The operator method for solving the fractional Fokker-Planck equation

The operator method has been used to solve the fractional Fokker-Planck equation (FPE) which recently formulated as a model for the anomalous transport process. Two classes of special interest of fractional F-P equations coming from plasma physics and charged particle transport problem has been considered. It is shown that the mean square-displacement 〈x²(t)〉 satisfy the universal power law characterized the anomalous time evolution i.e. .     

EPR study of Mn-implanted single crystal plates of TiO2 rutile

Single crystals of Mn-implanted TiO₂ rutile have been investigated by electron paramagnetic resonance (EPR) technique at room temperature. We have observed an EPR signal on Mn⁴⁺ ions (S=) in the manganese-implanted single crystal TiO₂ plates. Besides, weaker EPR signals due to Fe³⁺(S=, L=0) and Cr³⁺(S=) ions have also been observed. Characteristic six-line splitting of the manganese EPR lines due to hyper-fine interaction with ⁵⁵Mn nuclei (spin I=) has also been observed. Analysis of the EPR spectra shows that the manganese, iron and chromium ions substitute for Ti⁴⁺ ions in the TiO₂ rutile host. Two structurally equivalent groups of the centers have been observed in the EPR spectra in correspondence with two octahedral positions of the Ti ions in the rutile structure. Spin Hamiltonian parameters for the crystal field of orthorhombic symmetry on the Mn⁴⁺, Fe³⁺ and Cr³⁺ centers have been obtained as result of computer modelling.     

Pseudopotentials, correlations, and hierarchy states in quantum hall systems: When the composite Fermion picture works and why

A system of Fermions in a shell of angular momentum l can form a set of multiplets of total angular momentum L. The composite Fermion (CF) picture picks out the lowest lying energy multiplets by selecting from this set a subset that is “Laughlin correlated”, i.e. which maximally avoids pair orbits with the largest pair angular momentum L^′ (or smallest relative angular momentum R=2l−L^′). We demonstrate that Laughlin correlations occur only when the pseudopotential V(L^′) (the interaction energy of a pair as a function of L^′) increases with L^′ more rapidly, than the eigenvalue of L^′2 at the value of L^′ (or R) avoided in the Laughlin correlated state. This requirement is not satisfied for QEs and QHs of the Laughlin ν=1/3 and ν=1/5 states at R=1 and R=3 respectively. At and , clustering of QPs gives lower energy than Laughlin correlations. Novel spin polarized incompressible states at ν=4/11 and ν=4/13 cannot be explained as a second generation in the CF hierarchy.     

Nonperturbative equation of state of quark-gluon plasma: Applications

The vacuum-driven nonperturbative factors L_i for quark and gluon Green’s functions are shown to define the nonperturbative dynamics of QGP in the leading approximation. EoS obtained recently in the framework of this approach is compared in detail with known lattice data for μ = 0 including P/T⁴, ε/T⁴, . The basic role in the dynamics at T ? 3T_c is played by the factors L_i which are approximately equal to the modulus of Polyakov line for quark L_fund and gluon L_adj. The properties of L_i are derived from field correlators and compared to lattice data, in particular the Casimir scaling property follows in the Gaussian approximation valid for small vacuum correlation lengths. Resulting curves for P/T⁴, ε/T⁴, are in a reasonable agreement with lattice data, the remaining difference points out to an effective attraction among QGP constituents.     

Universality of 3D percolation exponents and first-order corrections to scaling for conductivity exponents

By using a fast, Nested Dissection algorithm we compare the results of finite-size scaling at p_c and of “p” scaling () on large cubic random resistor networks [up to 500×500×500]. The “p” scaling for conductivity of both site and bond networks leads to an exponent t=2.00(1). The finite-size scaling leads to the ratio of this conductivity exponent to the coherence length exponent ν: t/ν=2.283(3). Combining these results we estimate ν=0.876(6), in excellent agreement with a value proposed by Ballesteros et al. The first-order correctional exponent ω is found to be ω=1.0(2).     

The nature of the rectilinear diameter singularity

The rigorous explanation for the term |t|^2β in the rectilinear diameter equation is given (t=(T_c−T)/T_c, β is the critical exponent for the asymptotic form of the equation of state). The optimal order parameter, for which the branches of binodal are symmetric, is constructed within the canonical formalism. It is shown that the ratio of the amplitudes for the diameter singularity of the order parameter before |t|^1−α and |t|^2β, where α determines the behavior of the heat capacity and β is the critical exponent of the order parameter, takes the universal character modulo non-universal factor which depends on the thermodynamic class of the corresponding states. The analysis of entropy for argon and water leads to β=0.33 and the corresponding amplitude ratio .     

