A generalized stochastic liouville equation. Non-Markovian versus memoryless master equations

The interrelation between the well-known non-Markovian master equation and the new memoryless one used in the previous paper is clarified on the basis of damping theory. The latter equation is generalized to include cases in which the Hamiltonian or the Liouvillian is a random function of time, and is written in a form feasible for perturbational analysis. Thus, the existing stochastic theory in which those cases mentioned above are discussed is equipped with a more tractable basic equation. Two problems discussed in the previous paper, i.e., the random frequency modulation of a quantal oscillator and the Brownian motion of a spin, are treated from the viewpoint of the stochastic theory without such explicit consideration of external reservoirs as was taken in the previous paper.     

Linear response theory for systems obeying the master equation

Linear response theory is developed for systems whose time dependence is described by a master equation. The fluctuation dissipation theorem expressing the linear response of the system in terms of fluctuation properties of the system in equilibrium is derived. The time-dependent Ising spin system in interaction with a heat bath, the Glauber model, is discussed as a particular case of the formalism.The work of one of the authors (D.B.) was supported in part by the National Science Foundation under Contract GP 10536.On leave of absence from the Institute of Physics, Belgrade, Yugoslavia.     

非广延反应扩散系统的广义主方程

依据非平衡统计及密度算子方程，通过计算概率分布函数的时间变化，得到了非广延反应扩散系统在压力作用时其特征函数满足的广义主方程，其中非广延反应扩散系统的压力在Tsallis统计的框架下给出.与唯象理论中的主方程比较，新得到的方程不仅依赖于非广延参量q而且有更多的非线性项，因此更具有普遍性.当新的方程应用到单分子反应模型时, 非广延性对系统的稳定性将产生重要的作用, 特别是当系统接近临界状态时. 关键词： 主方程 非广延反应扩散系统 涨落     

On the standard quantum Brownian equation and an associated class of non-autonomous master equations

It is shown that the standard quantum Brownian equation (QBE) can violate positivity not only past the thermal correlation time, but at arbitrarily long times at high system frequencies. In an effort to improve the standard QBE, exact operator solutions are provided for a class of non-autonomous master equations. These exact solutions are used to derive sufficient positivity conditions for the coefficients of the master equations.     

Time-domain master equation for pulse evolution and laser mode-locking

Using the well-known analogy between the space and time domains we derive a temporal master equation (ME) operator which can be applied to any cavity containing dispersive and filtering elements, phase or amplitude modulators, and one nonlinear element. The cavity properties are described in terms of 2 × 2 `KIJL' matrices. We show that this ME correctly reproduces the cavity mode structure in the linear limit. Numerical simulation of an actively mode-locked Fabry–Perot laser with the nonlinear medium at an end mirror gives results in excellent agreement with those found using the more conventional Huygens' integral method. Using a simple perturbation approach based on the nonlinear Schrödinger equation (NLS) we also show that the field in this laser is soliton-like, and give analytic expressions for the soliton parameters.     

Non-equilibrium effects upon the non-Markovian Caldeira–Leggett quantum master equation

We obtain a non-Markovian quantum master equation directly from the quantization of a non-Markovian Fokker–Planck equation describing the Brownian motion of a particle immersed in a generic environment (e.g. a non-thermal fluid). As far as the especial case of a heat bath comprising of quantum harmonic oscillators is concerned, we derive a non-Markovian Caldeira–Leggett master equation on the basis of which we work out the concept of non-equilibrium quantum thermal force exerted by the harmonic heat bath upon the Brownian motion of a free particle. The classical limit (or dequantization process) of this sort of non-equilibrium quantum effect is scrutinized, as well.     

Generalized master equations for continuous-time random walks

An equivalence is established between generalized master equations and continuous-time random walks by means of an explicit relationship between(t), which is the pausing time distribution in the theory of continuous-time random walks, and(t), which represents the memory in the kernel of a generalized master equation. The result of Bedeaux, Lakatos-Lindenburg, and Shuler concerning the equivalence of the Markovian master equation and a continuous-time random walk with an exponential distribution for(t) is recovered immediately. Some explicit examples of(t) and(t) are also presented, including one which leads to the equation of telegraphy.This study was partially supported by ARPA and monitored by ONR Contract No. (N00014-17-C-0308).For continuity, the reader is directed to the article entitled Random Walks on Lattices. IV. Continuous Time Walks and Influence of Absorbing Boundaries, by E. W. Montroll and H. Scher, which will appear in Volume 9, Number 2, of this journal, and which should precede the following article. Regrettably, the two articles were inadvertently switched during processing.     

