Quasi-exactly solvable Fokker-Planck equations

We consider exact and quasi-exact solvability of the one-dimensional Fokker-Planck equation based on the connection between the Fokker-Planck equation and the Schrödinger equation. A unified consideration of these two types of solvability is given from the viewpoint of prepotential together with Bethe ansatz equations. Quasi-exactly solvable Fokker-Planck equations related to the sl(2)-based systems in Turbiner’s classification are listed. We also present one sl(2)-based example which is not listed in Turbiner’s scheme.     

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto(GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective,reliable and does not require any restrictive assumptions for nonlinear terms.     

耗散子运动方程与福克-普朗克量子主方程

发展非线性耦合环境下精确、非微扰的量子耗散方法仍然是一个巨大的挑战.本文针对线性和二次耦合热浴模型,介绍了两种刻画系统与环境耦合动力学的有效方法.一个是耗散子运动方程（DEOM）理论,最近已被扩展到处理非线性耦合环境.另一个是推广的福克-普朗克量子主方程（FP-QME）方法,将在这项工作中基于DEOM推导给出.本文对这两种方法进行了详细的比较,并重点介绍了所涉及的准粒子图像、物理含义以及实现方案.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

The Non-Compact Quantum Dilogarithm and the Baxter Equations

A review of the recent formulation of the quantum discrete Liouville model in the strongly coupled regime (corresponding to the Virasoro central charge 1相似文献   

非线性简并光学参量放大系统的量子起伏

解析求解了P表象中考虑抽运吃空后的非线性含时简并光学参量放大系统获得压缩光所满足的福克尔-普朗克（Fokker-Planck）方程,并且计算了它在阈值附近及远离阈值的量子起伏。研究表明：若略去与η成正比的项,则通解很自然地过渡到线性近似解。计及与η成正比的项后,该解与Drummond等在阈值附近的微扰展开理论相比,基本相符。但在远离阈值处,微扰展开理论不适用,而该结果在阈值附近以及远离阈值的整个区域均是适用的。当μ→0压缩很小时,起伏趋近于真空起伏;而μ〉〉1,压缩增大时,趋近1／1＋μ线性理论。     

A two-dimensional Fokker-Planck equation degenerating on a straight line

By an example of a two-dimensional hydrodynamic system, second-order Langevin equations with two correlated noise sources are investigated. It is shown that the asymptotic expression (t) for the stationary distribution functionP depends on the order in which the limiting transitions;t andN ₂₂0 (N ₂₂ is the power of one of the noises) are made. Using the method of local expansions in trigonometric form, approximate expressions are written for the distribution functionP at small but finiteN ₂₂ tending atN ₂₂0 to the known exact solution.     

Application of a singular perturbation expansion to the solution of certain Fokker-Planck equations

We show that a singular perturbation expansion for the solution of a parabolic equation can be applied to some Fokker-Planck equations arising in the analysis of the effects of noise on laser operations. A generalization to the approximate solution of the Smoluchowski equation, when diffusion is a small effect, is given.     

Geometrical derivation of the intrinsic Fokker-Planck equation and its stationary distribution

Starting from an intrinsic Langevin equation, we give a geometrical derivation of the Fokker-Planck equation. We also present a method for obtaining a stationary distribution and for deriving potential conditions when the diffusion matrix is singular.     

Preparing a Fast Pauli Decomposition for Variational Quantum Solving Linear Equations

Solving linear equations is a basic and significant mathematical task, and it can be executed by variational quantum algorithm (VQA) with quantum advantages by leveraging near-term quantum device and classical optimizer. In the above algorithm, the coefficient matrix should be decomposed first but slowly with the traditional method, for realizing an effective quantum circuit. In this paper, a general framework for preparing a fast Pauli decomposition for solving arbitrary linear equations using VQA is proposed. This method has a simpler form and reduces the complexity of matrix decomposition compared with the traditional one. Moreover, the concrete tables of two and three qubits cases are given for looking up quickly and the instances of Toeplitz matrix, Yule–Walker, and arbitrary equations are demonstrated. Finally, numerical simulations are given to verify this method. This work provides a more convenient and faster preparatory phase for solving linear equations using VQA.     

A WKB treatment of diffusion in a multidimensional bistable potential

We extend to the case of a finite set of stochastic variables whose distributionP obeys a nonlinear Fokker-Planck equation our previous treatment of diffusion in a bistable potentialU, in the limit of small, constant diffusion coefficient. This is done with the help of an extended WKB approximation due to Gervais and Sakita. The treatment is valid if there exists a well-defined most probable path connecting the minima ofU, and if the valley ofU along that path has a slowly varying width, and weak curvature and twisting. We find that: (i) the final approach to equilibrium is governed by Eyring's generalization of the Kramers high-viscosity rate, which we rederive; (ii) for intermediate times, if the initial distribution is concentrated in the region of instability (close vicinity of the saddle point ofU),P has, along the most probable path, the behavior described by Suzuki's scaling statement for a one-dimensional system. In a second part of this time domain,P enters the diffusive regions around the minima ofU and relaxes toward local longitudinal equilibrium on a time comparable with Suzuki's time scale. The time for relaxation toward transverse local equilibrium may, depending on the initial conditions, compete with these longitudinal times.We dedicate this work to our colleague, Yuri Orlov.     

