45 research outputs found
Математическое моделирование колебаний струны с подвижной опорой в вертикальной плоскости
У статті розглядаються вільні коливання струни, лівий кінець якої є нерухомим, а правий має можливість переміщуватись у вертикальній площині за деяким законом. Постановка цієї крайової задачі виникла через необхідність побудови адекватної математичної моделі коливань електричного дроту з урахуванням поздовжніх переміщень одного з його кінців. Рухлива права опора являє собою маятникову підвіску електричного дроту у вигляді гірлянди ізоляторів. Мета дослідження — вивести та обґрунтувати граничні умови, які мають місце під час коливань струни з урахуванням переміщень її правої опори, та розв’язати відповідну крайову задачу. Методика розв’язання крайової задачі полягає в тому, що розв’язок хвильового рівняння відшукується в такому вигляді, що осереднене за часом його
середньоквадратичне відхилення від виведеної крайової умови має бути мінімальним. Головний результат і висновок дослідження: зміною власної частоти основного тону коливань струни через переміщення її правої опори у вертикальній площині в діапазоні зміни тих параметрів системи, що розглядаються, можна знехтувати.We consider the natural oscillations of the string whose left end is fixed and the right one has the ability to move in a vertical plane by a defined law. We should address this issue to construct an adequate mathematical model of the electric wire taking into account longitudinal displacement of one of its ends. The right-hand mobile support is a commuting pendant with electric wire in the form of insulator strings. The purpose of research is to show and justify the boundary conditions occurring during vibration of a string considering the movements of its right pillar and solve the corresponding boundary problem. The technique for solving the boundary value problem lies in the state that the solution of the wave equation should be searched in such a form that its time-average standard deviation derived from the boundary condition should be minimal. Emphasized should be the obtained research results: change of fundamental frequency of basic tone of string vibrations caused by its right pillar movement in vertical plane within the range of parameters change of the considered system can be neglected.В статье рассматриваются собственные колебания струны, левый конец которой является неподвижным, а правый имеет возможность перемещаться в вертикальной плоскости по определенному закону. Постановка этой краевой задачи возникла из-за необходимости построения адекватной математической модели колебаний электрического провода с учетом продольных перемещений одного из его концов. Подвижная правая опора представляет собой маятниковую подвеску электрического провода в виде гирлянды изоляторов. Цель исследования — вывести и обосновать краевые условия, которые имеют место во время колебаний струны с учетом перемещений ее правой опоры, и решить соответствующую краевую задачу. Методика решения краевой задачи состоит в том, что решение волнового уравнения отыскивается в таком виде, что осредненное по времени его среднеквадратическое отклонение от выведенного краевого условия должно быть минимальным. Основной результат и вывод исследования: изменением собственной частоты основного тона колебаний струны из-за перемещения ее правой опоры в вертикальной плоскости в диапазоне изменения тех параметров системы, которые рассматриваются, можно пренебречь
Fluctuations in subsystems of the zero temperature XX chain: Emergence of an effective temperature
The zero-temperature XX chain is studied with emphasis on the properties of a
block of spins inside the chain. We investigate the quantum fluctuations
resulting from the entanglement of the block with the rest of the chain using
analytical as well as numerical (density matrix renormalization group) methods.
It is found that the rest of the chain acts as a thermal environment and an
effective temperature can be introduced to describe the fluctuations. We show
that the effective temperature description is robust in the sense that several
independent definitions (through fluctuation dissipation theorem, comparing
with a finite temperature system) yield the same functional form in the limit
of large block size (). The effective temperature can also be shown
to satisfy the basic requirements on how it changes when two bodies of equal or
unequal temperatures are brought into contact.Comment: 19 pages, 7 figure
Tree Tensor Network State with Variable Tensor Order: An Efficient Multireference Method for Strongly Correlated Systems
[Image: see text] We study the tree-tensor-network-state (TTNS) method with variable tensor orders for quantum chemistry. TTNS is a variational method to efficiently approximate complete active space (CAS) configuration interaction (CI) wave functions in a tensor product form. TTNS can be considered as a higher order generalization of the matrix product state (MPS) method. The MPS wave function is formulated as products of matrices in a multiparticle basis spanning a truncated Hilbert space of the original CAS-CI problem. These matrices belong to active orbitals organized in a one-dimensional array, while tensors in TTNS are defined upon a tree-like arrangement of the same orbitals. The tree-structure is advantageous since the distance between two arbitrary orbitals in the tree scales only logarithmically with the number of orbitals N, whereas the scaling is linear in the MPS array. It is found to be beneficial from the computational costs point of view to keep strongly correlated orbitals in close vicinity in both arrangements; therefore, the TTNS ansatz is better suited for multireference problems with numerous highly correlated orbitals. To exploit the advantages of TTNS a novel algorithm is designed to optimize the tree tensor network topology based on quantum information theory and entanglement. The superior performance of the TTNS method is illustrated on the ionic-neutral avoided crossing of LiF. It is also shown that the avoided crossing of LiF can be localized using only ground state properties, namely one-orbital entanglement
