Showing papers on "Master equation published in 1989"
TL;DR: Ancona and Tiersten as mentioned in this paper presented a macroscopic description of conduction electrons in a semiconductor, in which the equation of state for the electron gas was generalized to include a dependence on the gradient of the density.
Abstract: In a recent paper [M. G. Ancona and H. F. Tiersten, Phys. Rev. B 35, 7959 (1987)] a macroscopic description of conduction electrons in a semiconductor was presented in which the equation of state for the electron gas was generalized to include a dependence on the gradient of the density. This generalization led to a new transport equation---often expressible as a generalized diffusion--drift-current equation---which has been shown to accurately describe some important quantum mechanical effects occurring in semiconductor structures. In the present paper sufficient microscopic conditions under which the density-gradient--dependent equation of state does represent lowest-order effects of quantum mechanics are established using methods of quantum statistical mechanics. A microscopic derivation of the transport equation is also given.
478 citations
TL;DR: In an open quantum system, dissipation can cause decorrelation on a time scale significantly shorter than the relaxation time which characterizes the approach of the system to thermodynamic equilibrium, and it is demonstrated that the density matrix decays rapidly toward a mixture of ``approximate eigenstates'' of the ``pointer observable,'' which commutes with the system-environment interaction Hamiltonian.
Abstract: The effect of the environment on a quantum system is studied on an exactly solvable model: a harmonic oscillator interacting with a one-dimensional massless scalar field. We show that in an open quantum system, dissipation can cause decorrelation on a time scale significantly shorter than the relaxation time which characterizes the approach of the system to thermodynamic equilibrium. In particular, we demonstrate that the density matrix decays rapidly toward a mixture of ``approximate eigenstates'' of the ``pointer observable,'' which commutes with the system-environment interaction Hamiltonian. This observable can be regarded as continuously, if inaccurately, monitored by the scalar field environment. Both because in a harmonic oscillator the state of the system rotates in the phase space and because the effective environment ``measurement'' is weak, the system, on the short ``collision'' time scale (1/\ensuremath{\Gamma}), maintains a coherence in this pointer observable on time scales of order [\ensuremath{\gamma}/\ensuremath{\Omega}ln(\ensuremath{\Gamma}/\ensuremath{\Omega}${)]}^{1/2}$ and on longer time scales settles into a mixture of coherent states with a dispersion approximately consistent with the vacuum state. The master equation satisfied by the exact solution differs from the other master equations derived both for the high-temperature limit and for T=0. We discuss these differences and study the transition region between the high- and low-temperature regimes. We also consider the behavior of the system in the short-time ``transient'' regime. For T=0, we find that, in the long-time limit, the system behaves as if it were subject to ``1/f noise.'' The generality of our model is considered and its predictions are compared with previous treatments of related problems. Some of the possible applications of the results to experimentally realizable situations are outlined. The significance of the environment-induced reduction of the wave packet for cosmological models is also briefly considered.
460 citations
TL;DR: In this paper, an equation of state for the steady state plasma sheet, whose solution has the form of a mathematical catastrophe, is presented, along with dynamic terms describing the transition across the thermal catastrophe.
Abstract: Equations are presented for the physical mechanisms involved in the resonant absorption of Alfven waves in the plasma sheet boundary layer. It is shown that energy absorbed by the plasma-sheet particles is a function of the central plasma sheet temperature. The heating curve, when coupled with convective transport, yielded an equation of state for the steady state plasma sheet, whose solution has the form of a mathematical catastrophe. The master equation includes dynamic terms describing the transition across the thermal catastrophe, making it possible to evaluate the time scale for the catastrophe to occur.
139 citations
TL;DR: In this paper, the authors give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process, and a discrete ordinate numerical transport solution is generated for this realization.
