Wigner-Type Theorems for Projections

The Wigner theorem, in its Uhlhorn’s formulation, states that a bijective transformation of the set of all one-dimensional linear subspaces of a complex Hilbert space which preserves orthogonality is induced by either a unitary or an antiunitary operator. There exist in the literature many Wigner-type theorems and the purpose of this paper is to prove in an algebraic setting a very general Wigner-type theorem for projections (idempotent linear mappings). As corollaries, Wigner-type theorems for projections in real locally convex spaces, infinite dimensional complex normed spaces and Hilbert spaces are obtained.     

The Orthomodular Poset of Projections of A Symmetric Lattices

In a Hilbert space, there exists a natural correspondence between continuous projections and particular pairs of closed subspaces. In this paper, we generalize this situation and associate to a symmetric lattice L a subset P(L) of L× L, called its projection poset. If L is the lattice of closed subspaces of a topological vector space then elements of P(L) correspond to continuous projections and we prove that automorphisms of P(L) are determined by automorphisms of the lattice L when this lattice satisfies some basic properties of lattices of closed subspaces. Primary: 06C15, Secondary: 03G12 81P10.     

Automorphisms of An Orthomodular Poset of Projections

By using a lattice characterization of continuous projections defined on a topological vector space E arising from a dual pair, we determine the automorphism group of their orthomodular poset Proj(E) by means of automorphisms and anti-automorphisms of the lattice L of all closed subspaces of E. A connection between the automorphism group of the ring of all continuous linear mappings defined on E and the automorphism group of the orthoposet Proj(E) is established.     

Quantum Implication Algebras

Quantum implication algebras without complementation are formulated with the same axioms for all five quantum implications. Previous formulations of orthoimplication, orthomodular implication, and quasi-implication algebras are analyzed and put in perspective to each other and our results.     

Automorphism Groups of Orthomodular Lattices Obtained from Quadratic Spaces

We study the automorphism group of some orthomodular lattices, obtained from a quadratic space over a field K. We show how this group is linked to the semi-orthogonal group and with the group of all similarity transformations of the quadratic space. When the field K is finite, the cardinality of the automorphism group is given. AMS subject classification (1991): 06C15, 15A63, 20D45.     

Central Limit Theorems for Nonlinear Hierarchical Sequences of Random Variables

Let a random variable x ₀ and a function f:[a, b] ^k [a, b] be given. A hierarchical sequence {x _n :n=0, 1, 2,...} of random variables is defined inductively by the relation x _n =f(x _{n–1, 1}, x _{n–1, 2}..., x _{n–1, k} ), where {x _{n–1, i} :i=1, 2,..., k} is a family of independent random variables with the same distribution as x _n–1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.     

On the Most Compact Regular Lattices in Large Dimensions: A Statistical Mechanical Approach

In this paper I will consider the computation of the maximum density of regular lattices in large dimensions using an approach based on statistical mechanics. The starting point will be some theorems of Rogers, which are virtually unknown in the community of physicists. Using his approach one can find many similarities (and differences) with the problem of computing the entropy of a system of hard spheres. The relation between the two problems is investigated in detail. Some conjectures are presented: further investigation is needed in order to check their consistency.     

Lattice dynamics of impurity clusters. Application to pairs

A general solution is obtained for the lattice dynamics of a cluster ofn-impurity atoms using the double-time Green’s function formalism. The cluster is characterized byn-mass defect andm-force constant change parameters. It is shown that this general solution for the Green’s function for then-impurity cluster can also be expressed in terms of the Green’s function for the (n−1)-impurity cluster. As an application, the cluster impurity modes for a pair are calculated using the Debye model for the host lattice dynamics. The splitting of the high frequency local modes and nearly zero frequency resonant modes due to pairs show an oscillatory behaviour on varying the distance of separation between the two impurity atoms. These oscillations are most prominent for two similar impurities and get damped for two dissimilar impurities or if one of the impurities produces a force constant change. The predictions of the calculation provide qualitative explanation of the data obtained from the infrared measurements of the resonant modes in mixed crystal system of KBr_1−c Cl _c : Li⁺ and KBr_1−c I _c : Li⁺.     

Multiple-Relaxation-Time Lattice Boltzmann Approach to Richtmyer-Meshkov Instability

The aims of the present paper are twofold. At first, we further study the Multiple-Relaxation-Time (MRT) Lattice Boltzmann (LB) model proposed in [Europhys. Lett. 90 (2010) 54003]. We discuss the reason why the Gram-Schmidt orthogonalization procedure is not needed in the construction of transformation matrix M; point out a reason why the Kataoka-Tsutahara model [Phys. Rev. E 69 (2004) 035701(R)] is only valid in subsonic flows.The von Neumann stability analysis is performed. Secondly, we carry out a preliminary quantitative study on the Richtmyer-Meshkov instability using the proposed MRT LB model. When a shock wave travels from a light medium to a heavy one, the simulated growth rate is in qualitative agreement with the perturbation model by Zhang-Sohn. It is about half of the predicted value by the impulsive model and is closer to the experimental result. When the shock wave travels from a heavy medium to a light one, our simulation results are also consistent with physical analysis.     