Entanglement Measures for Single- and Multi-Reference Correlation Effects
Electron correlation effects are essential for an accurate ab initio
description of molecules. A quantitative a priori knowledge of the single- or
multi-reference nature of electronic structures as well as of the dominant
contributions to the correlation energy can facilitate the decision regarding
the optimum quantum chemical method of choice. We propose concepts from quantum
information theory as orbital entanglement measures that allow us to evaluate
the single- and multi-reference character of any molecular structure in a given
orbital basis set. By studying these measures we can detect possible artifacts
of small active spaces.Comment: 14 pages, 4 figure
Magnetic and quadrupolar order in a one-dimensional ferromagnet with cubic crystal-field anisotropy
The zero temperature phase diagram of a one-dimensional S=2 Heisenberg
ferromagnet with single-ion cubic anisotropy is studied numerically using the
density-matrix renormalization group method. Evidence is found that although
the model does not involve quadrupolar couplings, there is a purely quadrupolar
phase for large values of the anisotropy. The phase transition between the
magnetic and quadrupolar phases is continuous and it seems to be characterized
by Ising critical exponents.Comment: 11 pages, 7 figures, REVTeX, accepted in Phys. Rev. B (scheduled on
June 99
Opening of the Haldane Gap in Anisotropic Two- and Four-Leg Spin Ladders
We study the opening of the Haldane gap in two-leg and four-leg anisotropic
spin ladders using bosonization and renormalization group methods, and we
determine the phase diagram as a function of the interchain coupling and the
relative anisotropy. It is found that the opening of the Haldane gap is
qualitatively different for the two cases considered. For the two-leg ladder
the Haldane gap opens for arbitrarily small interchain coupling, independent of
the relative anisotropy, and the Haldane phase exists in a large region of
parameter space. For the four-leg ladder the opening of the Haldane gap is
strongly dependent on both the interchain coupling as well as the relative
anisotropy, and the Haldane phase exists only in a narrow region about the
isotropic antiferromagnet.Comment: 15 pages, 6 Postscript figure
The density-matrix renormalization group
The density-matrix renormalization group (DMRG) is a numerical algorithm for
the efficient truncation of the Hilbert space of low-dimensional strongly
correlated quantum systems based on a rather general decimation prescription.
This algorithm has achieved unprecedented precision in the description of
one-dimensional quantum systems. It has therefore quickly acquired the status
of method of choice for numerical studies of one-dimensional quantum systems.
Its applications to the calculation of static, dynamic and thermodynamic
quantities in such systems are reviewed. The potential of DMRG applications in
the fields of two-dimensional quantum systems, quantum chemistry,
three-dimensional small grains, nuclear physics, equilibrium and
non-equilibrium statistical physics, and time-dependent phenomena is discussed.
This review also considers the theoretical foundations of the method, examining
its relationship to matrix-product states and the quantum information content
of the density matrices generated by DMRG.Comment: accepted by Rev. Mod. Phys. in July 2004; scheduled to appear in the
January 2005 issu
The phase diagram of magnetic ladders constructed from a composite-spin model
White's density matrix renormalization group ({DMRG}) method has been applied
to an composite-spin model, which can also be considered as a
two-leg ladder model. By appropriate choices of the coupling constants this
model allows not only to study how the gap is opened around the gapless
integrable models, but also to interpolate continuously between models with
different spin lengths. We have found indications for the existence of several
different massive phases.Comment: 30 pages, 8 Postscript figure
Entanglement entropy in aperiodic singlet phases
We study the average entanglement entropy of blocks of contiguous spins in
aperiodic XXZ chains which possess an aperiodic singlet phase at least in a
certain limit of the coupling ratios. In this phase, where the ground state
constructed by a real space renormalization group method, consists
(asymptotically) of independent singlet pairs, the average entanglement entropy
is found to be a piecewise linear function of the block size. The enveloping
curve of this function is growing logarithmically with the block size, with an
effective central charge in front of the logarithm which is characteristic for
the underlying aperiodic sequence. The aperiodic sequence producing the largest
effective central charge is identified, and the latter is found to exceed the
central charge of the corresponding homogeneous model. For marginal aperiodic
modulations, numerical investigations performed for the XX model show a
logarithmic dependence, as well, with an effective central charge varying
continuously with the coupling ratio.Comment: 18 pages, 9 figure