizing measures for ϕ. We give here a dynamical proof of this result and we improve it. We prove that for any Hölder continuous function (not necessarily locally constant), f, the unique equilibrium state associated with f + βϕ converges to some measure with maximal f-pressure among the maximizing measures. Moreover we also identify the limit measure.

Fractal dimension of a random invariant set

In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.

Ergodicity and weak-mixing of homogeneous extensions of measure-preserving transformations with applications to Markov shifts

Given a homogeneous extensionS′ of a measure-preserving transformationT, we provide necessary and sufficient conditions for the ergodicity and weak-mixing ofS′ in terms of functional equations. We then apply our findings to the case whenT is a Markov shift and the associated skewing function ofS′ depends on a finite number of coordinates. In this case, we obtain a simplification to the appropriate functional equations.

QUANTUM DISCRETE GAUGE MODELS WITH BOSONIC AND FERMIONIC DEGREES OF FREEDOM

A general approach to quantizing discrete models (i.e. having a finite number of coordinates) with quadratic first-class constraints is presented in the framework of gauging linear canonical symmetries. Also, it is proposed how a natural superextension of matrix field theories (viz. orthosymplectic "zero-dimensional" matrix field theories) might emerge in this approach.

Inclusion of short-range order in a mean field theory for the disordered magnetic lattice gas

The disordered magnetic lattice gas (DMLG) as a unifying description of many simpler random spin systems has been investigated in an attempt to devise a mean field theory which goes beyond the infinitely-long-ranged model by incorporating short-range order (SRO). The authors have shown rigorously that the local thermodynamic properties of the DMLG on a Cayley tree of finite coordination number z are identical to the thermodynamic properties of the DMLG in a pair approximation obtained by using the method of the distribution function. Further, a modified pair approximation for the DMLG is presented which is exactly solvable. It is formulated for general random bond distribution functions, and is then examined for the special case of Gaussian distributions.

Some remarks on operator ranges

Let X be a Banach space and let X* be its conjugate space. A system (x. e, ~), 7 E F, xy ~ X, f7 ~ X*, F being some set, is said to be biorthogonal if fr(xfl) = 0 for ~ ~ fl and = 1 for 7 = fl, and fundamental if the closed linear hull [xr: 7 E F] = X. A fundamental biorthogonal system in which for each x E X, x ~ 0, there exists an index 7 E F such that f~,(x) ~ 0 is called a Markushevich basis (abbreviated: an M-basis). We say that a linear subspace Y CX is an operator range if there exist a Banach space Z and a bounded linear operator T: Z -, X such that TZ = Y. By et(F ) we shall denote the space of absolutely summable sequences x(7), 7 E F. The space el(FO), F 0 C F, will be identified with a subspace of el(F ). All the spaces are assumed to be real. THEOREM 1. Assume that there exists a continuous linear injective operator T: et(F ) --X with a dense range. Then X has two dense operator ranges, intersecting only at zero. Proof. Let Fi: i E J, be a partition of F into countable subsets: F = Ui~JFi. In each space el(Fi) there exist dense operator ranges Yi, Zi, Yi N Z i = 0 [1]. We denote by Y the subspace consisting of the absolutely convergent series ~iEJYi' Yi E Yi, and by Z the subspace consisting of the absolutely convergent series ZiEJZi, z i ~ Z i. Then TY and TZ will be the required operator ranges. COROLLARY 1. A Banach space X with a fundamental biorthogonal system (xe, fy, 7 E F) has two dense operator ranges, intersecting only at zero. Proof. Obviously, we can assume that Ix.rH = 1. Let e r be the unit vectors of the space el(F ). We set Te e = ~. The mapping T extends to a bounded linear operator from el(F ) into X. It is easy to see that it is injective. COROLLARY 2. In the space e| there exist two dense operator ranges, intersecting only at zero. Indeed, e~(F) has a fundamental biorthogonal system [2]. Theorem 1 and the corollaries are connected with the question from [3] regarding the existence in an arbitrary Banach space, in particular in e,o, of two operator ranges, intersecting only at zero. This question is related to the investigation of supportless convex sets. We recall the definitions. A subset A of a normed space is said to be supportless if no nonzero continuous linear functional attains its supremum and infimum on A. A normed space is said to be supportless if it contains a closed, bounded, convex, supportless set. We say that a Banach space is weakly compactly generated if it is the closed linear hull of a weakly compact subset. By the symbol c0(F ) we shall denote the space of those sequences x(7 ) such that for any e > 0 there exist only a finite number of coordinates exceeding e in absolute value, with the supremum norm. LEMMA 1. In the space et(F ) there exists a dense operator range which intersects the linear hull lin(e~,: 7 E F) of the unit vectors of el(F ) at zero. Proof. Let Fi: i E J, be a partition of F into countable subsets. In each el(Fi) there exists a dense operator range ~5 i, U i n fin(%: ), E Fi) = 0; this follows from the results of [1]. Then the subspace U, consisting of the absolutely convergent series ~iEjui, u i E Ui, is the required one. THEOREM 2. In each weakly compactly generated space X there exists a dense, supportless operator range. Proof. Let Y be a reflexive space and let R be a dense, injective operator from Y into X [4]. Let (Yr' ~)~,~r be an Mbasis in Y, lY~,I = 1 [5, p. 693] and let % be the standard unit vectors of the space el(F ). We set S(eT) = Yr The mapping S can be extended to a continuous linear injective operator from el(F ) into Y. According to Lemma 1, we select a dense operator range U C el(F ), U n l ine,re r = 0. Then the operator T = RS is weakly compact, iniective and, since S is a conjugate operator, A = T(Bu) (B U is the el-ball of the subspace U) is a closed subset of the normed space Z = (TU, I IX). We mention that the operator S* maps Y* into the space c0(F ) C eo.(F) = el(F)*. Therefore, if a is a support point of the set A with support

