Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State

The present paper deals with the study of conditional entropy and its properties in a quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. First, we obtained a pseudo-metric on the family of all partitions of the couple (B,s), where B is a Boolean algebra and s is a state on B. This pseudo-metric turns out to be a metric (called the Rokhlin metric) by using a new notion of s-refinement and by identifying those partitions of (B,s) which are s-equivalent. The present theory has then been extended to the quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. Applying the theory of commutators and Bell inequalities, it is shown that the couple (L,s) can be equivalently replaced by a couple (B,s ₀), where B is a Boolean algebra and s ₀ is a state on B.     

An analytic relation for the thickness of accretion flows

We take the vertical distribution of the radial and azimuthal velocities into account in spherical coordinates, and find that the analytic relation c _s0/ν _K Θ=[(γ−1)/2γ]^1/2 is valid for both geometrically thin and thick accretion flows, where c _s0 is the sound speed on the equatorial plane, ν _K is the Keplerian velocity, Θ is the half-opening angle of the flow, and γ is the adiabatic index.

     

Moments and Distributions of Trajectories in Slow Random Monads

Given a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter, 0≤s≤1. We fix a point ⊙∈B and consider the sequence M ^t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis _n , Rec _n and Tra _n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis _n and Tra _n converge in law to geometric distributions and Rec _n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s=1 is trivial, so let 0≤s<1. For any natural number k there is a number such that the k-th moments of Vis _n , Rec _n and Tra _n do not exceed this number for all n. About Vis _n : for any natural k the k-th moment of Vis _n is an increasing function of n. So it has a limit when n→∞ and for all n it is less than this limit. About Rec _n : for any k the k-th moment of Rec _n tends to one when n tends to infinity. About Tra _n : for any k the k-th moment of Tra _n has a limit when n tends to infinity.     

Finsler spaces with Riemannian geodesics

In Finsler spaces the intervalds=F(x ⁱ ,dx ⁱ ) is an arbitrary function of the coordinatesx ⁱ and coordinate incrementsdx ⁱ withF homogeneous of degree one in thedx ⁱ . It is shown that for Riemannian spacesds _R ²=g _ij dx ⁱ dx ⁱ which admit a non trivial covariantly constant tensorH _i .(x ^k ) there is an associated Finsler space with the same geodesic structure. The subset of such Finsler spaces withH _i .(x ^k ) a vector or second rank decomposable tensor is determined.     

Clarifying the nature of the Johari-Goldstein β-relaxation and emphasising its fundamental importance

ABSTRACT

Since 1998 the primitive relaxation time τ ₀(T,P) of the Coupling Model (CM) and the Johari-Goldstein (JG) β-relaxation time τ_JG (T,P) are shown approximately equal in many glass-formers. The CM relation between τ ₀(T,P) and τ_α (T,P) at any T and P is exact. Additionally from the CM relation τ_α (T,P)/τ ₀(T,P) is exactly invariant to variations of T and P while τ_α (T,P) is kept constant, and τ ₀ is exactly a function of ρ ^γ/T like τ_α . However, since τ_JG (T,P) ≈ τ ₀(T,P), the exact invariance of τ_α (T,P)/τ ₀(T,P) leads to approximate invariance of τ_α (T,P)/τ_JG (T,P), and τ_JG is approximately a function of ρ ^γ/T. Notwithstanding, the CM prediction of the approximate relations between τ_β and τ_α were mistaken as exact relations by some researchers. In this paper, we remove this misunderstanding by demonstrating via simulations and experiments that the JG β-relaxation is comprised of processes with different length-scales and degrees of cooperativity, and the process is heterogeneous. The distribution of processes makes τ_JG (T,P) equivocal, because it is just a single relaxation time used to represent the different processes within the distribution, which may change on varying T and P, at constant τ_α (T,P). The problem is compounded if the β-relaxation is not resolved, and fitting procedure used to extract τ_JG (T,P) and τ_α (T,P). Despite the relations of τ_JG (T,P) to τ_α (T,P) are approximate, we show these properties of τ_JG (T,P) are truly remarkable, fundamental, general, and important.     

