Showing papers on "Master equation published in 1974"

Markovian master equations

Abstract: We give a rigorous proof that under certain technical conditions the memory effects in a quantum-mechanical master equation become negligible in the weak coupling limit. This is sufficient to show that a number of open systems obey an exponential decay law in the weak coupling limit for a rescaled time variable. The theory is applied to a fairly general finite dimensional system weakly coupled to an infinite free heat bath.

Asymptotic solutions of continuous-time random walks

Abstract: The continuous-time random walk of Montroll and Weiss has a complete separation of time (how long a walker will remain at a site) and space (how far a walker will jump when it leaves a site). The time part is completely described by a pausing time distributionψ(t). This paper relates the asymptotic time behavior of the probability of being at sitel at timet to the asymptotic behavior ofψ(t). Two classes of behavior are discussed in detail. The first is the familiar Gaussian diffusion packet which occurs, in general, when at least the first two moments ofψ(t) exist; the other occurs whenψ(t) falls off so slowly that all of its moments are infinite. Other types of possible behavior are mentioned. The relationship of this work to solutions of a generalized master equation and to transient photocurrents in certain amorphous semiconductors and organic materials is discussed.

Generalized-master-equation theory of excitation transfer

Abstract: A theory of the time dependence of resonance transfer of excitation energy between molecules is developed in terms of memory functions appearing in the transition rates of a generalized master equation (GME). The memory can be computed explicitly and, due to the coarse-graining operation incorporated in our derivation of the GME, the accuracy of the memory function depends only on the amount of detailed information one has, or wishes to include, about the spectrum and dynamics of the system. The formalism yields a unified description of coherent motion at short times and diffusive transport at long times, and for the case of transfer between and among identical molecules provides a generalized approach to the theory of exciton transport. Memory functions for transfer between anthracene molecules are obtained as an illustration of the theory. The connection between the new formalism and existing exciton-transport theories is indicated and its relation to the theory of non-Markoffian random walks is presented.

Investigation of metastable states and nucleation in the kinetic Ising model

Abstract: The relaxation of a two-dimensional Ising ferromagnet after a sudden reversal of the applied magnetic field is studied from various points of view, including nucleation theories, computer experiments, and a scaling theory, to provide a description for the metastable states and the kinetics of the magnetization reversal. Metastable states are characterized by a "flatness" property of the relaxation function. The Monte Carlo method is used to simulate the relaxation process for finite $L\ifmmode\times\else\texttimes\fi{}L$ square lattices ($L=55, 110, 220 \mathrm{and} 440, \mathrm{respectively}$); no dependence on $L$ is found for these systems in the range of magnetic fields calculated. The metastable states found for small enough fields terminate at a rather well-defined "coercive field," where no singular behavior of the susceptibility can be detected, within the accuracy of the numerical calculation. In order to explain these results an approximate theory of cluster dynamics is derived from the master equation, and Fisher's static-cluster model, generalizing the more conventional nucleation theories. It is shown that the properties of the metastable states (including their lifetimes) derived from this treatment are quite consistent with the numerical data, although the details of the dynamics of cluster distributions are somewhat different. This treatment contradicts the mean-field theory and other extrapolations, predicting the existence of a spinodal curve. In order to elucidate the possible analytic behavior of the coercive field we discuss a generalization of the scaling equation of state, which includes the metastable states in agreement with our data.

Time-dependent projection-operator approach to master equations for coupled systems

Abstract: In this paper we derive master equations for two or more systems coupled to each other, perhaps strongly, by using a generalization of the usual projection-operator technique to include time-dependent projection operators. The coupled systems may be either similar or dissimilar and classical or quantum mechanical. Whereas the customary approaches to coupled systems are best able to treat situations in which some of the systems are "baths" with a specified density operator or phase-space probability density, our approach allows us to treat situations where it is necessary or convenient to treat the coupled systems on an equal footing. In our scheme the "relevant" part of the full density operator is considered to be the uncorrelated part of the full density operator and is a symmetric functional of the reduced density operators of each of the coupled subsystems. The "irrelevant" part of the density operator is then the part describing correlations between the coupled systems. Our formalism is particularly useful where systems are coupled to one another predominantly in a self-consistent fashion. First, we develop exact master equations for two coupled systems, taking as our prototype the dynamical problem of quantum optics, where a spatially extended collection of two-level atoms interact with a multimode optical field. We then generalize our results to $N$ coupled systems, taking as our prototype the kinetics of a classical nonideal gas interacting through two-body forces, and derive exact master equations for the system. We then consider as examples several approximate theories resulting from our exact equations. In the case of the imperfect gas we investigate the low-density limit and show how Bogoliubov's form of the Boltzmann equation emerges from our formalism, as well as corrections due to Klimontovich. We consider as special cases of our exact quantum-optical equations the equations in the first Born approximation, with and without memory, and show how several existing quantum-optical master equations are contained in our general results. As a second example in quantum optics, we consider the case where the predominant behavior of the system is described by the self-consistent-field or coupled Bloch and Maxwell equations and derive a first-order perturbation description for deviations from self-consistent-field behavior.

Stochastic models of the interconversion of three or more chemical species

Abstract: The phenomenological rate equations for reactions in which three or more chemical species are simultaneously interconverting are derived from a microscopic stochastic model. Particular attention is focused on the establishment of long chemical relaxation times, and on an important orthogonality property which guarantees that the principle of detailed balancing is obeyed. By developing a quantum mechanical analog, the mathematical origins of both of the above properties are related to a resonance phenomenon associated with three or more wells separated by high energy barriers. The quantum analog is itself equivalent to a stochastic master equation, the rate constants of which are analytically determined. These are shown to contain the expected Arrhenius factors and to obey the principle of the independent coexistence of reactions.