Abstract: We give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process. A Monte Carlo procedure is used to generate a physical realization of the statistics, and a discrete ordinate numerical transport solution is generated for this realization. The ensemble averaged solution, as well as the variance, is obtained by averaging a large number of such calculations. Reflection and transmission results are given for several problems in both rod and planar geometry. In a separate development, two coupled transport equations are derived which formally described transport in a random binary mixture for arbitrary mixing statistics. Closing these equations by approximating their coupling terms in a low order and intuitive way leads to a model for stochastic transport previously obtained via the master equation. The present derivation, based upon approximating exact equations, allows in principle the opportunity to develop more accurate models by making higher order approximations in the coupling terms.
112 citations
TL;DR: In this article, the authors calculate the asymptotic behavior of the moments of the first-passage time and survival probability for random walks on an exactly self-similar tree, and on a quasi-self-similar comb, by applying an exact decimation approach to the master equations.
Abstract: The authors calculate the asymptotic behaviour of the moments of the first-passage time and survival probability for random walks on an exactly self-similar tree, and on a quasi-self-similar comb, by applying an exact decimation approach to the master equations. For the hierarchical comb, a transition from ordinary to anomalous diffusion occurs at R=2, where R is the ratio of teeth length in successive iterations of the structure. In the anomalous regime (R>2), the positive integer moments of the first-passage time, (tq), scale as Ltau q, with tau q=1+(2q-1)ln R/ln 2, where L is the linear distance from input to output. The asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations. They find that the characteristic time, t*, in the asymptotic exponential decay of the survival probability, exp(-t/t*), scales as t* approximately Ltau *=, with tau *=ln R2/ln 2, i.e. tau * is distinct from tau 1. However, substantial corrections to this asymptotic form for tau * exist, and these are needed to account for the recent simulation data of Havlin and Matan (1988).
92 citations
TL;DR: The theory generalizes the derivation of the conventional superradiance master equation for molecules in vacuum and yields all local field and screening factors for both the dispersive and the dissipative couplings from a single, unified starting point.
Abstract: A reduced equation of motion that describes the excited-state dynamics of interacting two-level impurity molecules in a dielectric host crystal is derived starting from a microscopic model for the total system. Our theory generalizes the derivation of the conventional superradiance master equation for molecules in vacuum; the role of photons in the conventional theory is played by polaritons (mixed crystal-radiation excitations) in our approach. Our final equation thus contains dispersive and superradiant polariton-mediated intermolecular interactions. The effect of the dielectric host is completely contained within a rescaling of these interactions with the transverse dielectric function \ensuremath{\epsilon}(\ensuremath{\omega}) of the crystal taken at the impurity's transition frequency. Our theory yields all local field and screening factors for both the dispersive and the dissipative couplings from a single, unified starting point. Known scaling laws for the spontaneous-emission rate and the instantaneous dipole-dipole interaction are extended to the frequency region where the dispersion of \ensuremath{\epsilon}(\ensuremath{\omega}) is important.
83 citations
TL;DR: In this article, the exact equation of motion for the reduced density matrix of a system weakly coupled to a bath is obtained using projection operator techniques, which reduces to a generalized master equation when the bath relaxation is faster than the relaxation of the system induced by the weak interaction with the bath.
Abstract: The equation of motion for the reduced density matrix of a system weakly coupled to a bath is obtained using projection operator techniques. The exact equation of motion reduces to a generalized master equation when the bath relaxation is faster than the relaxation of the system induced by the weak interaction with the bath. The equation separates into streaming or systematic terms and dissipative terms which are separately equal to zero at equilibrium. We find both statistical and dynamical system frequency shifts; the statistical shifts are present in equilibrium but the dynamical shifts affect the time-dependence, only. The general results are applied to the two-level system model for tunneling in condensed phases.
73 citations
TL;DR: It is shown that amplification destroys quantum coherence more rapidly than does attenuation and the exact solution to the master equation of a nonlinear oscillator, subject to damping or amplification is presented.
Abstract: The exact solution to the master equation of a nonlinear oscillator, subject to damping or amplification, is presented. The effect of such incoherent processes on the quantum-coherence properties and recurrences is obtained. It is shown that amplification destroys quantum coherence more rapidly than does attenuation.
