Analytical properties of the anisotropic cubic Ising model

We combine an exact functional relation, the inversion relation, with conventional high-temperature expansions to explore the analytic properties of the anisotropic Ising model on both the square and simple cubic lattice. In particular, we investigate the nature of the singularities that occur in partially resummed expansions of the partition function and of the susceptibility.     

Quantum damped harmonic oscillator on non-commuting plane

In the present paper we study the quantum damped harmonic oscillator on non-commuting two-dimensional space. We calculate the time evolution operator and we find the exact propagator of the system. We investigate as well the thermodynamic properties of the system using the standard canonical density matrix. We find the statistical distribution function and the partition function. We calculate the specific heat for the limiting case of critical damping, where the frequencies of the system vanish. Finally we study the state of the system when the phase space of the second dimension becomes classical. We find that these systems have some singularities and zeros for low temperatures.     

多维耦合受迫量子谐振子的普遍解

运用广义线性量子变换的普遍理论求解多维耦合受迫量子谐振子，给出了系统演化算符的矩阵元、波函数、力学量期望值和配分函数的严格表达式. 关键词： 多维耦合受迫量子谐振子 演化算符矩阵元 配分函数     

虚时路径积分的广义四阶拆分方法及其谐振子分析

虚时路径积分方法是计算实际体系量子热力学性质的有效工具. 本文重点分析讨论了基于二阶和四阶拆分的虚时路径积分的有效性,给出谐振子体系配分函数统一的解析形式,进而得到其热力学性质和主要误差的表达式. 通过去除谐振子配分函数的主要误差,广义对称四阶拆分中的自由参数得以固定,以此计算实际体系的热力学量可以达到期望的三阶精度.     

Information and Entropy in Quantum Brownian Motion

We compare the thermodynamic entropy of a quantum Brownian oscillator derived from the partition function of the subsystem with the von Neumann entropy of its reduced density matrix. At low temperatures we find deviations between these two entropies which are due to the fact that the Brownian particle and its environment are entangled. We give an explanation for these findings and point out that these deviations become important in cases where statements about the information capacity of the subsystem are associated with thermodynamic properties, as it is the case for the Landauer principle.     

Construction of a natural partition of incomplete horseshoes

We present a method for constructing a partition of an incomplete horseshoe in a Poincare map. The partition is based only on the unstable manifolds of the outermost fixed points and eventually their limits. Consequently, this partition becomes natural from the point of view of asymptotic scattering observations. The symbolic dynamics derived from this partition coincides with the one derived from the hierarchical structure of the singularities of the scattering functions.     

Wu's Equations and Quasi-Hypergeometric Functions

We investigate analytic solutions to Wu's equations and their symmetry property, singularities and monodromy, relating them to quasi hypergeometric functions. We finally give the monodromy theorem for them in one, two and arbitrary dimensional cases (see Theorem 1, 2 and 3). Received: 22 October 1999 / Accepted: 28 June 2001     

Statistics of flexible rings with excluded volume

A grand partition function of a system of closed self-avoiding lines on a lattice is studied. The examples of such lines are closed linear singularities: dislocations, vortex lines, etc. The partition function is expressed in the form of an integral over fictitious fields so that the behaviour of the whole system and a single ring in the system may be qualitatively described. A length distribution of rings, a mean radius of a ring of length L in the system and an effective interaction of the points of a probe ring, arising from the interaction of the probe ring with other rings, have been derived. An impact of broken rings is discussed.     

Propagator of a Charged Particle with a Spin in Uniform Magnetic and Perpendicular Electric Fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and some special and limiting cases are outlined.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

Exact solution for a non-Markovian dissipative quantum dynamics

We provide the exact analytic solution of the stochastic Schr?dinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's function; in the case of a free particle and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.     

Bounds for effective parameters of heterogeneous media by analytic continuation

We give a mathematical formulation of a method for obtaining bounds on effective parameters developed by D. Bergman and G. W. Milton. This method, in contrast to others used before, does not rely on a variational principle, but exploits the properties of the effective parameter as an analytic function of the component parameters. The method is at present restricted to two-component media.     

