基于散射量子行走的完全图上结构异常搜索算法

完全图K_N 上某个顶点连接到图G将破坏其对称性. 为加速定位这类结构异常, 基于散射量子行走模型设计搜索算法, 首先给出了算法酉算子的定义, 在此基础上利用完全图的对称性, 将算法的搜索空间限定为一个低维的坍缩图空间. 以G为一个顶点的情况为例, 利用硬币量子行走模型上的研究结论简化了坍缩图空间中酉算子的计算, 并借助矩阵扰动理论分析算法演化过程. 针对星图S_N 上结构异常的研究表明, 以星图中心节点为界将整个图分为左右两个部分, 当且仅当两部分在N→∞时具有相同的特征值, 搜索算法可以获得量子加速. 本文说明星图上的分析方法和结论可以推广至完全图的坍缩图上. 基于此, 本文证明无论完全图连接的图G结构如何, 搜索算法均可在O(√N) 时间内定位到目标顶点, 成功概率为1-O(1√N), 即量子行走搜索该类异常与经典搜索相比有二次加速.     

Uniqueness of one-dimensional continuum Gibbs states

We investigate one-dimensional continuum grandcanonical Gibbs states corresponding to finite range superstable many-body potentials. Absence of phase transitions in the sense of uniqueness of the tempered Gibbs state is proved for potentials without hard-core by first proving uniqueness of the Gibbs measures for related hard-core potentials and then taking an appropriate limit of those Gibbs measures.     

Transport Efficiency of Continuous-Time Quantum Walks on Graphs

Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.     

A characterization of Gibbs states of lattice boson systems

We consider lattice boson systems interacting via potentials which are superstable and regular. By using the Wiener integral formalism and the concept of conditional reduced density matrices we are able to give a characterization of Gibbs (equilibrium) states. It turns out that the space of Gibbs states is nonempty, convex, and also weak-compact if the interactions are of finite range. We give a brief discussion on the uniqueness of Gibbs states and the existence of phase transitions in our formalism.     

Dobrushin uniqueness techniques and the decay of correlations in continuum statistical mechanics

Uniqueness of Gibbs states and decay properties of averaged, two point correlation functions are proved for many-body potentials in continuum statistical mechanical models via Dobrushin uniqueness techniques.     

Canonical versus Grand-Canonical Free Energies and Phase Diagrams of a Bipolaronic Superconductor Model

We continue the discussion of a bipolaronic superconductor (resp. an anisotropic antiferromagnet in quasispin formulation) as formulated in a previous work, based on a quantum-statistical, microscopic mean-field model. The grand-canonical thermodynamic limit is compared with the canonical thermodynamic limit in terms of a net of perturbations, becoming singular in the infinite lattice limit. A generalized thermostatistical framework is elaborated which covers model potentials with infinite parts. The function of the limiting free energy density in selecting the (stable) phases with broken symmetry is graphically illustrated. The phase diagrams for the two types of ensembles are shown to differ in the region where both the gauge symmetry and the invariance under sublattice exchange are broken. In particular, the type of the phase transitions, the order of the critical points, and the shape of some phase boundaries are found to depend on the ensemble, which clarifies certain controversial topics for these models. The uniqueness of the limiting Gibbs states with free boundary conditions in all thermodynamic phase regions is proved, and their decomposition into pure phase states in terms of a symmetric measure is evaluated. The field operators of the condensed particles are determined in the representations over the limiting Gibbs states.     

Approximate energies and thermal properties of a position-dependent mass charged particle under external magnetic fields

We solve the Schr?dinger equation with a position-dependent mass(PDM) charged particle interacted via the superposition of the Morse-plus-Coulomb potentials and is under the influence of external magnetic and Aharonov–Bohm(AB) flux fields. The nonrelativistic bound state energies together with their wave functions are calculated for two spatially-dependent mass distribution functions. We also study the thermal quantities of such a system. Further, the canonical formalism is used to compute various thermodynamic variables for second choosing mass by using the Gibbs formalism. We give plots for energy states as a function of various physical parameters. The behavior of the internal energy, specific heat, and entropy as functions of temperature and mass density parameter in the inverse-square mass case for different values of magnetic field are shown.     

Convergence of grand canonical Gibbs measures

We prove existence of Gibbs states for a large class of continuum many-body potentials through a limit process of hard-core approximations to the potential. Dobrushin uniqueness techniques [1] for the decay of correlations are then extended to a very general class of continuum potentials.This research was partially supported by the Council on Research at Louisiana State University     

Conditional Expectations Relative to a Product State and the Corresponding Standard Potentials

For a lattice system with a finite number of Fermions and spins on each lattice point, conditional expectations relative to an even product state (such as Fermion Fock vacuum) are introduced and the corresponding standard potential for any given dynamics, or more generally for any given time derivative (at time 0) of strictly local operators, is defined, with the case of the tracial state previously treated as a special case. The standard potentials of a given time derivative relative to different product states are necessarily different but they are shown to give the same set of equilibrium states, where one can compare states satisfying the variational principle (for translation invariant states) or the local thermodynamical stability or the Gibbs condition, all in terms of the standard potential relative to different even product states.     