69 citations
TL;DR: A master equation containing the coherent field exactly is derived, revealing the presence of four universal dynamical regimes and a transition, induced by the noise, in which coherence and localization are destroyed.
Abstract: We study theoretically and experimentally the effect of noise on Rydberg atoms passing through a strong coherent microwave field. We derive a master equation containing the coherent field exactly. Its solution reveals the presence of four universal dynamical regimes: (i) an initial classical regime, (ii) a subsequent coherent localized regime, (iii) a transition, induced by the noise, in which coherence and localization are destroyed, and (iv) the final regime, where equidistribution over the quasienergy states is reached.
66 citations
TL;DR: In this article, a generalized Master equation is derived for rotational tunnelling, where the effects of coupling to lattice vibrations are embraced in a memory function non-local in time.
Abstract: A novel approach to the temperature dependence of rotational tunnelling is presented. By means of a projection operator technique a generalized Master equation is derived, where the effects of coupling to lattice vibrations are embraced in a memory function non-local in time. After an expansion in a power series in the coupling Hamiltonian, the Master equation is used for the calculation of two-times correlation functions of scattering operators, the Fourier transforms of which give the scattering function. The theory allows for all features of the spectrum obtained by neutron scattering methods, in particular for those of the central peak and the librational excitations. Furthermore, it is not confined to the low temperature regime, but rather covers the whole range of temperatures of experimental interest.
65 citations
TL;DR: In this article, the authors describe the evolution and implementation of a set of quantum balance equations for examining transport in mescoscopic structures, and describe how they can be applied to the problem of transport analysis.
Abstract: This study describes the evolution and implementation of a set of quantum balance equations for examining transport im mescoscopic structures
TL;DR: This result enables both angular momentum and weak collision effects to be accurately incorporated into the solution of the master equation for chemical activation reactions in the falloff regime.
Abstract: A new solution to the master equation relating the rate coefficients for unimolecular, recombination (association) and chemical activation reactions, incorporating weak collision effects, is presented. The solution establishes conditions for the validity of the commonly used procedure of relating the recombination rate coefficient, throughout the falloff regime, to the reverse single‐channel unimolecular rate coefficient via the equilibrium constant. In addition, a relationship between the rate coefficient for stabilization in a chemical activation reaction and the reverse multichannel unimolecular dissociation rate coefficient is derived. This result, in conjunction with recently developed methods for fully incorporating angular momentum conservation into the solution of the master equation for unimolecular dissociation, enables both angular momentum and weak collision effects to be accurately incorporated into the solution of the master equation for chemical activation reactions in the falloff regime. A...
TL;DR: In this paper, a Schrodinger-type equation is considered in relation to p-adic quantum mechanics and the appropriate notion of differential operators is discussed, and a new set of vacuum states for the padic quantum harmonic oscillator is presented.
Abstract: A Schrodinger-type equation is considered in relation to p-adic quantum mechanics. We discuss the appropriate notion of differential operators. A solution of the Schrodinger-type equation is given and a new set of vacuum states for the p-adic quantum harmonic oscillator is presented. The correspondence principle with the standard quantum mechanics is also discussed.
TL;DR: If the pump noise is eliminated, it is found that the photon number noise in a micromaser and a laser can be significantly reduced below the shot-noise level.
Abstract: We discuss in detail the influence that the statistical properties of the pump source have on maser and laser dynamics. We derive a general master equation for the radiation field that is valid for a wide range of different pump mechanisms. If the pump noise is eliminated, we find that the photon number noise in a micromaser and a laser can be significantly reduced below the shot-noise level. In contrast, the phase fluctuations for both maser and laser are unaffected by the noise contribution of the pump.
TL;DR: The different conditions that could be imposed on a Markovian master equation for a nonlinear oscillator weakly coupled to a thermal reservoir are formulated and the existence of different stages of evolution associated with different time scales of the Hamiltonian dynamics is predicted.
