Showing papers on "Master equation published in 1991"

Quantum-mechanical derivation of the Bloch equations: Beyond the weak­ coupling limit

Abstract: Two nondegenerate quantum levels coupled off‐diagonally and linearly to a bath of quantum‐mechanical harmonic oscillators are considered. In the weak‐coupling limit one finds that the equations of motion for the reduced density‐matrix elements separate naturally into two uncoupled pairs of linear equations for the diagonal and off‐diagonal elements, which are known as the Bloch equations. The equations for the populations form the simplest two‐component master equation, and the rate constant for the relaxation of nonequilibrium population distributions is 1/T1, defined as the sum of the ‘‘up’’ and ‘‘down’’ rate constants in the master equation. Detailed balance is satisfied for this master equation in that the ratio of these rate constants is equal to the ratio of the equilibrium populations. The relaxation rate constant for the off‐diagonal density‐matrix elements is known as 1/T2. One finds that this satisfies the well‐known relation 1/T2=1/2T1. In this paper the weak‐coupling limit is transcended by de...

Derivation of a master equation for charge-transfer processes in atom-surface collisions

Abstract: The time-dependent Anderson model for multiple levels interacting with a continuum is studied using the slave-boson Green's-function technique, with an eye to understanding charge-transfer phenomena for an atomic species moving outside a metallic surface. It is shown that in the finite-temperature and low-velocity limit, the equations for the occupation numbers satisfy a master equation. Application to charge-exchange processes in atom-surface-scattering experiments shows that the presence of strong intra-atomic correlation effects can drastically change the charge-transfer dynamics.

Turbulent motion and the structure of chaos : a new approach to the statistical theory of open systems

