First passage time problems for a class of master equations with separable kernels

We discuss first passage time problems for a class of one-dimensional master equations with separable kernels. For this class of master equations the integral equation for first passage time moments can be transformed exactly into ordinary differential equations. When the separable kernel has only a single term the equation for the mean first passage time obtained is exactly that for simple diffusion. The boundary conditions, however, differ from those appropriate to simple diffusion. The equations for higher moments differ slightly from those for simple diffusion. Analysis is presented, of a generalization of a model of a random walk with long-range jumps first investigated by Lindenberg and Shuler. Since the equations can be solved exactly one can study the behavior of boundary conditions in the continuum limit. The generalization to a larger number of terms in the separable kernel leads to higher order equations for the first passage time moments. In each case, boundary conditions can be found directly from the original master equation.     

Resummation with Wilson lines off the light cone

I review the resummation formalism for organizing large logarithms in perturbative expansion of collinear subprocesses through the variation of Wilson lines off the light cone. A master equation is derived, which involves the evolution kernel resulting from this variation. It is then demonstrated that all the known single- and double-logarithm summations for a parton distribution function or a transverse-momentum-dependent parton distribution can be reproduced from the master equation by applying appropriate soft-gluon approximations to the evolution kernel. Moreover, jet substructures, information which is crucial for particle identification at the Large Hadron Collider and usually acquired from event generators, can also be calculated in this formalism.     

Evolution equations of the expected phase space ion densities and correlation functions in a D+T plasma

Summary In this paper we formualte a master equation approach describing a D+T thermonuclear plasma in a lumped phase space. From the first moments of this master equation and performing the pass to the continuous limit the evolution equations for the expected phase space ion densities emerge. Also we have obtained the evolution equations of the equal time correlation and covariance functions. Finally we have deduced the hydrodynamic equations that arise from a master equation approach.     

Nakajima-Zwanzig's generalized master equation: Evaluation of the kernel of the integro-differential equation

A method is presented, which allows the exact elimination of the projection operator from the kernel of the Nakajima-Zwanzig generalized master equation without using perturbational expansions. Expressions for kernels of generalized master equations using several frequently occuring types of projection operators are derived explicitly. The application of this method for the exact derivation of generalized master equations describing the coherent and the coupled coherent and incoherent exciton motion is proposed. As another application, the derivation of the Smoluchowski equation is suggested.     

Increasing the frequency of a free electron laser by means of a linearly polarized magnetic field

An approximation method is proposed to obtain time dependent solutions of discrete master equations and Fokker-Planck equations. We restrict ourselves to one-dimensional one-step master equations and to one-dimensional F-P equations. Approximate evolution equations for certain statistical averages, e.g. for statistical moments, are formulated and applied for two examples of nonequilibrium phase transitions.     

A Possible Time-Dependent Generalization of the Bipartite Quantum Marginal Problem

In this work, we study an inverse dynamical problem for a bipartite quantum system governed by the time local master equation: to find the class of generators which give rise to a certain time evolution with the constraint of fixed reduced states (marginals). The compatibility of such choice with a global unitary evolution is considered. For the nonunitary case, we propose a systematic method to reconstruct examples of master equations and address them to different physical scenarios.     

Influence of Initial Correlations on Evolution of a Subsystem in a Heat Bath and Polaron Mobility

A regular approach to accounting for initial correlations, which allows to go beyond the unrealistic random phase (initial product state) approximation in deriving the evolution equations, is suggested. An exact homogeneous (time-convolution and time-convolutionless) equations for a relevant part of the two-time equilibrium correlation function for the dynamic variables of a subsystem interacting with a boson field (heat bath) are obtained. No conventional approximation like RPA or Bogoliubov’s principle of weakening of initial correlations is used. The obtained equations take into account the initial correlations in the kernel governing their evolution. The solution to these equations is found in the second order of the kernel expansion in the electron–phonon interaction, which demonstrates that generally the initial correlations influence the correlation function’s evolution in time. It is explicitly shown that this influence vanishes on a large timescale (actually at \(t\rightarrow \infty \)) and the evolution process enters an irreversible kinetic regime. The developed approach is applied to the Fröhlich polaron and the low-temperature polaron mobility (which was under a long-time debate) is found with a correction due to initial correlations.     

Linearized master equations in the correlation-patterns representation

The evolution operator in the correlation-patterns representation is studied in the linear approximation. The linearized master evolution equations satisfied by the vacuum and the correlations are derived. The equations are applicable to inhomogeneous systems. The structure of the introduced operators is discussed.     

Rate processes in condensed media: Basic equations

A general theory of rate processes is developed. Starting from the first principles, the non-Markovian and Markovian type equations governing relaxation processes are derived. Under certain conditions (which are specified) these equations may be approximately reduced to master equations.The theory is applied to two specific models. In one of them the electron-nuclear system is represented by two intersecting electronic energy hyper-surfaces with a continuum of degrees of freedom plus a small perturbation causing transitions between these electronic states. The equations determining the time behaviour of the electronic subsystem in the general case do not coincide with master equations and the time evolution of the system has mixed oscillatory decaying behaviour.Another model takes into account a possible competition between electronic and vibrational relaxations. The corresponding kinetic equations are derived.     

完全变光滑长度SPH法及其实现

提出完全变光滑长度SPH法及其算法实现.方程组基于对称形式核函数近似,SPH密度演化方程与变光滑长度方程隐式关联;在Springel提出的全守恒SPH方程组基础上,通过将分散核近似形式改进为对称核近似形式得到SPH动量方程和能量方程.采用迭代求解密度演化方程和变光滑长度方程,显式求解SPH动量方程和能量方程,增加的计算量相对很少.给出三个1D激波管算例和2D Sedov算例验证方法的有效性.数值结果表明,算法保持动量和能量的守恒律,解决了传统SPH法中由于变光滑长度影响带来的计算误差,且在模拟2DSedov问题时能得到比Springel方法更准确的压强峰值位置和中心压强值.特别适合于模拟爆炸与冲击、大变形大扭曲等密度梯度和光滑长度梯度剧烈变化的问题.     

