Directional quantum random walk induced by coherence

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker's position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker's position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker's position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker's position distribution.     

Two-dimensional Quantum Random Walk

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of Pemantle and Wilson (J. Comb. Theory, Ser. A 97(1):129–161, 2002).     

用一种分数阶算法研究非马尔可夫过程中阻尼与涨落的竞争机制

基于分数阶朗之万方程和随机行走理论, 建立了一种用于研究非马尔可夫系统中随机变量随时间演化的数值模拟算法, 称之为分数阶随机行走模拟法. 进一步运用此算法分别数值研究了无阻尼有涨落、 有阻尼无涨落和阻尼与涨落兼备三种情况下, 受欠扩散分数阶朗之万方程约束的随机变量随时间的演化行为. 结果显示阻尼和涨落存在竞争关系: 高斯型涨落的影响会随着时间的增长被"抹平", 从而凸显阻尼使系统趋于平衡的作用; 而长尾型涨落则由于包含"小概率大贡献"事件, 使得长时间演化之后系统变量仍以一定概率出现突然变化. 关键词： 非马尔可夫 欠扩散 阻尼与涨落 分数阶朗之万方程     

An effective Hamiltonian approach to quantum random walk

In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative (see Chandrashekar, Sci. Rep. 3, 2829 (18)). We showed that in the case of two-step walk, the time evolution operator effectively can have multiplicative form. In the case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the graphene Hamiltonian, the walk has been studied on a graphene lattice and we conclude the preference of additive approach over the multiplicative one.     

Revisiting the European sovereign bonds with a permutation-information-theory approach

In this paper we study the evolution of the informational efficiency in its weak form for seventeen European sovereign bonds time series. We aim to assess the impact of two specific economic situations in the hypothetical random behavior of these time series: the establishment of a common currency and a wide and deep financial crisis. In order to evaluate the informational efficiency we use permutation quantifiers derived from information theory. Specifically, time series are ranked according to two metrics that measure the intrinsic structure of their correlations: permutation entropy and permutation statistical complexity. These measures provide the rectangular coordinates of the complexity-entropy causality plane; the planar location of the time series in this representation space reveals the degree of informational efficiency. According to our results, the currency union contributed to homogenize the stochastic characteristics of the time series and produced synchronization in the random behavior of them. Additionally, the 2008 financial crisis uncovered differences within the apparently homogeneous European sovereign markets and revealed country-specific characteristics that were partially hidden during the monetary union heyday.     

Large Time Behavior and Convergence Rate for Quantum Filters Under Standard Non Demolition Conditions

A quantum system \({\mathcal S}\) undergoing continuous time measurement is usually described by a jump-diffusion stochastic differential equation. Such an equation is called a quantum filtering equation (or quantum stochastic master equation) and its solution is called a quantum filter (or quantum trajectory). This solution describes the evolution of the state of \({\mathcal S}\) . In the context of quantum non demolition measurement, we investigate the large time behavior of this solution. It is rigorously shown that, for large time, this solution behaves as if a direct Von Neumann measurement has been performed at time 0. In particular the solution converges to a random pure state which can be directly linked to the wave packet reduction postulate. Using the theory of Girsanov transformation, we obtain the explicit rate of convergence towards this random state. The problem of state estimation (used in experiment) is also investigated.     

光散射法测量微粒粒径分布的一种反演遗传算法

本文提出一种基于改进遗传算法的随机及演方法,该方法具有更强的全局收敛性,同时也具有对粒径分布形状不敏感、抗噪声能力强等优点,适合于工业现场应用．     

A Kinetic Model of Quantum Jumps

A new class of models describing the dissipative dynamics of an open quantum system S by means of random time evolutions of pure states in its Hilbert space is considered. The random evolutions are linear and defined by Poisson processes. At the random Poissonian times, the wavefunction experiences discontinuous changes (quantum jumps). These changes are implemented by some non-unitary linear operators satisfying a locality condition. If the Hilbert space of S is infinite dimensional, the models involve an infinite number of independent Poisson processes and the total frequency of jumps may be infinite. We show that the random evolutions in are then given by some almost-surely defined unbounded random evolution operators obtained by a limit procedure. The average evolution of the observables of S is given by a quantum dynamical semigroup, its generator having the Lindblad form.⁽¹⁾ The relevance of the models in the field of electronic transport in Anderson insulators is emphasised.     

Hausdorff Dimension in Stochastic Dispersion

We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order away from the origin, there is an uncountable set of measure zero of points, which escape to infinity at the linear rate. In this paper we prove that this set of linear escape points has full Hausdorff dimension.     

Analysis on the evolution process of BFW-like model with discontinuous percolation of multiple giant components

Recently, the modified BFW model on random graphs [W. Chen, R.M. D’Souza, Phys. Rev. Lett. 106 (2011) 115701], which shows a discontinuous percolation transition with multiple giant components, has attracted much attention from physicists and statisticians. In this paper, by establishing the evolution equations on the modified BFW model, the evolution process and steady-states on both random graphs and finite-dimensional lattices are analyzed. On a random graph, by varying the edge accepted rate α$\alpha$, the system stabilizes in a steady-state with different numbers of giant components. Moreover, a close correspondence is built between the values of α$\alpha$ and the number of giant components in steady-states, the efficiency of which is verified by the numerical simulations. Then, the sizes of giant components for different evolution strategies can be obtained by solving some constraints derived from the evolution equations. Meanwhile, a similar analysis is expanded to finite-dimensional lattices, and we find the BFW (α$\alpha$) model on a finite-dimensional lattice has different steady-states from those on a random graph, but they have the same evolution mechanism. The analysis of the evolution process and steady-state is of great help to explain the properties of discontinuous percolation and the role of nonlocality.     