Abstract: I.1. The Criteria of the Relative Degree of Order of the States of Open Systems.- I.2. Connection Between the Statistical and the Dynamic Descriptions of Motions in Macroscopic Open Systems. The Constructive Role of Dynamic Instability of Motion.- I.3. The Transition from Reversible to Irreversible Equations. The Gibbs Ensemble in the Statistical Theory of Nonequilibrium Processes.- I.4. The Role of Fluctuations at Different Levels of Description. Fluctuation Dissipation Relations.- I.5. Brownian Motion in Open Systems. Molecular and Turbulent Sources of Fluctuations.- I.6. Laminar and Turbulent Motion.- 1. Evolution of Entropy and Entropy Production in Open Systems.- 1.1. Chaos and Order. The Controlling Parameters. Physical Chaos. Evolution and Self-Organization in Open systems.- 1.2. Boltzmann-Shannon-Gibbs Entropy.- 1.3. Entropy Distribution Function.- 1.4. The Gibbs Ensemble. Smoothing over the Physically Infinitesimal Volume. Entropy Including Fluctuations. Space (Time) and Phase Averages. Local Ergodicity Condition.- 1.5. The Kinetic Boltzmann Equation for Statistical and Smoothed Distribuions. Physically Infinitesimal Scales. The Constructive Role of the Dynamic Instability of Motion of Atoms in a Gas.- 1.6. The Role of Nonequilibrium Fluctuations in a Boltzmann Gas. Molecular and Turbulent Sources of Fluctuations.- 1.7. The Kinetic Equations for the N-Particle Distribution Functions. The Leontovich Equation.- 1.8. Boltzmann's H-Theorem for Smoothed (Pulsating) and Deterministic Distributions.- 1.9. Entropy and Entropy Production for Smoothed and Deterministic Distributions.- 1.10. The Gibbs' Theorem.- 1.11. H-Theorem for Open Systems. Kulback Entropy.- 1.12. Evolution in the Space of Controlling Parameters. The S-Theorem.- 1.13. The S-Theorem. Local Formulation.- 1.14. The Comparison of the Relative Degree of Order of States on the Basis of the S-Theorem Using Experimental Data.- 1.15. Dynamic and Statistical Descriptions of Complex Motions. K-Entropy, Lyapunov Indices. Nonlinear Characteristics of the Trajectory Divergence.- 1.16. Criteria of Dynamic Instability of Motion in Statistical Theory.- 1.17. Entropy as Measure of Diversity in Biological Evolution.- 1.18. The Principle of Minimum Entropy Production in Self-Organization Processes.- 2. Transition From the Reversible Equations of Mechanics to the Irreversible Equations of the Statistical Theory.- 2.1. Two Types of Reversible Processes. Symmetry Properties of Distribution Functions.- 2.2. Liouville Equation and Vlasov Equation. The First Moments and the "Collisionless" Approximations.- 2.3. Reversible Equations in Quantum Statistical Theory.- 2.4. Two Types of Dissipative Kinetic Equations for N-Particle Distributions.- 2.5. Measure of Deficiency (Incompleteness) of the Statistical Description.- 2.6. The Hierarchy of Equations of Fluid Mechanics.- 3. Fluctuation Dissipation Relations.- 3.1. Examples of Fluctuation Dissipation Relations.- 3.2. FDR for N-Particle Distribution Functions.- 3.3. Thermodynamic Form of the FDR. The Callen-Welton Formula.- 3.4. FDR for a Boltzmann gas. The Fluctuative Representation of Boltzmann Collision Integral.- 3.5. FDR for Large-Scale (Kinetic) Fluctuations.- 3.6. Examples of FDR for Large-Scale Fluctuations.- 3.7. System of Quantum Atoms Oscillators.- 3.8. Fluctuation Dissipation Relations in Hydrodynamics.- 3.9. Two Ways of Defining Kinetic Coefficients.- 3.10. The Molecular Langevin Source in the Difffusion Equation.- 3.11. Connection between the Intensities of Langevin Sources and the Correlator of of Phase Density Fluctuations.- 3.12. Natural Flicker Noise ("1/f Noise"). FDR for Flicker Noise.- 3.13. Natural Flicker Noise and Superconductivity.- 4. Brownian Motion.- 4.1. Fokker-Planck and Langevin Equations.- 4.2. Three Definitions of the Langevin and Fokker-Planck Equations.- 4.3. The Fokker-Planck Equations in the Statistical Theory of Nonequilibrium Processes.- 4.4. Transition to the Fokker-Planck Equation from the Smoluchowski equation (the Chapman-Kolmogorov Equation) and from the Master Equation.- 4.5. Langevin.Sources in Kinetic Equations.- 4.6. Langevin Sources in Fokker-Planck and Einstein-Smoluchowski Equations.- 4.7. Turbulent Langevin Sources and Fluctuation Dissipation Relations in Hydrodynamics.- 4.8. Brownian Motion in Systems with a Variable Number of Particles..- 5. The Boltzmann-Gibbs-Shannon Entropy As Measure of the Relative Degree of Order in Open Systems.- 5.1. Van der Pol Generator.- 5.2. Generator with Inertial Nonlinearity.- 5.3. Invariant Measures. Examples of Gibbs Distributions for Open Systems.- 5.4. Generalized Van Der Pol Generators. Bifurcations of the Limiting Cycle Energy and the Period of Oscillations.- 5.5. Dynamic and Statistical Distributions.- 5.6. Comparison Between the Degrees of Order in the Bifurcation Points and in the State of Dynamic Chaos.- 5.7. Evolution of Entropy in Systems with Two Controlling Parameters.- 5.8. A Medium of Linked Generators. The Kinetic Approach in the Theory of Self-Organization.- 5.9. A System of Van der Pol Generators with Common Feedback. Associative Memory and Pattern Recognition.- 5.10. Kinetic Description of Chemically Reacting Systems.- 5.11. A Medium of Bistable Elements. The Kinetic Approach in the Theory of Phase Transitions.- 6. Turbulent Motion. The Structure of Chaos.- 6.1. Characteristic Features of Turbulent Motion. The Main Problems.- 6.2. Incompressible Fluid. Reynolds Equations. Reynolds Stresses.- 6.3. Well-Developed Turbulence. Turbulent Viscosity.- 6.4. Semiempirical Prandtl-Karman Theory of Turbulence.- 6.5. Onset of Turbulence in Steady Couette and Poiseuille Flows.- 6.6. Entropy Production in Laminar and Turbulent Flows.- 6.7. The Principle of Least Dissipation and the Principle of Minimum Entropy Production in Self-Organization Processes.- 6.8. Evolution of Entropy in the Transition from Laminar to Turbulent Flow.- 6.9. Kinetic Description of Hydrodynamic Motion.- Conclusion.- References.