Abstract: The different conditions that could be imposed on a Markovian master equation for a nonlinear oscillator weakly coupled to a thermal reservoir are formulated. They concern preservation of trace and positivity of a density matrix, return to a proper equilibrium state, and the detailed balance condition. It is shown that only one of the known master equations satisfies all of these conditions. Then the validity of the Markovian approximation is reanalyzed using certain non-Markovian weak-coupling approximations, and the existence of different stages of evolution associated with different time scales of the Hamiltonian dynamics is predicted. The consequences of these facts for the description of a damped nonlinear oscillator are discussed.
Book•
31 Mar 1989
TL;DR: In this article, the authors present a dynamic theory of decision processes in the context of urban mobility and show that the transition rates of individuals in an urban environment can be modeled as a discrete choice process.
Abstract: 1. Introduction.- 2. A Dynamic Theory of Decision Processes.- 2.1 The Panel Data-Based Discrete Choice Approach.- 2.2 The Master Equation View in Dynamic Choice Processes.- 2.3 The Decision Process.- 2.3.1 Decision Space and Decision Configuration.- 2.3.2 Individual Decision Processes and Conditional Probability.- 2.3.3 Individual and Configurational Transition Rates.- 2.3.4 Some Functional Forms of the Individual Transition Rates.- 2.4 The Equations of Motion.- 2.4.1 The Master Equation for the Decision Configuration.- 2.4.2 The Translation Operator.- 2.4.3 The Mean Value Equations of the Dynamic Decision Theory.- 2.5 Parameter Estimation.- 2.5.1 Parameter Estimation via Comparison of Transition Rates.- 2.5.2 Parameter Estimation via Comparison of Decision Configurations.- 2.5.3 The Dependence of Trend Parameters on Motivating Factors.- 2.5.4 Scheme of Model Building for Dynamic Decision Processes.- 2.6 Selection Criteria for the Examples.- 3. Shocks in Urban Evolution.- 3.1 Introduction.- 3.2 A Stochastic Model on Shocks in Urban Evolution.- 3.2.1 The Configurational Transition Rates.- 3.2.2 The Master Equation.- 3.2.3 The Individual Transition Rates.- 3.2.4 The Stationary Solution of the Master Equation.- 3.2.5 Equations of Motion for Mean Values and Variances.- 3.2.6 Estimation of the Utility Function and Mobility.- 3.2.7 Regression of Trend Parameters on Socio-Economic Data.- 4. Intra - Urban Migration.- 4.1 Introduction.- 4.2 A Stochastic Model on Intra-Urban Dynamics.- 4.2.1 The Configurational Transition Rates.- 4.2.2 The Master Equation.- 4.2.3 The Stationary Solution of the Master Equation.- 4.2.4 The Mean Value Equations.- 4.2.5 The Parameter Estimation Procedure.- 4.2.6 Empirical Testing of the Land Use Density-Rent Model.- 5. Inter-Regional Migration.- 5.1 Introduction.- 5.2 The Stochastic Migration Model.- 5.2.1 The Individual Transition Rates for the Migration Process.- 5.2.2 The Configurational Transition Rates.- 5.2.3 The Stochastic Equations of Motion.- 5.2.4 The Stationary Solution of the Migratory Master Equation.- 5.2.5 Quasi-Deterministic Equations of Motion.- 5.2.6 The Stationary Solution of the Quasi-Deterministic Equations.- 5.2.7 Determination of Utilities and Mobilities from Empirical Data.- 5.3 Comparative Analysis of Inter-Regional Migration.- 5.3.1 Choice of Comparable Socio-Economic Variables.- 5.3.2 The Global Mobility under Comparative Aspects.- 5.3.3 The Regional Utilities and Preferences of the Federal Republic of Germany.- 5.3.4 Comparison of the Variance of Utilities.- 5.3.5 Comparison of the Migratory Stress.- 6. Chaotic Evolution of Migratory Systems.- 6.1 Introduction.- 6.2 The Migratory Master Equation and Mean Value Equations for Interacting Populations.- 6.2.1 Inter-Group and Intra-Group Interactions of Individuals.- 6.2.2 The Master Equation for Interacting Subpopulations.- 6.2.3 The Deterministic Equations for Interacting Subpopulations.- 6.2.4 The Exact Stationary Solution of the Deterministic Equations.- 6.3 Chaotic Behaviour of Migratory Trajectories.- 6.3.1 A Numerical Simulation.- 6.3.2 Lyapunov Exponents and Fractal Dimensions.- 6.4 Conclusion.