Selfconsistent Electron Energy Distribution Functions in Non Equilibrium oxygen: Effects on the Dissociation Rate

Electron energy distribution functions (edf) in non equilibrium oxygen have been calculated by solving the Boltzmann equation coupled to a system of vibrational master equations. The results show the importance of both superelastic vibrational collisions and of the presence of oxygen atoms in affecting edf. The coupling between the Boltzmann equation and the system of vibrational master equations brings to a temporal evolution of edf, which progressively changes from a cold molecular gas situation (all molecules in the ground vibrational level) to a vibrationally excited molecular gas and finally to a gas composed by oxygen molecules and oxygen atoms. All electronic rate coefficients follow a temporal evolution. due to the corresponding evolution of edf. Finally the present results are used for discussing the dissociation rate of molecular oxygen in electrical discharges.     

From microscopic interactions to macroscopic laws of cluster evolution

We derive macroscopic governing laws of growth velocity, surface tension, mobility, critical nucleus size, and morphological evolution of clusters, from microscopic scale master equations for a prototype surface reaction system with long range adsorbate-adsorbate interactions.     

On closed evolution equations for the moments of a macroscopic variable

Closed equations for the time evolution of the mean value of a macroscopic variable and the second moment around it are derived exactly from the master equation. Comparison with the method of system size expansion is made.     

Decoherence of Two-qubits Coupled with Reservoirs Studied with New Ket-Bra Entangled State Method

For the first time we define a so-called Ket-Bra Entangled State (KBES) for two-qubits coupled with reservoirs by introduce an extra fictitious mode vector, and convert the corresponding master equation into Schrödinger-like equation by virtue of this state. Via this approach we concisely obtain the dynamic evolution of two uncoupled qubits each immersed in local thermal noise. Based on this, the decoherence evolution for the extended Werner-like states is derived and how purity and temperature influence the concurrence is analyzed. This KBES method may also be applied to tackling master equations of limited atomic level systems.     

Quantum Evolution beyond the Markovian Semigroup — Generalizing the Stenholm–Barnett Approach

We provide conditions for the memory kernel governing the time-nonlocal quantum master equation which guarantee that the corresponding dynamical map is completely positive and trace-preserving. This approach gives rise to the new parametrization of dynamical maps in terms of two completely positive maps – so-called legitimate pair. In fact, these new parameterizations are a natural generalization of Markovian semigroup. Interestingly our class contains recently studied models like semi-Markov evolution and collision models.     

Solution of the Master Equation in the Amplitude Damping Model Under the Action of Linear Resonance Force

The amplitude damping model master equations of density operators under the action of linear resonance force can be concisely solved by virtue of thermo entangled state representation and the technique of integration within an ordered product of operators. We obtain the infinitive operator-sum representation of density operators. This approach may also be effective to treat other master equations. Further, the evolution of the coherent state in this model is discussed. The results show that the coherent state maintains its original coherence character in the amplitude damping model under the action of linear resonance force.     

Collisional Depolarization of the Luminescence of Asymmetric Top Molecules in the Gas Phase

To describe the collisional depolarization of the luminescence of asymmetric top polyatomic molecules, integral and integrodifferential forms of master equations, in which the effect of collisions is determined by the conditional probabilities of an instantaneous error of rotational phase variables, have been obtained. A symmetry analysis of the master equations has been performed, and it has been shown that in the general case the evolution of optically induced anisotropy is given by five independent relaxation components. The kinetic equations derived are initial equations for specific calculations of the anisotropy relaxation in various collisional models. They make it possible to study the influence of the angular momentum transfer efficiency on the orientational relaxation of anisotropy in a wide range of buffer medium densities: from free rotation to binary collisions in the gas and then to rotational diffusion.     

Nonlinear relaxation and fluctuations of damped quantum systems

The paper reexamines the treatment of irreversible quantum systems by master equations. Shortcomings of the conventional theory of quantum Markov processes pointed out by Talkner are analyzed. It is shown that a frequently used quantum regression hypothesis is not correct, in general. A new generalized master equation determining the relaxation to equilibrium is derived by means of time-dependent projection operator techniques. It is shown that this master equation also determines the time evolution of equilibrium correlations and response functions. The Markovian approximation is discussed, and a new type of Markovian limit, the Brownian motion limit, is introduced besides the weak coupling limit. The shortcomings of the conventional approach are resolved by deriving new formulae for the time evolution of the correlation and response functions of a quantum Markov process. The symmetries of the process are emphasized, and it is shown how the fluctuation-dissipation theorem and the detailed balance symmetry emerge from the master equation approach.     

An interacting particle model with compact hierarchical structure

We present a model interacting particle system with a population of fixed size in which particles wander randomly in space, and pairs interact at a rate determined by a reaction kernel with finite range. The pairwise interaction randomly selects one of the particles (the victim) and instantly transfers it to the position of the other (the killer), thus maintaining the total number. The special feature of the model is that it possesses a closed hierarchical structure in which the statistical moments of the governing master equation lead to closed equations for the reduced distribution functions (the concentration, pair correlation function, and so on). In one spatial dimension, we show that persistent spatial correlations (clusters) arise in this model and we characterize the dynamics in terms of analytical properties of the pair correlation function. As the range of the reaction kernel is increased, the dynamics varies from an ensemble of largely independent random walkers at small range to tightly bound clusters with longer-range reaction kernels.     

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.