Semi-ab initio calculations of superposition model and crystal field parameters for Co ions using the exchange charge model

This paper attempts for the first time to establish a reliable linkage between the two well-known and independent models of crystal field (CF), namely the exchange charge (ECM) and superposition models (SM). Our approach aims to show that the SM parameters can be reliably extracted from the distance dependence of the CF invariants for Co²⁺ as derived from the ECM through some semi-ab initio calculations which involved a single fitting parameter and a set of newly constructed procedures. Complete sets of the numerical values of SM parameters and t_k for Co²⁺ in its own host lattices of Li₂Co₃(SeO₃)₄, CoSO₄·H₂O, CoSeO₄·H₂O, and Co(OH)₂ are obtained and they are found to be around 13,000-16,000 cm⁻¹ for , 4100-5700 cm⁻¹ for , 4.1-5.0 for t₂ and 6.2-6.5 for t₄. The present results generally agree with but should be much better than those incomplete sets of results found by previous researchers using the conventional fitting approach. Plausible explanations for some noticeable discrepancies are also discussed together with the effects of different CF contributions on values of the SM parameters.     

Mound morphology of the 2+1 -dimensional Wolf-Villain model caused by the step-edge diffusion effect

The mound morphology of the 2+1-dimensional Wolf-Villain model is studied by numerical simulation. The diffusion rule of this model has an intrinsic mechanism, i.e., the step-edge diffusion, to create a local uphill particle current, which leads to the formation of the mound. In the simulation, a noise reduction technique is employed to enhance the local uphill particle current. Our results for the dynamic exponent 1/z and the roughness exponent α obtained from the surface width show a dependence on the strength of the step-edge diffusion. On the other hand, λ(t), which describes the separation of the mounds, grows as a function of time in a power-law form in the regime where the coalescence of mounds occurs, λ(t)∼tⁿ, with n≈0.23-0.25 for a wide range of the deposition conditions under the step-edge diffusion effect. For m=1, a noise reduction factor of unity, the behavior of λ(t) in the saturated regime is also simulated. We find that the evolution behavior of λ(t) in the whole process can be described by the standard Family-Vicsek scaling.     

Judd-Ofelt parameters of Er in transparent TeO2-based nanocrystallized glasses

Transparent Er³⁺-doped bulk nanocrystallized (size of nanocrystals: ∼40 nm) glasses of 15K₂O·15Nb₂O₅·70TeO₂·0.5Er₂O₃ and 10BaO·10Gd₂O₃·80TeO₂·0.5Er₂O₃ are prepared, and the Judd-Ofelt parameters, (t=2, 4, 6), of Er³⁺ are evaluated from optical absorption spectra. The change in the molar polarizability due to the nanocrystallization is small in both samples, but a clear decrease in the mean atomic volume due to the nanocrystallization, i.e. more close atom packing, is observed. In both systems, a large decrease is observed in the parameter due to the nanocrystallization, indicating that the degree of the site symmetry of Er³⁺ ions in nanocrystallized glasses is much higher than that in the precursor glasses. The decrease in the and parameters due to the crystallization is small, suggesting that the covalency of Er³⁺-O bonds in nanocrystals is not so different from that in the precursor glasses.     

Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels and noise term for the anharmonic coordinates . For zero temperature, i.e. for , we prove that the support of the Fourier transform of and of the time averaged velocity-velocity correlation functions of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency we find that the energy E_T transferred to the harmonic system up to time T is proportional to T^α. If equals one of the phonon frequencies ω_ν, it is α=2. We prove that there is a zero measure set L such that for in its full measure complement R?L, it is α=0, i.e. there is no energy dissipation. Under certain conditions L contains a subset L^′ such that for the dissipation rate is nonzero and may be subdissipative (0≤α<1) or superdissipative (1<α≤2), compared to ordinary dissipation (α=1). Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a Cantor set C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for , related to . For the small denominators do not lead to divergencies such that is a smooth and bounded function in t.     