Solution of the Fokker-Planck equation for the shock wave problem

An eigenexpansion solution of the time-independent Brownian motion Fokker-Planck equation is given for a situation in which the external acceleration is a step function. The solution describes the heavy-species velocity distribution function in a binary mixture undergoing a shock wave, in the limit of high dilution of the heavy species and negligible width of the light-gas internal shock. The diffusion solution is part of the eigenexpansion. The coefficients of the series of eigenfunctions are obtained analytically with transcendentally small errors of order exp(–1/M), whereM 1 is the mass ratio. Comparison is made with results from a hypersonic approximation.     

Exact time evolution and second-order quantum master equations for two interacting qubits

Time evolution of two interacting qubits under the influence of thermal reservoirs is considered. When there is only one excitation in the whole system, an exact reduced dynamics can be obtained. The result is compared with those obtained by the time-convolutionless and time-convolution quantum master equations in the second order approximation.     

Critical fluctuation universality in chemically oscillatory systems: A soluble master equation

Multiple time scale arguments are used to show that near a Hopf bifurcation to a chemical oscillation the dynamics of the system reduces to that of a classic soluble limit cycle system. A birth and death master equation is then introduced and the spectrum of the resulting transition operator is shown to be complex. Exact solutions of the master equation are obtained both for the steady and (for a rather general class of systems) excited states. Thus a simple basis of universality of critical properties in chemical oscillations is provided.Research supported in part by a grant from the National Science Foundation.     

Half-range completeness for the Fokker-Planck equation with an external force

This paper deals with a generalization of a classical result obtained by R. Beals and V. Protopopescu for the Fokker-Planck equations to the case in which a constant external force is present.     

Approximate any 1-state solutions of the Dirac equation for modified deformed Hylleraas potential by using the Nikiforov-Uvarov method

<正>We investigate the spin and pseudospin symmetries of the Dirac equation under modified deformed Hylleraas potential via a Pekeris approximation and the Nikiforov-Uvarov technique.A tensor interaction of Coulomb form is considered and its degeneracy-removing role is discussed in detail.The solutions are reported for an arbitrary quantum number in a compact form and useful numerical data are included.     

求Emden-Fowler方程积分的分析力学方法

研究求Emden-Fowler方程积分的分析力学方法,包括Hamilton-Poisson方法,Lagrange-Noether方法,Lie-Hojman方法以及Hamilton-Jacobi方法等.     

The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary

The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance x_M beyond the wall. We find x_M=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.     

q-Gaussians in the porous-medium equation: stability and time evolution

The stability of q-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, , the porous-medium equation, is investigated through both numerical and analytical approaches. An analysis of the kurtosis of the distributions strongly suggests that an initial q-Gaussian, characterized by an index q_i, approaches asymptotically the final, analytic solution of the porous-medium equation, characterized by an index q, in such a way that the relaxation rule for the kurtosis evolves in time according to a q-exponential, with a relaxation index q_rel ≡q_rel(q). In some cases, particularly when one attempts to transform an infinite-variance distribution (q_i ≥ 5/3) into a finite-variance one (q < 5/3), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary states to the ultimate thermal equilibrium state.     

两位中国女物理学家对非平衡统计物理学的重要贡献

介绍了两位中国女物理学家王明贞和王承书在20世纪中叶对统计物理发展所作出的重要贡献，王明贞与乌伦贝克(Uhlenbeck)1945年在Review of Modem Physics上发表的有关布朗运动的综述文章。不仅详尽地分析了耦合谐振子的布朗运动特性，而且对随机过程做了完整的科学分类，至今仍被科学界采用，王承书对气体分子动理论所作的大量研究结果，特别是她与乌伦贝克提出的处理多原子分子气体输运系数的王承书-乌伦贝克方程，以及她对麦克斯韦气体的线性化玻尔兹曼方程碰撞算子本征值与本征函数的结果，已经成为现代动理学理论的经典内容，两位中国女物理学家通过她们在物理学史上留下的科学工作，为当代中国妇女攀登科学研究高峰树立了榜样。     

New Approach for Solving Master Equations in Quantum Optics and Quantum Statistics by Virtue of Thermo-Entangled State Representation

By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation <η |, which can arrange master equations of density operators ρ(t) in quantum statisticsas state-vector evolution equations due to the elegant properties of <η |. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of thedecay constant κ we find that the matrix element of ρ (t) at time t in <η| representation is proportional to that of the initial ρ₀ in the decayed entangled state <ηe^-κt| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ρ=∫(d²η /π)< η | ρ> D(η), which is different from all the previous known epresentations.     

Simultaneous Brownian motion of N particles in a temperature gradient

A system of N Brownian particles suspended in a nonuniform heat bath is treated as a thermodynamic system with internal degrees of freedom, in this case their velocities and coordinates. Applying the scheme of nonequilibrium thermodynamics, one then easily obtains the Fokker-Planck equation for simultaneous Brownian motion of N particles in a temperature gradient. This equation accounts for couplings in the motion as a result of hydrodynamic interactions between particles.