Decay of a thermofield-double state in chaotic quantum systems

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite N for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.     

Molecular evolution under fitness fluctuations

Molecular evolution is a stochastic process governed by fitness, mutations, and reproductive fluctuations in a population. Here, we study evolution where fitness itself is stochastic, with random switches in the direction of selection at individual genomic loci. As the correlation time of these fluctuations becomes larger than the diffusion time of mutations within the population, fitness changes from an annealed to a quenched random variable. We show that the rate of evolution has its maximum in the crossover regime, where both time scales are comparable. Adaptive evolution emerges in the quenched fitness regime (evidence for such fitness fluctuations has recently been found in genomic data). The joint statistical theory of reproductive and fitness fluctuations establishes a conceptual connection between evolutionary genetics and statistical physics of disordered systems.     

利用随机相位实现Duffing系统的混沌控制

基于线性随机系统Khasminskii球面坐标变换, 计算了谐和激励中含有随机相位的Duffing 方程的最大Lyapunov指数. 依据平均最大Lyapunov指数符号的变化, 分析随机相位对非线性系统动力学行为的影响.说明随机相位可以产生混沌亦可抑制混沌, 从而可以作为混沌控制的一种方法. 结合对相图、Poincaré截面、时间历程图的分析, 说明上述方法是有效的. 关键词： 随机相位 混沌控制 最大Lyapunov指数 Poincaré截面     

Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets

We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian (Spitzer in Adv. Math. 5:246–290, 1970) and the stochastic chains with memory of variable length which have finite state space (Rissanen in IEEE Trans. Inf. Theory 29(5):656–664, 1983). These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erdös-Rényi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.     

Super-Hydrodynamic Limit in Interacting Particle Systems

This paper is a follow-up of the work initiated in (Arab J Math, 2014), where we investigated the hydrodynamic limit of symmetric independent random walkers with birth at the origin and death at the rightmost occupied site. Here we obtain two further results: first we characterize the stationary states on the hydrodynamic time scale as a family of linear macroscopic profiles parameterized by their mass. Then we prove that beyond hydrodynamics there exists a longer time scale where the evolution becomes random. On such a super-hydrodynamic scale the particle system is at each time close to the stationary state with same mass and the mass fluctuates performing a Brownian motion reflected at the origin.     

Measurement of persistence in 1D diffusion

Using a novel NMR scheme we observed persistence in 1D gas diffusion. Analytical approximations and numerical simulations have indicated that for an initially random array of spins undergoing diffusion, the probability p(t) that the average spin magnetization in a given region has not changed sign (i.e., "persists") up to time t follows a power law t(-straight theta), where straight theta depends on the dimensionality of the system. Using laser-polarized 129Xe gas, we prepared an initial "quasirandom" 1D array of spin magnetization and then monitored the ensemble's evolution due to diffusion using real-time NMR imaging. Our measurements are consistent with analytical and numerical predictions of straight theta approximately 0.12.     

The quantum cat map on the modular discretization of extremal black hole horizons

Based on our recent work on the discretization of the radial \(\hbox {AdS}_2\) geometry of extremal BH horizons, we present a toy model for the chaotic unitary evolution of infalling single-particle wave packets. We construct explicitly the eigenstates and eigenvalues for the single-particle dynamics for an observer falling into the BH horizon, with as time evolution operator the quantum Arnol’d cat map (QACM). Using these results we investigate the validity of the eigenstate thermalization hypothesis (ETH), as well as that of the fast scrambling time bound (STB). We find that the QACM, while possessing a linear spectrum, has eigenstates, which are random and satisfy the assumptions of the ETH. We also find that the thermalization of infalling wave packets in this particular model is exponentially fast, thereby saturating the STB, under the constraint that the finite dimension of the single-particle Hilbert space takes values in the set of Fibonacci integers.     

Metastability and Spinodal Points for a Random Walker on a Triangle

We investigate time-dependent properties of a single-particle model in which a random walker moves on a triangle and is subjected to nonfocal boundary conditions. This model exhibits spontaneous breaking of a Z ₂ symmetry. The reduced size of the configuration space (compared to related many-particle models that also show spontaneous symmetry breaking) allows us to study the spectrum of the time evolution operator. We break the symmetry explicitly and find a stable phase, and a metastable phase which vanishes at a spinodal point. At this point, the spectrum of the time evolution operator has a gapless and universal band of excitations with a dynamical critical exponent z=1. Surprisingly, the imaginary parts of the eigenvalues E _j(L) are equally spaced, following the rule . Away from the spinodal point, we find two time scales in the spectrum. These results are related to scaling functions for the mean path of the random walker and to first passage times. For the spinodal point, we find universal scaling behavior. A simplified version of the model which can be handled analytically is also presented.