Eigenvalues of graphs with twofold symmetry

There are many cases in which the spectrum of a graph contains the complete spectrum of a smaller graph. The larger (composite) graph and the smaller (component) graph are said to be subspectral. It is shown here that whenever a composite graph G has a twofold symmetry operation which defines two equivalent sets of vertices r and s, it is possible to construct two subspectral components G ₊ and G _-, whose eigenvalues, taken jointly, comprise the full spectrum of G. The following rules are given for constructing the components. (1) Draw the r set of vertices and all the edges connecting the members of the set. Then examine in G the vertices through which r and s are connected (the so-called bridging vertices). (2) If a bridging vertex r ₁ is connected to its symmetry-equivalent partner s ₁, then r ₁ is weighted +1 in G ₊ and -1 in G _-. (3) If r ₁ is connected to a vertex s ₂ which is symmetry-equivalent to a second bridging vertex r ₂ in r, then the weight of the edge between r ₁ and r ₂ in G (+1 if they are connected, zero if they are not) is increased by one unit in G ₊ and decreased by one unit in G _-. The derivation of these rules is shown, and the relationship between the spectrum of G and the spectra of G ₊ and G _- is explained in terms of the symmetry properties of the adjacency matrix of G.     

空间网络上的随机游走

本文以一维均匀环为基础, 通过添加有限数量的长程连接构造出了一维有限能量约束下的空间网络, 环上任意节点i与j之间存在一条长程连接的概率满足p_ijα d_ij^-α (α≥ 0),其中d_ij为节点i与j之间的网格距离, 并且所有长程连接长度总和受到总能量∧=cN(c≥ 0)的约束, N为网络节点总数.通过研究该空间网络上的随机游走过程,存在最优幂指数α₀ 使得陷阱问题的平均首达时间最短.进一步研究发现,平均首达时间与网络规模N之间存在着幂律关系, 随着网络规模N和总能量∧的增加,最优幂指数α₀单调增加,并趋近最优值1.5.     

An exact formulation of the Blume-Emery-Griffiths model on a two-fold Cayley tree model

A two-fold Cayley tree graph with fully q-coordinated sites is constructed and the spin-1 Ising Blume-Emery-Griffiths model on the constructed graph is solved exactly using the exact recursion equations for the coordination number q = 3. The exact phase diagrams in (kT/J, K/J ) and (kT/J, D/J) planes are obtained for various values of constants D/J and K/J, respectively, and the tricritical behavior is found. It is observed that when the negative biquadratic exchange (K) and the positive crystal-field (D) interactions are large enough, the tricritical point disappears in the (kT/J, K/J) plane. On the other hand, the system always exhibits a tricritical behavior in the phase diagram of (kT/J, D/J) plane. Received 8 June 2001 and Received in final form 28 September 2001     

An image processing study of a reentrant discotic cholesteric – biaxial cholesteric phase transition

In this work, we study and characterize the cholesteric sequence of phases (Ch_Dr ? Ch_B ? Ch_D), where the first Ch_Dr is the reentrant cholesteric discotic phase, Ch_B is the cholesteric biaxial phase and the second Ch_D is the cholesteric discotic phase. This sequence of phases is studied through polarized light microscopy and image processing technique, where, for the first time, the domains and borders of these transitions are established and characterized. They are also investigated and optically characterized throughout their textures.     

Photophoresis of coarse aerosol particles with nonuniform thermal conductivity

The photophoresis of a coarse solid spherical aerosol particle in a one-component gas of nonuniform temperature is examined with consideration of the inertial effects in the hydrodynamic equations and the temperature jump in the Knudsen layer. The problem is solved in the spherical coordinates r, Θ, and ϕ. The photophoresis of a homogeneous particle is considered first. Then the results are generalized to an inhomogeneous particle. A particle whose thermal conductivity χ _i varies as a function of r is chosen as a model which describes a broad class of natural and artificially produced aerosol particles. It is shown that the error can be significant if the variable internal thermal conductivity χ _i =χ _i (r) of the particle is ignored and only the value of the thermal conductivity on its surface χ _i (a) is considered, on the assumption that the particle is homogeneous. It is also shown that a particle with a variable internal thermal conductivity χ _i =χ _i (r) and a density of heat sources within it q _i (r,Θ) can be regarded as a homogeneous particle with a thermal conductivity γχ _i (a) and a heat-source density m(r)q _i (r,Θ). Recurrence formulas for gand m(r) in the general case are presented. Analytical expressions for γ and m(r) are found for a model particle with pronounced inhomogeneity. Zh. Tekh. Fiz. 68, 1–6 (April 1998)     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