- 7. Spatial Interaction Models and their Micro-Foundation.- 7.1 Introduction to Spatial Urban Theory.- 7.2 A Service System as the Basis of the Model.- 7.3 A Master Equation Approach.- 7.3.1 The Total Transition Rates for the Service Sector Model.- 7.3.2 Consumer Dynamics.- 7.3.3 Decision Processes of Developers, Retailers and Land Owners.- 7.4 The Quasi-Deterministic Equations to the Dynamic Service Sector Model.- 7.5 The Stationary Solution of the Service Sector Model.- 7.5.1 Relation between Expenditure Flows and Transportation Costs.- 7.6 Dynamic Simulations and their Interpretation.- 7.6.1 The Influence of Low Transportation Costs.- 7.6.2 The Influence of High Transportation Costs.- 7.7 Concluding Comments.- 8. Further Applications and Extensions.- 8.1 Knowledge, Innovation, Productivity.- 8.1.1 Knowledge as an Endogeneous Input of the Growth Process.- 8.1.2 Regional Decisions about the Investment Ratio.- 8.1.3 Decision Processes Concerning the Percentage Share of Research Investment.- 8.1.4 Some Conjectures.- 8.2 Economic Cycles.- 8.2.1 Short-Term Cycles.- 8.2.2 Long-Term Cycles.- 8.3 Housing and Labour Market.- 8.4 Concluding Remarks.- 9. Appendix: The Master Equation.- 9.1 Deterministic and Probabilistic Description of Systems.- 9.2 Some General Concepts of Probability Theory.- 9.3 The Derivation of the Master Equation.- 9.4 The Stationary Solution of the Master Equation for Detailed Balance.- 9.5 The Stationary Solution of the Master Equation of Chapter 4.- 9.6 The Stationary Solution of the Master Equation of Chapter 5.- 9.7 The Embedding of Random Utility Theory.- 9.7.1 The Multinomial Logit Model.- 9.7.2 The Multinomial Logit Model as Limiting Case of our Dynamic Theory.- 9.8 The Construction of Configurational Transition Rates via Panel Data.- References.
TL;DR: In this paper, the authors investigated relaxation to equilibrium in the random energy model with the use of the master equation within the assumption that transition state between any two energy levels has the same energy.
Abstract: Relaxation to equilibrium in the random energy model is investigated with the use of the master equation within the assumption that transition state between any two energy levels has the same energy. Relaxation of physical quantities of the model after drastic decrease of temperature is a two-stage process when final temperature is lower than the temperature of freezing transition. At the first stage there occurs a logarithmic relaxation of the extensive part of energy to equilibrium (in the thermodynamic limit) value. The second stage is connected with the freezing transition.
TL;DR: In this article, the Langer theory of spinodal decomposition in binary solutions is extended to the case of ternary alloys, and a system of diffusion equations in the Cahn-Hilliard approximation and equations of motion for the three independent partial structure functions are derived.
TL;DR: In this paper, a semiclassical method of incorporating the effect of the hindered dipole rotation into an RRKM calculation of the microscopic rate coefficients k(E,J) was presented for unimolecular dissociation and recombination reactions.
Abstract: Two new techniques are presented for calculating the pressure dependence of ion/molecule association rates: (i) a semiclassical method of incorporating the effect of the hindered dipole rotation into an RRKM calculation of the microscopic rate coefficients k(E,J), and (ii) a solution for the master equation for unimolecular dissociation and recombination reactions, which incorporates angular momentum (J) conservation and is applicable when the moments of inertia of reactant and activated complex differ by a large amount. These techniques provide the optimal currently available means for calculating the pressure dependence of rate coefficients for ion–molecule reactions, which are highly sensitive to J‐conservation effects. The method may be used for reliable estimates and fitting of experimental fall‐off data. The new technique shows that Troe’s solution of the low‐pressure J‐conserving master equation is accurate for the nonequilibrium population distribution but overestimates (by up to a factor of 2) th...