Killing vector fields in self-dual, Euclidean Einstein spaces with Λ¬=;0

Abstract: Killing equations for self‐dual, Euclidean Einstein spaces with nonvanishing cosmological term Λ are considered. It is shown that in this case one can reduce ten Killing equations to one master equation. Some important solutions of the master equation are found and one of these solutions is analyzed in detail as it leads to metrics defined by the field equation well known in Euclidean or complex relativity. Ernst potentials for the solutions considered are found.

Class of exactly solvable master equations describing coupled nonlinear oscillators

Abstract: Through use of the notation of thermofield dynamics, an exact solution of a class of master equations describing coupled nonlinear oscillators is presented.

Comparison of deterministic and stochastic kinetics for nonlinear systems

Abstract: The deterministic kinetics of chemical reactions is compared with a stochastic description for the cubic Schlogl model with a single stable steady state, which has a nonlinear reaction mechanism. We solve numerically the birth‐death master equation for this system for various numbers of particles (N=20–160). For small systems with tens of particles the deviation of the first moment of the stochastic distribution from the deterministic temporal variation of concentration can be substantial in the initial relaxation towards a stationary state. The relaxation of the master equation is faster than that of the deterministic equation. The maximum deviation in trajectories decreases as the parameters in the kinetic model are altered towards a linear mechanism. The maximum deviation differs from N1/2 as N decreases, but approaches N1/2 as N increases. Deviations from deterministic temporal evolution due to fluctuations depend on the extent of nonlinearity of the reaction. The variance of a stationary distribution...

Reversible diffusion-influenced reactions: comparison of theory and simulation for a simple model

Abstract: Computer simulations of a simple model of a reversible diffusion-influenced reaction are used to test various approximate theoretical treatments. The model is a random walk in continuous time ofN particles on a one-dimensional lattice. The particles can be trapped reversibly at the origin. They move independently, except that only one particle at a time can occupy the origin. The theory is formulated in general terms using master equations for the probability distribution of occupancy numbers of different lattice sites. The general theoretical problem is not solved, although some exact consequences are presented. Several approximation schemes are described and tested by comparison with the simulations.

Theory of the nondegenerate two-photon micromaser

Abstract: We present the theory of a microscopic maser operating on a nondegenerate two-photon transition between two states of same parity. We start from a three-level Hamiltonian, and discuss both the one-photon resonant cascade and the two-photon regime, when the intermediate state is far off resonance. In the semiclassical limit, we derive rate equations for the field intensities and get the corresponding steady-state values. The vacuum is stable in the two-photon regime but becomes unstable for zero detuning (a clear signature of a one-photon process). Solutions displaying true two-photon operation (intermediate state unpopulated) even with zero detuning are displayed. Using the full quantum density-matrix approach, we derive a master equation for the field and discuss both operating regimes, showing that the spontaneous emission may turn the vacuum unstable in the two-photon case. In the off-resonance two-photon regime, there is a strong correlation between the fluctuations of the intensities of the two modes. We find a squeezing factor of 50% for the fluctuations in the difference of intensities.

Dynamics of the infinite-ranged Potts model

Abstract: We formulate a theory of single-spin-flip dynamics for the infinite-rangeq-state Potts model. We derive a Fokker-Planck equation, without diffusive term, from a phenomenological master equation. It describes the approach to equilibrium of the time-dependent probability density and thus generalizes Griffiths' (1966) result for the Ising model. We particularly compare the dynamic evolutions ofq=2 andq=3 systems when sinusoidal external fields are applied. In the caseq=2 we find evidence of a nonequilibrium phase transition and forq=3 period doubling bifurcations are observed, yielding a good estimate of Feigenbaum's universal exponent.