Optical Bistability in V Medium by Squeezed Vacuum Input

In this paper we present the analytic expressions of partition lines in the parametric space of a piece-wise smooth mapping describing an electronic relaxation oscillator. The supercritical regions with permission of period-doubling bifurcation, prohibition of period-doubling bifurcation, and complete phase-locking are discussed. Among them the region, where period-doubling bifurcation is prohibited but chaos is permitted is reported for the first time to our knowledge. These kinds of phenomena and regions can be observed in a lot of similar systems.     

一个张弛振子的超临界子区域

解析导出了描述一种电子张弛振子的分段光滑圆映象在参数空间超临界区域内显示几种不同动力学行为的子区域分界线.这些子区域分别是：允许倍周期分岔发生的区域,禁止倍周期分岔发生的区域,以及完全锁相区域.类似的现象和子区域可以在许多分段光滑系统中观察到. 关键词：     

Harmonically Confined, Semiflexible Polymer in a Channel: Response to a Stretching Force and Spatial Distribution of the Endpoints

We consider an inextensible, semiflexible polymer or worm-like chain which is confined in the transverse direction by a parabolic potential and subject to a longitudinal force at the ends, so that the polymer is stretched out and backfolding is negligible. Simple analytic expressions for the partition function, valid in this regime, are obtained for chains of arbitrary length with a variety of boundary conditions at the ends. The spatial distribution of the end points is also analyzed.     

The driven oscillator in the photon-number probability representation of quantum mechanics

We study the evolution of the driven harmonic oscillator in the probability representation of quantum mechanics. We use the photon-number tomographic-probability-distribution function to describe the quantum states of the system. We give a general review of the photon-number tomographic framework, including a discussion on the connection with other representations of quantum mechanics. We find tomograms of coherent states as well as excited states of the harmonic oscillator in an explicit form. We discuss the time evolution of the photon-number tomograms and transforms of the propagators for different representations of quantum mechanics. We obtain the propagator for the photon-number tomographic-distribution function for the case of the driven oscillator in an explicit form.     

Effective boundary field theory for a Josephson junction chain with a weak link

We show that a finite Josephson junction (JJ) chain, ending with two bulk superconductors, and with a weak link at its center, may be regarded as a condensed matter realization of a two-boundary sine-Gordon model. Computing the partition function yields a remarkable analytic expression for the DC Josephson current as a function of the phase difference across the chain. We show that, in a suitable range of the chain parameters, there is a crossover of the DC Josephson current from a sinusoidal to a sawtooth behavior, which signals a transition from a regime where the boundary term is an irrelevant operator to a regime where it becomes relevant.     

Investigations on finite ideal quantum gases

Recursion formulae of the N-particle partition function, the occupation numbers and its fluctuations are given using the single-particle partition function. Exact results are presented for fermions and bosons in a common one-dimensional harmonic oscillator potential, for the three-dimensional harmonic oscillator approximations are tested. Applications to excited nuclei and Bose–Einstein condensation are discussed.     

Spectroscopy of a driven solid-state qubit coupled to a structured environment

We study the asymptotic dynamics of a driven spin-boson system where the environment is formed by a broadened localized mode. Upon exploiting an exact mapping, an equivalent formulation of the problem in terms of a quantum two-state system (qubit) coupled to a harmonic oscillator which is itself Ohmically damped, is found. We calculate the asymptotic population difference of the two states in two complementary parameter regimes. For weak damping and low temperature, a perturbative Floquet-Born-Markovian master equation for the qubit-oscillator system can be solved. We find multi-photon resonances corresponding to transitions in the coupled quantum system and calculate their line-shape analytically. In the complementary parameter regime of strong damping and/or high temperatures, non-perturbative real-time path integral techniques yield analytic results for the resonance line shape. In both regimes, we find very good agreement with exact results obtained from a numerical real-time path-integral approach. Finally, we show for the case of strong detuning between qubit and oscillator that the width of the n-photon resonance scales with the nth Bessel function of the driving strength in the weak-damping regime.     

Fractal dimensions and homeomorphic conjugacies

We investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes. When the homeomorphic conjugacy is not Lipschitz continuous, we discuss how its singularities can affect the whole set of generalized fractal dimensions. We give applications to homeomorphisms that conjugate critical circle maps with irrational (golden mean) winding numbers. We present numerical studies corroborating the theoretical predictions.