Polymer Gas Approach to N-Body Lattice Systems

We give a simple proof, based only on combinatorial arguments, of the Kotecký–Preiss condition for the convergence of the cluster expansion. Then we consider spin systems with long-range N-body interactions. We prove directly, using the polymer gas representation, that the pressure may be written in terms of an absolutely convergent series uniformly in the volume when the interaction is summable in a suitable sense. We also give an estimate of this radius of convergence. In order to get the proof we use a method introduced by Cassandro and Olivieri in the early 1980s. We apply this method to various concrete examples.     

Entropy Function Approach to the Lattice Boltzmann Method

In this paper, we present the construction of the Lattice Boltzmann method equipped with the H-theorem. Based on entropy functions whose local equilibria are suitable to recover the Navier–Stokes equations in the framework of the Lattice Boltzmann method, we derive a collision integral which enables simple identification of transport coefficients, and which circumvents construction of the equilibrium. We discuss performance of this approach as compared to the standard realizations.     

Leading Branches of the Transfer-Matrix Spectrum for Lattice Spin Systems (Quasi-Particles of Different Species)

For lattice systems under high temperatures T with compact or finite spin we construct three invariant subspaces of the transfer-matrix, which can be interpreted as the spaces of states for quasi-particles of two different species and the space of states for two particles of the first species. We formulate a condition on a priori distribution guaranteeing that the spectrum of the transfer-matrix on these subspaces are not overlapping.     

Lattice bosons in a quasi-disordered environment

In this paper, we study non-interacting bosons in a quasi-disordered one-dimensional optical lattice in a harmonic potential. We consider the case of deterministic quasi-disorder produced by an Aubry–André potential. Using exact diagonalization, we investigate both the zero temperature and the finite temperature properties. We investigate the localization properties by using an entanglement measure. We find that the extreme sensitivity of the localization properties to the number of lattice sites in finite size closed chains disappear in open chains. This feature continues to be present in the presence of a harmonic confining potential. The quasi-disorder is found to strongly reduce the Bose–Einstein condensation temperature and the condensate fraction in open chains. The low temperature thermal depletion rate of the condensate fraction increases considerably with increasing quasi-disorder strength. We also find that the critical quasi-disorder strength required for localization increases with increasing strength of the harmonic potential. Further, we find that the low temperature condensate fraction undergoes a sharp drop to 0.5 in the localization transition region. The temperature dependence of the specific heat is found to be only marginally affected by the quasi-disorder.     

On the Model Pseudopotential Approach to Calculate the Lattice Mechanical Properties of Thorium

The applicability of model pseudopotential approach to investigate lattice mechanical properties of Thorium, a typical f-shell metal, has been examined in light of some recent experimental and theoretical studies. It is found that presently available model potentials do not account for s-d-f hybridization adequately. The equation of state is greatly affected by s-d-f hybridization. It is also observed that a model potential giving reasonably good phonon dispersion curves may not reproduce equally good density of phonon states and hence the related lattice mechanical properties. However, it is possible to have a model potential, which may give fairly good estimate of lattice dynamical properties.     

Multistable Spatiotemporal Dynamics in the Driven Frenkel-Kontorova Lattice

Spatiotemporal dynamics of the damped dc-driven Frenkel-Kontorova lattice is studied. Multistable topologies are shown. Intermittency of the dynamical contraction factor is found, and this behavior is a consequence of the collisions of kinks and antikinks. Fast kinks and antikinks are unstable. The transition from the localized kink to the whirling mode is found to be a temporal bifurcation cascade of generations of kink-antikink pairs and the collision-induced avalanche dynamics. Noise-induced topology transition is observed and discussed.``     

Multistable Spatiotemporal Dynamics in the Driven Frenkel-Kontorova Lattice

Spatiotemporal dynamics of the damped dc-driven Frenkel-Kontorova lattice is studied. Multistable topologies are shown. Intermittency of the dynamical contraction factor is found, and this behavior is a consequence of the collisions of kinks and antikinks. Fast kinks and antikinks are unstable. The transition from the localized kink to the whirling mode is found to be a temporal bifurcation cascade of generations of kink-antikink pairs and the collision-induced avalanche dynamics. Noise-induced topology transition is observed and discussed.     

Preliminary Results in the Use of Energy-Dependent Octagonal Lattices for Thermal Lattice Boltzmann Simulations

Thermal lattice Boltzmann simulations are prone to severe numerical instabilities. While octagonal velocity lattices increase the range of temperatures that can be successfully simulated, the ranges are insufficient for many applications. Second order interpolation is required to correlate diagonal streaming to the square spatial grid. Here, the role of energy-dependent octagonal lattices is examined, an idea spawned from Gauss–Hermite quadratures. A nontrivial allocation scheme is now required to ensure moment conservation in connecting to the spatial grid. For the energy-dependent lattices, it is shown that there are no lower bounds to the temperature, thus allowing for higher Reynolds number simulations. Simulations are presented and compared to theory (viscosity and sound speed dependence on temperature) showing excellent agreement.     

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h_s(φ,A) of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism φ, we prove a few results on that, define the entropy of a dynamical system h_s(Φ), and show its invariance. The concept of sufficient families is also given and we establish that h_s(Φ) comes out to be equal to the supremum of h_s(φ,A), where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,φ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B, s₀,φ), where B is a Boolean algebra and s₀ is a state on B.