TL;DR: In this article, it was shown that the intrinsic fluctuations in the N2O5 decay are larger than those implied by the master equation suggested by the macroscopic rate law.
Abstract: The master equation for a complex chemical reaction cannot always be reduced to a simpler master equation, even if there are fast and slow individual reaction steps. Nevertheless the elimination of intermediates can be carried out with the help of theΩ-expansion. This is illustrated with a well-known complex reaction: the dissociation of N2O5. It is shown that the intrinsic fluctuations in the N2O5 decay are larger than those implied by the master equation suggested by the macroscopic rate law.
TL;DR: In this paper, it is shown that when a ternary alloy with B2 order is characterized by two order parameters, the alloy may develop various combinations of the two-order parameters en route to a final equilibrium state.
TL;DR: In this article, it was shown that a binary alloy with an AB{sub 3} stoichiometry on a bcc lattice may develop various combinations of B2 and DO{ sub 3} order along its kinetic path toward equilibrium.
Abstract: It is shown that a binary alloy with an AB{sub 3} stoichiometry on a bcc lattice may develop various combinations of B2 and DO{sub 3} order along its kinetic path toward equilibrium. The temporal evolution of these two order parameters is analyzed with an activated-state rate theory. Individual vacancy jumps are treated in a master equation formalism that involves first-nearest-neighbor (1nn) and second-nearest-neighbor (2nn) interactions. In our formulation, a set of coupled differential equations is obtained describing the time-dependence of six order parameters. These equations were integrated numerically for a variety of interatomic interactions and initial conditions. It was found that the {ital relative} rates of B2 and DO{sub 3} ordering, and hence the path of the alloy through the space spanned by the B2 and DO{sub 3} order parameters, depend on the relative strengths of the interatomic interaction potentials and on the temperature. For very strong 2nn interactions, a transient B32 structure was observed to develop at early times, although this phase disappeared as equilibrium was approached.
TL;DR: In this paper, the master equation for the poloidal spectrum of electromagnetic energy in a toroidal plasma is derived, which provides a theoretical framework for a global description of the propagation and absorption of lower hybrid waves (or any weakly damped small wavelength mode) in present-day Tokamaks.
Abstract: The master equation for the poloidal spectrum of electromagnetic energy in a toroidal plasma is derived. This equation provides a theoretical framework for a global description of the propagation and absorption of lower hybrid waves (or any weakly damped small wavelength mode) in present-day Tokamaks. Resonant toroidal couplings between the unperturbed cylindrical modes are the basic processes involved in this description. They lead to the destruction of correlations above a stochasticity threshold for the toroidal perturbation. A general method for computing the coupling coefficients is proposed and applied under typical current generation conditions. This normal mode energy diffusion provides a simple theory to address the so-called 'spectral gap' problem.
TL;DR: A stochastic theory for the formation of planar-atomic clusters by atomic diffusion is developed and an application is given by comparing the results of calculations with experimental data on interstitial-loop formation in ion-irradiated nickel.
Abstract: Atomic clustering into circular planar disks is an important process responsible for interstitial- loop formation in the bulk of irradiated materials, and the evolution of atomic planes during thin-film growth. We develop a stochastic theory for the formation of planar-atomic clusters by atomic diffusion. The theory accounts for the transient coupling between master equations representing small-size atomic clusters and a Fokker-Planck (FP) equation for larger ones. The FP equation is solved self-consistently, together with the master equations by the moments method. Equations for the rates of change of atomic species and for the nucleation rate of atomic clusters are simultaneously solved with appropriate equations for the average size and various moments of the distribution function. An application of the theory is given by comparing the results of calculations with experimental data on interstitial-loop formation in ion-irradiated nickel.
