Passivity and Microlocal Spectrum Condition

In the setting of vector-valued quantum fields obeying a linear wave-equation in a globally hyperbolic, stationary spacetime, it is shown that the two-point functions of passive quantum states (mixtures of ground- or KMS-states) fulfill the microlocal spectrum condition (which in the case of the canonically quantized scalar field is equivalent to saying that the two-pnt function is of Hadamard form). The fields can be of bosonic or fermionic character. We also give an abstract version of this result by showing that passive states of a topological *-dynamical system have an asymptotic pair correlation spectrum of a specific type. Received: 9 February 2000 / Accepted: 7 June 2000     

Evolution on a smooth landscape

We study in detail a recently proposed simple discrete model for evolution on smooth landscapes. An asymptotic solution of this model for long times is constructed. We find that the dynamics of the population is governed by correlation functions that although being formally down by powers ofN (the population size), nonetheless control the evolution process after a very short transient. The long-time behavior can be found analytically since only one of these higher order correlators (the two-point function) is relevant. We compare and contrast the exact findings derived herein with a previously proposed phenomenological treatment employing mean-field theory supplemented with a cutoff at small population density. Finally, we relate our results to the recently studied case of mutation on a totally flat landscape.     

Probing anomalous relaxation by coherent multidimensional optical spectroscopy

We propose to study the origin of algebraic decay of two-point correlation functions observed in glasses, proteins, and quantum dots by their nonlinear response to sequences of ultrafast laser pulses. Power-law spectral singularities and temporal relaxation in two-dimensional correlation spectroscopy signals are predicted for a continuous time random walk model of stochastic spectral jumps in a two-level system with a power-law distribution of waiting times psi(t) approximately t;{-alpha-1}. Spectroscopic signatures of stationary ensembles for 1相似文献   

Multipartite entanglement signature of quantum phase transitions

We derive a general relation between the nonanalyticities of the ground state energy and those of a subclass of the multipartite generalized global entanglement (GGE) measure defined by de Oliveira et al. [Phys. Rev. A 73, 010305(R) (2006)] for many-particle systems. We show that GGE signals both a critical point location and the order of a quantum phase transition (QPT). We also show that GGE allows us to study the relation between multipartite entanglement and QPTs, suggesting that multipartite but not bipartite entanglement is favored at the critical point. Finally, using GGE we were able, at a second-order QPT, to define a diverging entanglement length (EL) in terms of the usual correlation length. We exemplify this with the XY spin-1/2 chain and show that the EL is half the correlation length.     

Finite-size scaling for correlations of quantum spin chains at criticality

We study the finite-size scaling behavior of two-point correlation functions of translationally invariant many-body systems at criticality. We propose an efficient method for calculating the two-point correlation functions in the thermodynamic limit from numerical data of finite systems. Our method is most effective when applied to a two-dimensional (classical) system which possesses a conformal invariance. By using this method with numerical data obtained from exact diagonalizations and Monte Carlo simulations, we study the spin-spin correlations of the quantum spin-1/2 and-3/2 antifierromagnetic chains. In particular, the logarithmic corrections to power-law decay of the correlation of the spin-1/2 isotropic Heisenberg antiferromagnetic chain are studied thoroughly. We clarify the cause of the discrepancy in previous calculations for the logarithmic corrections. Our result strongly supports the field-theoretic prediction based on the mappings to the Wess-Zumino-Witten nonlinear -model or the sine-Gordon model. We also treat logarithmic corrections and crossover phenomena in the spin-spin correlation of the spin-3/2 isotropic Heisenberg antiferromagnetic chain. Our results are consistent with the Affleck-Haldane prediction that the correlation of the spin-3/2 chain exhibits a crossover to the same asymptotic behavior as in the spin-1/2 chain.     

Asymptotic behavior of correlation functions for electric potential and field fluctuations in a classical one- component plasma

The correlations of the electric potential fluctuations in a classical one-component plasma are studied for large distances between the observation points. The two-point correlation function for these fluctuations is known to decay slowly for large distances, even if exponential clustering holds for the charge correlation functions. In this paper the asymptotic behavior of the generalk-point electric potential correlation functions is analyzed. Each of these correlation functions can be split into a reducible part, which is given by a sum of products of lower-order correlation functions, and a remaining irreducible part. It is shown, on the basis of an exponential clustering hypothesis for the charge correlation functions, that for allk3 the irreducible parts of the electric potential correlation functions decay faster than any inverse power of the distance, if one or more of the observation points move far away from the others. Hence, the two-point electric potential correlation function is the only one with a slow algebraic decay. The same statement holds for the correlation functions of the electric field fluctuations.     