The Thermodynamic Evolution of the Cosmological Event Horizon

By manipulating the integral expression for the proper radius R _e of the cosmological event horizon (CEH) in a Friedmann-Robertson-Walker (FRW) universe we obtain an analytical expression for the change δR _e in response to a uniform fluctuation δρ in the average cosmic background density ρ. We stipulate that the fluctuation arises within a vanishing interval of proper time, during which the CEH is approximately stationary, and evolves subsequently such that δρ/ρ is constant. The respective variations 2πR _e δR _e and δE _e in the horizon entropy S _e and enclosed energy E _e should be therefore related through the cosmological Clausius relation. In that manner we find that the temperature T _e of the CEH at an arbitrary time in a flat FRW universe is E _e /S _e , which recovers asymptotically the usual static de Sitter temperature. Furthermore it is proven that during radiation-dominance and in late times the CEH conforms to the fully dynamical First Law T _e dS _e =PdV _e −dE _e , where V _e is the enclosed volume and P is the average cosmic pressure.     

多量子位Heisenberg XX链中的杂质纠缠

通过分析系统的杂质位与其余部分间的纠缠N1-A以及单个正常位与其余部分间的纠缠NL-A研究了匀强磁场作用下含杂质Heisenberg XX链的纠缠特性.研究表明三量子位时纠缠存在的临界温度依赖于杂质参数J1和匀强磁场B.研究发现,当量子位L为奇数时,纠缠N1-A随量子位的增加而增大,而L为偶数时则相反,并且量子位L为偶数时的纠缠大于量子位L为奇数时的纠缠;对NL-A,量子位L为奇数时,纠缠随杂质参数J1的变化与L=3类似,而L为偶数时纠缠随杂质参数|J1|的增加而增加.     

多量子位Heisenberg XX链中的杂质纠缠

通过分析系统的杂质位与其余部分间的纠缠N_1-A以及单个正常位与其余部分间的纠缠N_L-A研究了匀强磁场作用下含杂质Heisenberg XX链的纠缠特性.研究表明三量子位时纠缠存在的临界温度依赖于杂质参数J₁和匀强磁场B.研究发现,当量子位L为奇数时,纠缠N_1-A随量子位的增加而增大,而L为偶数时则相反,并且量子位L为偶数时的纠缠大于量子位L为奇数时的纠缠;对N_L-A, 量子位L为奇数时,纠缠随杂质参数J₁的变化与L=3类似,而L为偶数时纠缠随杂质参数|J₁|的增加而增加.     

Self-diffusion in liquid metals

A survey is made of all published data on self-diffusion in liquid metals and an examination is carried out on the temperature-dependence of D for studies covering a wide range of temperatures. Log D versus 1/T and D versus T represent the best data equally well, while D versus T ^1/2 and D versus T ² do not. The proper functional form is taken to be D=CT experimentally, and a simple free diffusion theory is developed from the standpoint of vibrational atom motion. An equation having no adjustable parameters results (D=k ² ? _D T/hk), in which κ is the vibrational force constant. Values calculated for D at the melting point are in good agreement with experimental values for potassium, sodium, copper and silver, but are below the measured values for lead, indium, zinc and tin.     

Zonal PANS: evaluation of different treatments of the RANS–LES interface

The partially Reynolds-averaged Navier–Stokes (PANS) model can be used to simulate turbulent flows either as RANS, large eddy simulation (LES) or DNS. Its main parameter is f_k whose physical meaning is the ratio of the modelled to the total turbulent kinetic energy. In RANS f_k = 1, in DNS f_k = 0 and in LES f_k takes values between 0 and 1. Three different ways of prescribing f_k are evaluated for decaying grid turbulence and fully developed channel flow: f_k = 0.4, f_k = k^3/2 _tot/? and, from its definition, f_k = k/k_tot where k_tot is the sum of the modelled, k, and resolved, k_res, turbulent kinetic energy. It is found that the f_k = 0.4 gives the best results. In Girimaji and Wallin, a method was proposed to include the effect of the gradient of f_k. This approach is used at RANS– LES interface in the present study. Four different interface models are evaluated in fully developed channel flow and embedded LES of channel flow: in both cases, PANS is used as a zonal model with f_k = 1 in the unsteady RANS (URANS) region and f_k = 0.4 in the LES region. In fully developed channel flow, the RANS– LES interface is parallel to the wall (horizontal) and in embedded LES, it is parallel to the inlet (vertical). The importance of the location of the horizontal interface in fully developed channel flow is also investigated. It is found that the location – and the choice of the treatment at the interface – may be critical at low Reynolds number or if the interface is placed too close to the wall. The reason is that the modelled turbulent shear stress at the interface is large and hence the relative strength of the resolved turbulence is small. In RANS, the turbulent viscosity – and consequently also the modelled Reynolds shear stress – is only weakly dependent on Reynolds number. It is found in the present work that it also applies in the URANS region.     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