Van Vleck paramagnetism of the trivalent Eu ions

Magnetic susceptibilities of Eu₂O₃, EuF₃ and EuBO₃ have been measured over the wide temperature range 5-650 K. The Van Vleck paramagnetism, with the ground state of ⁷F₀ (S=3, L=3), has been investigated comprehensively. The temperature independent paramagnetism emerges manifestly below approximately 100 K. The variation of the susceptibility with temperature for EuBO₃ is in satisfactory agreement with the coupling constant , where the spin-orbit interaction is λL·S for the Russell-Saunders coupling on the basis of Van Vleck theory with one parameter λ. The value of can fit the susceptibility data of EuF₃. The deviation from the theory arises in Eu₂O₃. This discrepancy originates mainly from the influence of the crystalline field. Susceptibility of Gd₂O₃, having the ground state of ⁸S_7/2 (S=7/2, L=0), is also presented as a magnetic standard compound in comparison with these results.     

The rule for a subdiffusive particle in an extremely diverse environment

We consider a subdiffusive continuous time random walker in an inhomogeneous environment. Each microscopic random time is drawn from a waiting time probability density function (WT-PDF) of the form: , 0<β?1. The parameter k is a random quantity also, and is drawn from a PDF, , 0?γ<1, for a cutoff parameter . We show that the effective WT-PDF, ψ(t), obtained by averaging φ(t;k) with p(k), exhibits a transition in the rule that governs the power of ψ(t). ψ(t) obeys, , and μ is given by two different formula. When, 1−γ>β, μ=β, but otherwise, μ=1−γ. The rule for the scaling of ψ(t) reflects the competition between two different mechanisms for subdiffusion: subdiffusion due to the heavily tailed φ(t;k) for individual jumps, and subdiffusion due to the collective effect of an environment made of many slow local regions. These two different mechanisms for subdiffusion are not additive, and compete each other. The reported transition is dimension independent, and disappears when the power β is also distributed, in the range, 0<β?1. Simulations exemplified the transition, and implications are discussed.     

4-benzyl pyridinium dihydrogenmonophosphate C6H5CH2C5H4NHH2PO4 : Single-crystal structure and its conductivity in polycrystalline form

The salt 4-benzyl pyridinium dihydrogenmonophosphate is monoclinic P2₁/c with the following unit cell dimensions: ; ; ; and β=97.328(11). Also, , D_x=1.403, , F(000)=560; ; and R=0.0495 and R_w=0.0964 for 3733 independent reflections. The structure consists of infinite parallel two-dimensional planes built of H₂PO₄⁻ anions and C₆H₅CH₂C₅H₄NH⁺ cations mutually connected by strong O-H ?O and N-H ?O hydrogen bonding. There are no contacts other than the normal Van der Waals interactions between the layers. The conductivity relaxation parameters associated with some H⁺ conduction have been determined from an analysis of the spectrum measured in a wide temperature range.     

Quantum mechanics of Klein-Gordon-type fields and quantum cosmology

With a view to address some of the basic problems of quantum cosmology, we formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (∂_t²+D)ψ(t)=0, where and D is a positive-definite operator acting in a Hilbert space . In particular, we determine all the positive-definite inner products on the space of the solutions of such an equation and establish their physical equivalence. This specifies the Hilbert space structure of uniquely. We use a simple realization of the latter to construct the observables of the theory explicitly. The field equation does not fix the choice of a Hamiltonian operator unless it is supplemented by an underlying classical system and a quantization scheme supported by a correspondence principle. In general, there are infinitely many choices for the Hamiltonian each leading to a different notion of time-evolution in . Among these is a particular choice that generates t-translations in and identifies t with time whenever D is t-independent. For a t-dependent D, we show that regardless of the choice of the inner product the t-translations do not correspond to unitary evolutions in , and t cannot be identified with time. We apply these ideas to develop a formulation of quantum cosmology based on the Wheeler-DeWitt equation for a Friedman-Robertson-Walker model coupled to a real scalar field with an arbitrary positive confining potential. In particular, we offer a complete solution of the Hilbert space problem, construct the observables, use a position-like observable to introduce the wave functions of the universe (which differ from the Wheeler-DeWitt fields), reformulate the corresponding quantum theory in terms of the latter, reduce the problem of the identification of time to the determination of a Hamiltonian operator acting in , show that the factor-ordering problem is irrelevant for the kinematics of the quantum theory, and propose a formulation of the dynamics. Our method is based on the central postulates of nonrelativistic quantum mechanics, especially the quest for a genuine probabilistic interpretation and a unitary Schrödinger time-evolution. It generalizes to arbitrary minisuperspace (spatially homogeneous) models and provides a way of unifying the two main approaches to the canonical quantum cosmology based on these models, namely quantization before and after imposing the Hamiltonian constraint.