TL;DR: In this article, a unified dynamical theory of rate processes such as electron transfer in solution, which interpolates between the nonadiabatic and the adiabatic limits, is presented.
Abstract: A unified dynamical theory of rate processes such as electron transfer in solution, which interpolates between the nonadiabatic and the adiabatic limits, is presented. The theory is based on expanding the rate perturbatively to fourth order in the nonadiabatic coupling V using the density matrix in Liouville space and performing a partial resummation. The present theory establishes a profound connection between rate theories and nonlinear optical spectroscopies. The rate to order is related to linear optics and the linear susceptibility x('). The rate to order is related to the third-order susceptibility x(~). This connection arises since the same dephasing mechanisms which affect the optical line shapes also control the dynamics of rate processes. The frequency-dependent reaction rate is calculated and the dielectric continuum model for polar solvation is extended to incorporate the microscopic solvation structure via the wave vector and frequency dependent dielectric function c(k,w). In this article we discuss some recent theoretical developmentsI4 which establish a general and fundamental connection between rate proce~ses~'~ and quantities being probed by nonlinear optical techniques.'s-21 This connection arises since the same solvation dynamics underlying reaction rates such as electron transfer is also responsible for dephasing processes which affect spectral line shapes (e.g., absorption, fluorescence, pump probe, and four-wave mixing). Information obtained in optical measurements such as femtosecond spectroscopy may therefore be used to predict reaction rates. The present theory is based on the use of projection operator techniques in Liouville space.'Z-% Bob Zwanzig was instrumental in developing these techniques and in pioneering the use of Liouville space (superoperator) methods in condensed-phase molecular dynamics. Our earlier resultsI4 are extended in this article in two ways. First we present a closed expression for the full frequency-dependent reaction rate (eq (11-1 2)). This expression allows us to define precisely the transition state even when simple rate equations do not hold and we need to use a generalized master equation instead. It is clearly shown how the transition state then occupies a larger volume in phase space. We further apply our rate theory to electron transfer (ET) in polar solvents and relate the solvent dynamics in ET processes to the complete wave vector and frequency-dependent dielectric function of the solvent c(k,o) (eq IV-7). This provides a natural extension of the conventional dielectric continuum theories.
TL;DR: In this paper, the age-state probability density obeys a system of age-dependent master equations with time and agedependent transition rates, and an alternative approach based on Haken's slaving principle is developed.
Abstract: Physical systems described by fast and slow relaxing variables are considered. The underlying microscopic equations are assumed to be Markovian. Considering that the macroscopic state of the system is characterized by the set of slow relaxing variables we show that the age-state probability density obeys a system of age-dependent master equations with time and age-dependent transition rates. If the time scales attached to the slow and fast relaxing variables are well differentiated there is a strong correlation between the age of a fluctuation state and the corresponding number of transition events. An alternative approach based on Haken's slaving principle is developed. For time homogeneous microscopic equations the adiabatic elimination of fast relaxing variables leads to a system of age-dependent master equations with time independent and age-dependent transition rates. These equations are equivalent to a generalized master equation with an inhomogeneous term.
TL;DR: A quantum theory of a two-level single-mode laser with injected atomic coherence is developed by generalizing the Scully-Lamb laser theory to a form appropriate for the analysis of a coherently pumped laser.
Abstract: A quantum theory of a two-level single-mode laser with injected atomic coherence is developed by generalizing the Scully-Lamb laser theory to a form appropriate for the analysis of a coherently pumped laser. We assume that the active atoms are prepared initially in a coherent superposition of the upper and lower levels, and we derive the master equation for the field density operator by treating the interaction of the laser field with many active atoms simultaneously. It is shown that the photon-number distribution can be exactly Poissonian. The laser operation is analyzed in terms of the Fokker-Planck equation for the laser field. Both the intensity and phase diffusion coefficients are phase sensitive and, for stable laser operation, become much smaller than those of an ordinary laser. Consequently, the injected atomic coherence reduces both the photon-number noise and phase noise simultaneously. The intensity diffusion coefficient can vanish exactly, and at the same time the phase diffusion coefficient can become very small. This leads to spontaneous-emission noise quenching in the photon-number distribution, and the laser field can become very close to a coherent state. A scheme to generate the proper form of the initial atomic coherence necessary for the quantum noise quenching is proposed and analyzed.