Causal Set Dynamics and Elementary Particles

A causal set is considered a finite, acyclic oriented graph with special restrictions: each vertex has two incident edges directed to this vertex and two incident edges directed from this vertex. This graph is called a causal graph. The vertex with incident edges is called an X-structure. Quantum measurements are discussed. A dynamics of the causal graph is a random sequence of elementary interactions of edges that is described by complex amplitudes. These amplitudes correspond to each pair of interacting edges. The edges are elementary particles. The mass of a particle is a probability of the interaction. An equation of particles is proposed. In a simple case this equation for X-structure is the Dirac's equation. The edges are fermions with the spin 1/2.     

Canonical and grand canonical Gibbs states for continuum systems

It is shown that for a large class of interactions any canonical Gibbs state satisfying a natural temperedness condition is a mixture of Gibbs states with appropriate activities, and vice versa. Some general results on Gibbs states and canonical Gibbs states are established. In particular, a differential characterization of Gibbs states is given.     

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs. It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type. For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1. We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. Received: 27 March 2000 / Accepted: 7 December 2000     

Discrete Time and Intrinsic Length in Quantum Mechanics

We consider the possibility that simultaneously time and intrinsic length can be regarded as discrete real parameters. We study the dynamics of the free particle. For both scattering and bound states there are configurations where the energy is bounded from above and from below even for positive wave-function solutions. For the case of continuous evolution we show that the wave equation with a linear scalar coupling describes an oscillator that has built-in hidden supersymmetry.     

Quantum search of many vertices on the joined complete graph

The quantum search on the graph is a very important topic. In this work, we develop a theoretic method on searching of single vertex on the graph [$Phys. Rev. Lett$. 114 110503 (2015)], and systematically study the search of many vertices on one low-connectivity graph, the joined complete graph. Our results reveal that, with the optimal jumping rate obtained from the theoretical method, we can find such target vertices at the time $O\left({\sqrt N } \right)$, where $N$ is the number of total vertices. Therefore, the search of many vertices on the joined complete graph possessing quantum advantage has been achieved.     

In one and two dimensions,every stationary measure for a stochastic Ising Model is a Gibbs state

It is shown that one and two dimensional (generalized) stochastic Ising models with finite range potentials have only Gibbs states as their stationary measures. This is true even if the stationary measure or the potential is not translation invariant. This extends previously known results which are restricted to translation invariant stationary measures and potentials. In particular if the potential has only one Gibbs state the stochastic Ising Model must be ergodic.Research supported in part by N.S.F. Grant MPS 74-18926Alfred P. Sloan Fellow     

On the Well-Posedness of the Spatially Homogeneous Boltzmann Equation with a Moderate Angular Singularity

We prove an inequality on the Kantorovich-Rubinstein distance–which can be seen as a particular case of a Wasserstein metric–between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity. Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the physically relevant cases of hard and moderately soft potentials. In the case of hard potentials, we relax the regularity assumption of [6], but we need stronger assumptions on the tail of the distribution (namely some exponential decay). We thus obtain the first uniqueness result for measure initial data. In the case of moderately soft potentials, we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy (for very moderately soft potentials), plus sometimes an additionnal moment condition. We thus improve significantly on all previous results, where weighted Sobolev spaces were involved.     

Entropy,limit theorems,and variational principles for disordered lattice systems

We study infinite volume limits and Gibbs states of disordered lattice systems with bounded and continuous potentials. Our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes. We find that many familiar results of invariant potentials, such as large deviation theorems, variational principles, and equivalence of ensembles, continue to hold for disordered models, with suitably modified statements.     

Equilibrium States for Interval Maps: Potentials with sup φ − inf φ < h top (f)

We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials φ with the ‘bounded range’ condition sup φ ? inf φ < h _top (f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of Perron-Frobenius operators. We demonstrate that this ‘bounded range’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context.     

A study of generalized second law of thermodynamics in magnetic universe in the light of non-linear electrodynamics

In this work, we have considered the magnetic universe in non-linear electrodynamics. The Einstein field equations for non-flat FRW model have been considered when the universe is filled with the matter and magnetic field only. We have discussed the validity of the generalized second law of thermodynamics of the magnetic universe bounded by Hubble, apparent, particle and event horizons using Gibbs? law and the first law of thermodynamics for interacting and non-interacting scenarios. It has been shown that the GSL is always satisfied for Hubble, apparent and particle horizons but for event horizon, the GSL is violated initially and satisfied at late stage of the universe.