Quantum Discord Behavior for Two Qubits Heisenberg XYZ Chain with Inhomogeneous Magnetic Field

We report the quantum correlation behavior (quantum discord (QD)) of a two-qubit anisotropic Heisenberg XYZ chain under an inhomogeneous magnetic field. It is shown that in the lower region of b QD can be enhanced evidently through increasing anisotropic parameter γ, while the effects of γ are disappear when b is strong enough. The role of J _z is nearly opposite to γ, there is a critical value of b _c , with b>b _c QD is improved with decreasing interaction J _z , while the value of QD is nearly invariable whatever changing J _z when b<b _c . In addition, one can get non-zero QD by properly tuning J _z and γ even at a higher temperature. When inhomogeneity is increased to b _c , the QD can exhibit a regrowth procession, and the regrowth value of QD can be larger than that of before dropping in the region of weak J _z . We also obtain the ground state QD properties. These investigations can imply us more control parameters on quantum correlation in solid state systems.     

Das Neutralgasprofil und die Elektronentemperatur in der Freifallsäule bei hohem Ionisationsgrad

The neutral gas flux from the wall into the column - connected with the ion flux to the wall - and the anisotropy of the velocity distribution of the neutral gas are included into the theory of the steady-state positive column. d/l_n and v_n0/v₀ occur as characteristic parameters. d denotes the radius of the column, l_n the mean free path of the neutral atoms for ionization, v_n0 the mean velocity of the neutral atoms re-emitted from the wall, v₀ = (2kT_e/M)^1/2, M the ion mass, T_e the electron temperature. On the axis the neutral gas density N_n is decreasing, if d/l_n is rising. At the wall N_n is increasing for v_n0 ? v₀, but it is almost constant or decreasing for v_n0 ? v₀ at the same time. In the plasma the total number of the atoms and the ions is taken as constant. In the case of small v_n0 the degree of ionization is high only for d ? l_n. However, it is already high for d ≈ l_n in the case of high v_n0. Therefore, the radial profiles of the neutral gas densities of different gases in a column can differ from each other. Almost full ionization can be reached near the axis. These results hold, too, if the initial velocity of the ions and a magnetic field are taken into account. d/l_n, the degree of ionization, and the electron temperature are given as functions of the electric power input and of the total number of the neutrals and the ions. The velocities of the re-emitted atoms depend on the accomodation coefficient of the ions recombining at the wall. With rising d/l_n an increasing number of the neutral atoms is re-emitted with superthermal velocities and the total number of the neutrals and the ions can be decreased. The anisotropy of the velocity distribution of the neutral atoms and the ions must be taken into account for the interpretations of spectroscopical measurements.     

Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D ± J Ising Spin Glass

The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and −1 bonds are studied for L × L square lattices with L ≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. When L is even, almost all domain walls have energy E _dw = 0 or 4. When L is odd, most domain walls have E _dw = 2. The probability distribution of the entropy, S _dw , is found to depend strongly on E _dw . When E _dw = 0, the probability distribution of |S _dw | is approximately exponential. The variance of this distribution is proportional to L, in agreement with the results of Saul and Kardar. For E _dw = k > 0 the distribution of S _dw is not symmetric about zero. In these cases the variance still appears to be linear in L, but the average of S _dw grows faster than L. This suggests a one-parameter scaling form for the L-dependence of the distributions of S _dw for k> 0. PACS: 75.10.Nr, 75.40.Mg, 75.60.Ch, 05.50.+q