Toward a theory of relaxation in correlated systems: diffusion in the phase space of a chaotic Hamiltonian

Abstract: Relaxation in correlated systems such as interacting ions, entangled polymer chains or viscous liquids requires a time-dependent relaxation rate for a full accounting of the observed phenomena. This has been demonstrated by the coupling scheme for relaxations in correlated systems. A fundamental theory of these necessarily involves non-integrable interactions or constraints which are known to produce chaos in Hamiltonian models. Van Kampen has shown that the transition rates in the master equation for a relaxing Hamiltonian system can be interpreted in terms of diffusion in phase space. In this paper, this approach is generalized to include chaotic Hamiltonian. It is demonstrated that general results from anomalous diffusion in fractals produce all the properties of the time-dependent relaxation rate required to explain the experiments. This leads to a conjecture on the non-Euclidean and possibly fractal phase sphase structure of relaxing chaotic Hamiltonians which would allow a basic understanding of the central results of the coupling scheme.

Kinetic six-vertex model as model of bcc crystal growth

Abstract: The growth of bcc crystals is studied using van Beijeren's mapping onto the six-vertex model. The growth-evaporation processes are described in terms of vertices. The time evolution is given by a master equation for the probability of the six-vertex configurations. The model, studied in the finite-size case by both Monte Carlo and analytic methods, applies to the (001) surface and its vicinal surfaces. Different growth modes (including nucleation) are found, depending on the strength of disequilibrium and on temperature, and the transition between them is investigated.

Solution of the Master Equation for an Attenuated or Amplified Nonlinear Oscillator with an Arbitrary Initial Condition

Abstract: Using the thermo-field dynamics notation, an elegant solution of the master equation for an attenuated or amplified nonlinear oscillator is presented. It is found that, in this notation, the solution of the master equation under consideration amounts to a simple modification of that for the linear oscillator.

Analysis of the projection-operator method: inconsistencies and their removal

Abstract: It is pointed out that the application of the projection-operator method leads to the violation of a consistency condition if the so-called Born approximation is made in the nonMarkovian master equation. Since only the validity of this consistency condition guarantees obtaining non-negative probabilities, its violation makes the applicability of the whole method in this approximation meaningless. Therefore, it is demonstrated that an additional Markov approximation restores the validity of the consistency condition in the master equation, and thus re-establishes the applicability of the projection-operator method. Furthermore, the possibility of obtaining a self-consistent projection-operator method is also examined.

Derivation of the master equation for the atomic density matrix for line polarization studies in the presence of magnetic field and depolarizing collisions in astrophysics

Abstract: In this paper, we derive in a coherent manner, starting from the basic equations of evolution of a quantum mechanical system, the master equation for the density matrix of an atom interacting with a bath of perturbers (electrons, protons) and photons, in the presence of a weak magnetic field (in the domain of sensitivity to the Hanle effect: 0.1 ≲ ωLτ ≲ 10). This paper has been inspired by astrophysical purposes: the interpretation of line polarization induced by anisotropic excitation of the levels, eventually modified by the local magnetic field (the Hanle effect); the polarization can be due to scattering of the incident anisotropic radiation, as in solar prominences, or to impact polarization, as in solar flares. The physical conditions are then those of numerous astrophysical media: any directions of polarization and magnetic field, two-level atom approximation not valid, weak radiation field (so that the bare atom description is convenient), weak density of perturbers (so that the impact approximation is valid). The master equation is derived in the framework of the impact approximation, using the S-matrix formalism without perturbation development in a first step; the Hanle effect is included. The impact approximation leads to a decoupling of the interactions with the perturbers and with the radiation, which are then additive and can be treated independently. The perturbation development is introduced in a second step. The population and coherence transfer probabilities are then obtained for a polarized neutral atom interacting with collisional charged perturbers on the one hand, in the frame of the semi-classical perturbational theory, and on the other hand for a polarized atom interacting with an anisotropic and eventually polarized incident radiation. The linear polarization of the emitted radiation is studied in the following paper which is devoted to the derivation of the transfer equation for polarized radiation in the presence of a magnetic field.