Diffusion in a bistable potential: A systematic WKB treatment

We study the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, in the limit of constant and small diffusion coefficient. A systematic WKB analysis of the lowest eigenmodes of the equivalent Schrödinger-like equation yields the following results: the final approach to equilibrium is governed by the Kramers high-viscosity rate, which is shown to be exact in this limit; for intermediate times, we show that Suzuki's scaling statement does give the correct behavior for the transition between the one-peak and the two-peak structure forP. However, the intermediate time domain also contains a second half, whereP enters the diffusive equilibrium regions, characterized by a time scale of the same order as Suzuki's time.     

Towards Quantum Control with Advanced Quantum Computing: A Perspective

We propose the combination of digital quantum simulation and variational quantum algorithms as an alternative approach to numerical methods for solving quantum control problems. As a hybrid quantum–classical framework, it provides an efficient simulation of quantum dynamics compared to classical algorithms, exploiting the previous achievements in digital quantum simulation. We analyze the trainability and the performance of such algorithms based on our preliminary works. We show that specific quantum control problems, e.g., finding the switching time for bang-bang control or the digital quantum annealing schedule, can already be studied in the noisy intermediate-scale quantum era. We foresee that these algorithms will contribute even more to quantum control of high precision if the hardware for experimental implementation is developed to the next level.     

A mini-tutorial on measure-valued Markov processes and nonlinear martingale problems—As a reply to McCauley's comment

One goal of this mini-tutorial is to provide an introduction into the theory of measure-valued Markov processes and nonlinear martingales defined by strongly nonlinear Fokker-Planck equations and to discuss the physical relevance of the associated processes. Another goal is to reply to McCauley's comment on T.D. Frank [Physica A 331, 391 (2004)]. The tutorial addresses in detail two approaches found in physics and mathematics. The first approach exploits a mapping between linear and nonlinear Fokker-Planck equations. The second approach exploits martingale theory. Several examples of Markov processes and martingales in quantum mechanical, nonextensive, and self-organizing systems defined by nonlinear Fokker-Planck equations are discussed.     

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.     

加偏置电场的双曲线量子阱中线性与三阶非线性光学吸收系数的计算

利用紧致密度矩阵近似方法,研究了加偏置电场双曲线量子阱中的线性与三阶非线性光学吸收系数. 得到了该系统中的线性与三阶非线性光学吸收系数的解析表达式.分析了势阱的形状、外加电场的大小以及入射光场的强度对吸收系数的影响规律. 文章以典型的AlxGa1－xAs/ GaAs双曲线量子阱为例作了数值计算.结果表明：随着势阱宽度的增加,系统的吸收系数将减小;随着外加电场的增加,系统的非对称性增加,系统的吸收系数将增加;随着外加光场强度的增加,系统的吸收系数将减小,并且当光强增加到一定值时会出现明显的饱和吸收现象,这一结论为进一步的实验研究提供了相应的理论依据.     

Nonlinear effects in white-noise driven spatial diffusion: General analytical results and probabilities of exceeding threshold

We consider a general nonlinear diffusion, typified by those deriving from Fitzhugh-Nagumo or Hindmarsh-Rose models of nerve-cell dynamics, perturbed also by 2-parameter white noise. In order to investigate the effects of the nonlinearity, we find for general boundary conditions the mean to order ?² and the four-point covariance to order ?³. The derivations involve multiple stochastic integrals in the plane. The mean and variance of the state variable are thus obtained and may be used to estimate the probabilities that a threshold value is exceeded as a function of space and time. A numerical example is given for a space-time white-noise driven diffusion with a cubic nonlinearity. From the asymptotic form of the covariance the spectral density of the process can also be obtained.     

Allowable Generalized Quantum Gates Using Nonlinear Quantum Optics

In this paper, we give an efficient physical realization of a double-slit duality quantum gate. Weak cross- Kerr nonlinearity is exploited here. The probability of success can reach 1/2. Asymmetrical slit duality control gate also can be constructed conveniently. The special quantum control gate could be realized easily in optical system by our current experimental technology.     

囚禁离子非线性J-C模型量子态保真度

用全量子理论研究驻波激光场与囚禁离子相互作用系统中量子态保真度,详细讨论离子质心在驻波激光场中位置及离子初始状态对保真度的影响.结果表明:随着囚禁离子从远离驻波激光场波节处向波节处移动,量子态保真度振荡频率越来越高,振荡幅度几乎不变,且保真度到达第一个极小值所用时间越来越短,但不会出现信息完全失真;随着囚禁离子处于基态概率增加,量子态保真度振荡频率几乎不变,振荡幅度越来越小,也不会出现信息完全失真;在信息储存或传递过程中,囚禁离子量子态失真比系统和驻波场量子态失真小.