Linear response and fluctuation theorems for nonstationary stochastic processes

Linear response theory is developed for nonstationary Markov processes. A generalized fluctuation theorem is derived which relates the response function to a correlation of fluctuations of the unperturbed nonstationary process. It is shown that it reduces to an ordinary fluctuation theorem relating the response function to the two-point correlation between fluctuations of state variables in the case of a Gaussian distribution function. The results are illustrated by explicit calculation for the class of nonstationary linear Fokker-Planck processes.Supported by Schweizerischer Nationalfonds.     

Controllable coupling and quantum correlation dynamics of two double quantum dots coupled via a transmission line resonator

We propose a theoretical scheme to generate a controllable and switchable coupling between two double-quantum-dot (DQD) spin qubits by using a transmission line resonator (TLR) as a bus system. We study dynamical behaviors of quantum correlations described by entanglement correlation (EC) and discord correlation (DC) between two DQD spin qubits when the two spin qubits and the TLR are initially prepared in X-type quantum states and a coherent state, respectively. We demonstrate that in the EC death regions there exist DC stationary states in which the stable DC amplification or degradation can be generated during the dynamical evolution. It is shown that these DC stationary states can be controlled by initial-state parameters, the coupling, and detuning between qubits and the TLR. We reveal the full synchronization and anti-synchronization phenomena in the EC and DC time evolution, and show that the EC and DC synchronization and anti-synchronization depends on the initial-state parameters of the two DQD spin qubits. It is shown that the initial quantum correlation may be suppressed completely when the evolution time approaches to the infinity in the presence of dissipation. These results shed new light on dynamics of quantum correlations.     

Exact relation for correlation functions in compressible isothermal turbulence

Compressible isothermal turbulence is analyzed under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation is derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case. The main difference resides in the presence of a new type of term which acts on the inertial range similarly as a source or a sink for the mean energy transfer rate. When isotropy is assumed, compressible turbulence may be described by the relation -2/3ε(eff)r = F(r)(r), where F(r) is the radial component of the two-point correlation functions and ε(eff) is an effective mean total energy injection rate. By dimensional arguments, we predict that a spectrum in k(-5/3) may still be preserved at small scales if the density-weighted fluid velocity ρ(1/3)u is used.     

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring. II. The Charged System and the Two-Component Burgers Equations

We further study the stochastic model discussed in ref. 2 in which positive and negative particles diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. We extend the analysis of this model to the case when the densities of the charged particles are not the same. The mean-field equations describing the model are coupled nonlinear differential equations that we call the two-component Burgers equations. We find roundabout weak solutions of these equations. These solutions are used to describe the properties of the stationary states of the stochastic model. The values of the currents and of various two-point correlation functions obtained from Monte-Carlo simulations are compared with the mean-field results. Like in the case of equal densities, one finds a pure phase, a mixed phase and a disordered phase.     

Asymptotic Structure of Constrained Exponential Random Graph Models

In this paper, we study exponential random graph models subject to certain constraints. We obtain some general results about the asymptotic structure of the model. We show that there exists non-trivial regions in the phase plane where the asymptotic structure is uniform and there also exists non-trivial regions in the phase plane where the asymptotic structure is non-uniform. We will get more refined results for the star model and in particular the two-star model for which a sharp transition from uniform to non-uniform structure, a stationary point and phase transitions will be obtained.     

Stationary Correlations for the 1D KPZ Equation

We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the initial condition the two-sided Brownian motion (BM) with respect to the space variable. Developing techniques for dealing with this initial condition in the replica analysis, we elucidate some exact nature of the height fluctuation for the KPZ equation. In particular, we obtain an explicit representation of the probability distribution of the height in terms of the Fredholm determinants. Furthermore from this expression, we also get the exact expression of the space-time two-point correlation function.     