01 Jan 1989
TL;DR: In this article, the authors generalize the Kramers theory to multi-dimensional diffusion equations with particular emphasis on the frictional coupling between reactive and non-reactive modes of the potential function.
Abstract: Discrete master equations can be obtained from diffusion-Smoluchowski equation in the presence of large barriers separating the potential minima, the treatment being equivalent to the derivation of the Kramers transition rates in the overdamped regime. The one-dimensional problem is considered as a test case to illustrate the projection procedure onto the subspace of localized functions. This method, however, generalizes the Kramers theory to intermediate potential barriers. The inertial effects are shortly discussed in relation to the numerical solutions for a bistable problem. The coupling between the overall rotation and the conformational transitions is analyzed in a molecule with one torsional degree of freedom. The generalization of the Kramers theory to multi-dimensional diffusion equations is presented with particular emphasis on the frictional coupling between reactive and non-reactive modes of the potential function. The model system of a linear chain of rotors is used to demonstrate that cooperative transitions during saddle point crossing arise as a consequence of the frictional coupling. The parametrization of the transition rates for an alkyl chain attached to a rigid core is summarized, together with the main results concerning the relaxation of conformer populations and the methylene rotational relaxation.
TL;DR: In this article, the relative rates of change of B2 and DO3 order parameters were temperature dependent; hence at different temperatures the alloy passed through different states of order en route to thermal equilibrium.
Abstract: Rapidly quenched powders of Fe3Al were subjected to thermal annealings at temperatures well below the critical temperatures for B2 and DO3 ordering. X-ray diffractometry was used to measure the subsequent evolution of B2 and DO3 long-range order. It was found that the relative rates of change of B2 and DO3 order parameters were temperature dependent; hence at different temperatures the alloy passed through different states of order en route to thermal equilibrium. These temperature dependences of “kinetic paths” can be understood in terms of a theory of kinetic paths based on the kinetic master equation. The theory indicates that the temperature dependence of the observed kinetic paths originates from having first-nearest-neighbor interactions that are stronger than second-nearest-neighbor interactions. This seems consistent with previous thermodynamic analyses of critical temperatures of Fe3Al.
TL;DR: A comparative study of various quasiprobability distributions in phase-sensitive quantum-optical systems and an application of these distributions to the two-photon correlated-emission laser shows that the intracavity field can be near-perfectly squeezed in the phase quadratures and the maximum quadrature squeezing is reached when the mean laser amplitude vanishes.
Abstract: We make a comparative study of various quasiprobability distributions in phase-sensitive quantum-optical systems. Starting from a general, linear master equation for the field, which emerges in different models of correlated-emission lasers, we derive the Fokker-Planck equations in the CJlauber-Sudarshan P, the antinormal ordering Q, the Wigner W, the complex P, and the positive P representations and find the steady-state solutions for the five distributions. Simple relations between the complex and positive P functions are discovered for the first time. Various moments calculated by using these distributions are found to be identical, as expected. An application of these distributions to the two-photon correlated-emission laser shows that the intracavity field can be near-perfectly squeezed in the phase quadrature and the maximum quadrature squeezing is reached when the mean laser amplitude vanishes. I. INTRODUCTION Recently, several mechanisms for the correlatedemission laser (CEL) have been considered. ' The correlated emission is based on using atoms prepared in a coherent superposition of the states between which the laser emission takes place. The initial atomic coherence can lead to the reduction in either phase or amplitude noise. It can even lead to the squeezing in one of the quadratures of the field. The microscopic theories of the (single-mode) CEL show that the dynamical equation for the density matrix for the field mode a can be written in the form'