Non-ergodicity of Two Particles Interacting via a Smooth Potential

We examine two point particles interacting via a smooth Lennard-Jones-type potential of finite range on a two-dimensional torus. We find situations under which this system contains a stable, elliptic periodic orbit and hence is not ergodic. This result is in contrast to the case of hard spheres interacting via inelastic collision, which are always ergodic for two particles, are conjectured to be ergodic for arbitrarily many particles, and can never contain elliptic periodic orbits.     

State Ensembles and Quantum Entropy

This paper considers quantum communication involving an ensemble of states. Apart from the von Neumann entropy, it considers other measures one of which may be useful in obtaining information about an unknown pure state and another that may be useful in quantum games. It is shown that under certain conditions in a two-party quantum game, the receiver of the states can increase the entropy by adding another pure state.     

Order Statistics and Ginibre's Ensembles

The moduli of the eigenvalues at the edge of Ginibre's complex and quaternion Gaussian random matrix ensembles are shown to respond to a limit theorem identical in nature to that for independent identically distributed sequences. This is a companion work to ref. 15 in which the limit law for the (scaled) spectral radius of these ensembles was identified.     

Non-equivalence between Heisenberg XXZ spin chain and Thirring model

The Bethe ansatz equations for the spin 1/2 Heisenberg XXZ spin chain are numerically solved, and the energy eigenvalues are determined for the antiferromagnetic case. We examine the relation between the XXZ spin chain and the massless Thirring model, and show that the spectrum of the XXZ spin chain has a gapless excitation while the regularized Thirring model calculated with the Bogoliubov transformation method has a finite gap. This finite gap spectrum is also confirmed by the Bethe ansatz solution of the massless Thirring model.Received: 28 October 2004, Published online: 21 January 2005PACS: 10.Kk, 03.70. + k, 11.30.-j, 11.30.Rd     

Gram Matrices of Mixed-State Ensembles

International Journal of Theoretical Physics - Gram matrices arise naturally in the consideration of pair-wise overlap of a family of vectors or quantum pure states, and play an important role in...     

Gibbs Ensembles of Nonintersecting Paths

We consider a family of determinantal random point processes on the two-dimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are non-translation-invariant. The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices.     

Gaussian-Random Ensembles of Pseudo-Hermitian Matrices

Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or well-known number sequences, may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enigmatic Riemann-zeros (1/2±it _n would rather correspond to some PT-symmetric (pseudo-Hermitian) Hamiltonian.     

Universality for Orthogonal and Symplectic Laguerre-Type Ensembles

We give a proof of the Universality Conjecture for orthogonal (β=1) and symplectic (β=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K _{n, β}, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β=2) Laguerre-type ensembles have been proved by the fourth author in Ref. 23. The varying weight case at the hard spectral edge was analyzed in Ref. 13 for β=2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in Refs. 7, 8 analogous results for Hermite-type ensembles. As in Refs. 7, 8 we use the version of the orthogonal polynomial method presented in Refs. 22, 25, to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from Ref. 23.     

Statistical Ensembles for Economic Networks

Economic networks share with other social networks the fundamental property of sparsity. It is well known that the maximum entropy techniques usually employed to estimate or simulate weighted networks produce unrealistic dense topologies. At the same time, strengths should not be neglected, since they are related to core economic variables like supply and demand. To overcome this limitation, the exponential Bosonic model has been previously extended in order to obtain ensembles where the average degree and strength sequences are simultaneously fixed (conditional geometric model). In this paper a new exponential model, which is the network equivalent of Boltzmann ideal systems, is introduced and then extended to the case of joint degree-strength constraints (conditional Poisson model). Finally, the fitness of these alternative models is tested against a number of networks. While the conditional geometric model generally provides a better goodness-of-fit in terms of log-likelihoods, the conditional Poisson model could nevertheless be preferred whenever it provides a higher similarity with original data. If we are interested instead only in topological properties, the simple Bernoulli model appears to be preferable to the correlated topologies of the two more complex models.     

Superbosonization of Invariant Random Matrix Ensembles

‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group U _n , O _n , or USp _n , giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner’s n-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type.     

Information and Distinguishability of Ensembles of Identical Quantum States

We consider a fixed quantum measurement performed over n identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon’s 12th theorem, we show that in the case of a single qubit, the number of distinguishable states is , where (α₁,α₂) is the angle interval from which the states are chosen. In the general case of an N-dimensional Hilbert space and an area Ω of the domain on the unit sphere from which the states are chosen, the number of distinguishable states is . The optimal distribution is uniform over the domain in Cartesian coordinates.     

Correlation Kernels for Discrete Symplectic and Orthogonal Ensembles

In [49] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlation kernels associated to the classical Meixner and Charlier orthogonal polynomials.     

Thermalization and Transport in Dense Ensembles of Triplet Magnetoexcitons

JETP Letters - In a dilute gas of triplet magnetoexcitons, complete thermalization does not occur because the energy and momentum cannot be conserved simultaneously. Relaxation to the lowest energy...     

Grand Canonical Ensembles of Sparse Networks and Bayesian Inference

Maximum entropy network ensembles have been very successful in modelling sparse network topologies and in solving challenging inference problems. However the sparse maximum entropy network models proposed so far have fixed number of nodes and are typically not exchangeable. Here we consider hierarchical models for exchangeable networks in the sparse limit, i.e., with the total number of links scaling linearly with the total number of nodes. The approach is grand canonical, i.e., the number of nodes of the network is not fixed a priori: it is finite but can be arbitrarily large. In this way the grand canonical network ensembles circumvent the difficulties in treating infinite sparse exchangeable networks which according to the Aldous-Hoover theorem must vanish. The approach can treat networks with given degree distribution or networks with given distribution of latent variables. When only a subgraph induced by a subset of nodes is known, this model allows a Bayesian estimation of the network size and the degree sequence (or the sequence of latent variables) of the entire network which can be used for network reconstruction.     

On the Reconstructability of Ensembles by Measurements

A formal division of ensembles into pure ensembles is interpreted in physical terms. If the division of an ensemble can be generated by measurement of all properties, then the ensemble is called reconstructable. It is investigated in which cases an ensemble is reconstructable. Furthermore the hypothesis of random phases is discussed.     

Controlled Coherent Quantum Tunneling of Atomic Ensembles

We investigate the coherent tunneling phenomenon of the laser-driven atomic ensembles confined in a well-separated double-well potential. By generalizing the Frohlich canonical transformation to adiabatically eliminate the light field variable, a BCS-like effective Hamiltonian is obtained to depict the residual interaction between the two atomic ensembles. The number of the tunneling collective low excitations and its relationship to the ratios gr/gl and Nr/Nl are given.     

Rhythmic Behavior Generated by Ensembles of Neurons

A method for analyzing the rhythmic behavior ofbiological neural networks is developed. Themathematical foundation, based on group theory and graphtheory, is explicitly constructed, and examples are given to clarify the method. An application ismade to the brainstem circuitry of the vestibularsystem. The physiological mechanisms involved ingenerating vestibular nystagmus are characterized, andpredictions are made about the phase relations ofidentified vestibular neurons with eye movements.Comparisons with other models of vestibular circuitryare discussed and suggestions are made for improvementsto previous models.     

Limit Shapes for Gibbs Ensembles of Partitions

We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation processes. We show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.