Exact Scaling Functions for One-Dimensional Stationary KPZ Growth

We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann–Hilbert problem related to the Painlevé II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore⁽¹⁾ obtained from the mode coupling approximation.     

Stationary Waves for the Discrete Boltzmann Equation in the Half Space with Reflective Boundaries

The present paper is concerned with stationary solutions for discrete velocity models of the Boltzmann equation with reflective boundary condition in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of stationary solutions satisfying the reflective boundary condition as well as the spatially asymptotic condition given by a Maxwellian state. First, the sufficient condition is obtained for the linearized system. Then, this result is applied to prove the existence theorem for the nonlinear equation through the contraction mapping principle. Also, it is shown that the stationary solution approaches the asymptotic Maxwellian state exponentially as the spatial variable tends to infinity. Moreover, we show the time asymptotic stability of the stationary solutions. In the proof, we employ the standard energy method to obtain a priori estimates for nonstationary solutions. The exponential convergence at the spatial asymptotic state of the stationary solutions gives essential information to handle some error terms. Then we discuss some concrete models of the Boltzmann type as an application of our general theory. Received: 7 July 1999 / Accepted: 3 November 1999     

Uniqueness of reconstruction of multiphase morphologies from two-point correlation functions

The restoration of the spatial structure of heterogeneous media, such as composites, porous materials, microemulsions, ceramics, or polymer blends from two-point correlation functions, is a problem of relevance to several areas of science. In this contribution we revisit the question of the uniqueness of the restoration problem. We present numerical evidence that periodic, piecewise uniform structures with smooth boundaries are completely specified by their two-point correlation functions, up to a translation and, in some cases, inversion. We discuss the physical relevance of the results.     

Markov dilations and quantum detailed balance

We construct quantum stochastic processes whose multi-time correlation functions, with suitable time ordering, can be obtained from a quantum dynamical semigroup. We prove that such a process defines a stationary Markov dilation of the associated semigroup if and only if (up to technicalities) the semigroup satisfies the quantum detailed balance condition with respect to its stationary state.     

Stochastic arbitrage return and its implication for option pricing

The purpose of this work is to explore the role that random arbitrage opportunities play in pricing financial derivatives. We use a non-equilibrium model to set up a stochastic portfolio, and for the random arbitrage return, we choose a stationary ergodic random process rapidly varying in time. We exploit the fact that option price and random arbitrage returns change on different time scales which allows us to develop an asymptotic pricing theory involving the central limit theorem for random processes. We restrict ourselves to finding pricing bands for options rather than exact prices. The resulting pricing bands are shown to be independent of the detailed statistical characteristics of the arbitrage return. We find that the volatility “smile” can also be explained in terms of random arbitrage opportunities.     

Generalized thermalization in an integrable lattice system

After a quench, observables in an integrable system may not relax to the standard thermal values, but can relax to the ones predicted by the generalized Gibbs ensemble (GGE) [M. Rigol et al., Phys. Rev. Lett. 98, 050405 (2007)]. The GGE has been shown to accurately describe observables in various one-dimensional integrable systems, but the origin of its success is not fully understood. Here we introduce a microcanonical version of the GGE and provide a justification of the GGE based on a generalized interpretation of the eigenstate thermalization hypothesis, which was previously introduced to explain thermalization of nonintegrable systems. We study relaxation after a quench of one-dimensional hard-core bosons in an optical lattice. Exact numerical calculations for up to 10 particles on 50 lattice sites (≈10(10) eigenstates) validate our approach.     

Escape driven by strongly correlated noise

We consider the colored-noise-driven archetypal bistability dynamics of the Ginzburg-Landau type. The focus is on the stationary behavior and the problem of escape from metastable states. The deterministic flow of the underlyingtwo-variable Fokker-Planck process is studied as a function of the noise correlation time . As a main result we find that the separatrix exhibits a cusp at asymptotically large noise color. The stationary probability is evaluated approximately (unified colored noise approximation) and compared with numerical exact results. The stationary probability forms the key input in the evaluation of the rate of escape. At very strong noise color, the escape path closely follows a nodal line, passing through the corresponding stable node. The asymptotic result for the escape rate at large is compared with exact calculations for the lowest, nonvanishing eigenvalue.