Dynamic stiffness for lateral buckling

The dynamic stiffness method can predict an infinite number of natural modes of a structural system by means of a finite number of coordinates. It has been successful for dynamic structural analysis under the influence of constant axial force. The influence of constant moment is considered here and the instability of the plane form of the cross-section of a thin beam member is of particular interest. It is found that the constant moment softens the flexural modes and hardens the torsional modes. As a result, the characteristic curve can be concave or convex.

Dynamic Stiffness Analysis of Follower Moments

Abstract: The dynamic stiffness method can predict an infinite number of natural modes of a conservative structure by means of a finite number of coordinates. The methos is extended to a nonconservative system characterized by follower moments to investigate the dynamic lateral buckling and flutter in this paper. Straight beam members with doubly symmetrical cross‐sections are of interest, and skeletal frames are taken as examples. Flexural and torsional modes are coupled. The fact that the applied moment softens the flexural modes, but hardens the torsional modes, makes the characteristic diagram much more complicated than that of follower forces. Isola loops are possible.

Short-distance behavior of renornalizable quantum field theories

We introduce particular coordinates to the infinite-dimensional theory space (consisting of all possible hamiltonians) of euclidean quantum field theory such that all but a finite number of coordinates vanish for renormalizable theories. With this choice we can analyze the structure of the renormalization group equations by counting scaling dimensions of the coordinates. Following Wegner's idea we combine the renormalization group with the operator product expansion to determine all possible non-analytic dependence on the renormalized coupling constants in the short-distance expansions of the n-point correlation functions.

Three characterizations of polyhedral Banach spaces

Three new criteria for the polyhedrality of a Banach space are given in terms of norms, locally dependent on a finite number of coordinates, tilings of the space, and coverings of the sphere by a sequence of cuts.

A simplified proof of the representation of functionals of diffusions

A simple proof is given for the stochastic integral representation of a Frechet differentiable functional of the paths of a given diffusion process. The proof begins when the functional depends on one coordinate, then passes to (by appropriate conditioning) a perhaps more difficult case—when the functional depends on a finite number of coordinates—and finally, by approximation, to the general case.

Q-partitioning of graphs with finite coordination number

The NP-complete optimisation problem of q-partitioning of graphs with finite connectivity is discussed. Using the effective local field method, the authors obtain the local field distributions with and without the continuous part and use them to calculate the ground-state energies of the three-state Potts spin glass on lattices with finite coordination number equal to three. This in turn gives analytic results for the optimal cost of the 3-partitioning of graphs. They perform simulations of actual 3-partitions of random graphs to compare with their theoretical results and obtain very good agreement. Ways for further improvement of the estimates are discussed.

A network model for deep bed filtration processes

AbstractEquations for the deep bed filtration problem are obtained as a special case of a general model formulated earlier. The filtration coefficient λ is expressed as an explicit function of system parameters such as fluid flow rate, pore density, and pore size distribution. For a unimodal pore size distribution it is found that λ remains constant both in space and time. In general, however, λ is shown to decrease. Explicit solutions to the problem, including particle density profiles and permeability, are obtained for the two cases of large values of the coordination number Z (capillary tube model), and for a bimodal pore size distribution with a finite coordination number.

Mean-field theory of spin-glasses with finite coordination number.

The mean-field theory of dilute spin-glasses is studied in the limit where the average coordination number is finite. The zero-temperature phase diagram is calculated and the relationship between the spin-glass phase and the percolation transition is discussed. The present formalism is applicable also to graph optimization problems.

Probabilistic Description of the Response of Linear Structures to Random Inputs

This paper considers a probabilistic description of the random response of linear structures which is described by a finite number of coordinates. Such a description differs from those available in the literature, where random response is usually either assumed Gaussian or is described on the basis of correlation and spectral theories of random functions; it is based on the response probability density distribution freed from the previous assumptions. The development of the response density function is the main task of this work. Given the characteristics of a structure and the density distribution of a random input, stationary or nonstationary, a response density distribution is derived.