Transport‐collisional master equations for termolecular recombination as a function of gas density

Abstract: Sets of transport‐collisional master equations are developed for the microscopic distribution n(R, E, L) of pairs over internal separation R, energy E, and orbital angular momentum L of (A–B) pairs in a background gas M of variable density. Expressions are also provided for the rate of recombination of A and B as a function of gas density in transport and collisional forms which, respectively, involve microscopic probabilities for association of dissociated (A–B) pairs and probabilities for collisional stabilization of bound pairs. Analytical solutions for the pair distributions n and microscopic probabilities for recombination are obtained in the classical absorption limit. They pertain to exact (A–B) trajectories under general symmetric interaction V(R) between A and B and are applied to ion–ion and electron–ion collisional recombination in a gas. A classical variational method is also presented. Useful expressions for the segments of hyperbolic and general trajectories enclosed by a sphere are derived ...

The low-density limit of quantum systems

Abstract: The present paper concludes a research programme developed, in the past three years, by various authors in Several papers. The basic idea of this programme is to expiain the irreversible behaviour of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system interacling with a reservoir and, in the first attempts to deal with this problem, only limits of observables of the system and deduced master equations were considered. In our approach we study the limits of quantities related to the whole compound syrtem. As a corollary we obtain an explanation of the physical origins of the quantum Brownian motion. In the present paper we study state at invene temperature p and fugacity r, through an interaction of the scattering type, i.e. one which preserves the total number of particles of the reservoir. We obtain a macroscopic equation, far the limit of the compound system, which is a quantum stochastic differential equation of the Poisson type, in the Frigerio and Maassen sense, whose coefficients are uniquely determined by the one-particle scattering operator of the original Hamiltonian system and whose driving noises are the Creation annihilation and number (or gauge) processes living in the space of a Fock quantum Brownian motion over the space L'(R, dt, where KO,, is an Hilbert space depending on the inverse temperature p, the one-particle reservoir dynamics, the free-system dynamics and the interaction. :he )ny&-si!g limit of2 system co-p!ed q-asi.f:ee bnsnn reje~~oi: 1" the eq:i!ibri:m

Non-equilibrium effects on forward and reverse rate coefficients of reversible reactions in solutions: A stochastic approach

Abstract: The kinetics of irreversible and reversible reactions in solution is studied by applying a stochastic approach in the sense of Teramoto and Shigesada. For sufficiently low reactant concentrations the random processes attributed to the reactions reduce to Markovian ones, i.e. rate equations follow as deterministic approximation form a master equation and rate coefficients from its transition rates. The forward and reverse rate constants for the reactions A+B⇌C, A+B⇌C+B and A+B⇌C+D derived in this way have the involved structure well-known from other papers. Our treatment does not confirm the objections to former results recently rised by Lee and Karplus [1].

Quasiprobability distributions for open quantum systems within the Lindblad theory

Abstract: The Lindblad master equation for the damped quantum harmonic oscillator is transformed into Fokker–Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal‐ordering Q representation and the Wigner W representation. It will be proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker–Planck equations in the steady state, it will be shown that the quasiprobability distributions are two‐dimensional Gaussians with widths determined by the diffusion coefficients.

Quantum Markovian master equation for a system of identical particles interacting with a heat reservoir

Abstract: The author's earlier quantum kinetic theory of systems in thermal contact with a heat reservoir is now applied to systems of identical particles. A statistical description of such systems is provided by a hierarchy of many-particle density matrices for which a system of “linked” equations is obtained. The quantum Markovian master equation is derived, which makes it possible to determine the single-particle density matrix. Consideration is given to three examples of the application area of this equation, viz. the latter is described for three cases in which particles experience (1) random walks in a crystal lattice, (2) migration in a homogeneous isotropic continuum, (3) thermoactivated transitions between quantum states. In each of these cases the density matrix is found to describe the stationary state of a system of identical particles.

Recent developments in the Virasoro master equation

Abstract: The Virasoro master equation collects all possible Virasoro constructions which are quadratic in the currents of affine Lie g. The solution space of this system is immense, with generically irrational central charge, and solutions which have so far been observed are generically unitary. Other developments reviewed include the exact C-function, the superconformal master equation and partial classification of solutions by graph theory and generalized graph theories. 37 refs., 1 fig., 1 tab.