On the master equation approach of vibrational relaxation in condensed media

Abstract: The purpose of this paper has been to examine the validity and limitations of the Pauli master equation and to demonstrate the use of the master equation approach in describing the vibrational relaxation in the condensed phase. The master equations for vibrational relaxation of a number of model systems have been derived. The calculation of rate constants of vibrational relaxation has been discussed. The solution of the master equations of multimode relaxation at low temperatures has been presented. The Brownian motion of an oscillator linearly coupled with the medium has been investigated and a model of the coupling between the vibrational relaxation and other unimolecular processes (like electronic relaxation, etc.) has been developed.

A master equation approach to nonlinear optics

Abstract: The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.

Nonequilibrium phase transitions in chemical reactions

Abstract: The far from equilibrium steady states of a simple nonlinear chemical system are analyzed. A standard macroscopic analysis shows that the nonlinearity introduces an instability which causes a transition analogous to a thermodynamic second-order phase transition. Fluctuations are introduced into this model through a stochastic master equation approach. The solution of this master equation in the steady state reveals that the system goes into a more ordered state above the transition point. An analogy is drawn with the nonequilibrium phase transition occurring in the laser at threshold.

A Markovian random coupling model for turbulence

Abstract: The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.

Theory of vibrational relaxation in isolated molecules

Abstract: A quantum mechanical theory of intramolecular energy transfer is presented which treats the approach to statistical equilibrium of the vibrations in isolated molecules. The theory is appropriate for nonreactive molecules which have at least 6–9 atoms and which are so highly excited that the vibrational interaction couples together many degrees of freedom at a time and causes the simultaneous exchange of a large number of quanta. A generalized master equation of the Van Hove type and a weak‐coupling master equation of the Pauli type are obtained for functions related to occupation probabilities of zero‐order states. The asymptotic behavior of the coarse‐grained occupation probabilities is examined and molecular ergodicity is proved. Convergent infinite expansions for the probabilities are also derived. These analytical expressions make it possible to study the intramolecular dynamics for all relevant times.

Exact diffusion equation for a model for superradiant emission

Abstract: The super-radiant master equation (SME) of Bonifacio et al. is analyzed using the coherent-atomic-state representation. We have succeeded in deriving an exact Fokker-Planck equation for the density function corresponding to the reduced atomic density operator in the diagonal atomic-state representation. A solution to the Fokker-Planck equation has been provided in an elementary fashion for arbitrary atomic states which are sufficiently removed from the state of complete inversion at time zero. The general solution for arbitrary initial conditions (including the initial state of complete inversion) has been obtained using the method of eigen-function expansions and the final result expressed in terms of an integral over the initial density function. The moments of the collective atomic operators are also discussed.

Master-equation approach to stochastic models of crystal growth

Abstract: A master-equation approach is formulated to obtain kinetic equations for stochastic models of the crystal-vapor interface. The properties of two Ising-type models are studied in stable and metastable states. General transition probabilities for the adsorption and evaporation of atoms at the interface are introduced, which may account for different types of dynamic behavior. Marked dependence of the interface kinetics upon the details of the transition probabilities is found, in contrast to the case of homogeneous systems.

Rotational relaxation of molecules in isotropic and anisotropic fluids

Abstract: A theory of rotational relaxation in isotropic and anisotropic liquids is presented. The Debye rotational diffusion model is generalized so as to include reorientations of arbitrary angle with the use of a nonlocal in orientation master equation for the orientational conditional probability. For isotropic media, we have previously demonstrated that spectral line shapes (Fourier transforms of time correlation functions) appropriate to, for example, Raman and ir line broadening spectroscopy, are always superpositions of Lorentzian lines. We present here an algebraic formulation which gives the linewidths of the Lorentzian lines in terms of the transition probability describing the reorientational motion. Several models and general trends for these linewidths are discussed in order to facilitate the comparison of experimental results and this theory. For anisotropic media, such as liquid crystals or small molecules ordered by liquid crystals, a nonlocal reorientational mechanism leads to a continuous spectru...

A comment on fluctuations around nonequilibrium steady states

Abstract: We study fluctuations around nonequilibrium steady states of some model nonlinear chemical systems. A previous result of Nicolis and Prigogine states that the mean square fluctuation computed from a master equation in the space of internal states of the reacting species is identical to that calculated from Einstein's fluctuation formula. Our analysis of fluctuations based on that master equation leads with the assumption of local equilibrium to a result identical to that obtained from a master equation for the total concentration of the reacting species, which is different from the equilibrium (Einstein relation) result. Nicolis and Prigogine approximated one term in their master equation, and a discussion of this approximation is presented. The master equation without this approximation yields at equilibrium the result expected on the basis of Einstein's formula.

Theory of intramolecular vibrational relaxation in large systems

Abstract: A general theory for intramolecular vibrational relaxation is proposed. The theory provides a master equation, analogous to the Pauli master equation in non-equilibrium statistical mechanics, for this process. The master equation contains a completely general time independent perturbation operator, which causes the system to relax to equilibrium. The choice of the three-phonon interaction operator as a perturbation is treated in detail, and a comparison with experimental data is made.

Stochastic Processes in Self-Excited Oscillation

Abstract: The amplitude and phase fluctuations in self-excited oscillation are studied phenomenologically using a van der Pol type stochastic equation. The second-order differential equation is converted into two first-order differential equations for the amplitude and the phase and their probability distributions are derived from the associated Fokker-Planck equations. The probability distribution of the amplitude around its most probable value is anomalously broadened and non-Gaussian near threshold of oscillation. The fluctuation in the oscillatory state is rather quasi-Gaussian. The growth of the time coherence of phase above threshold is accounted for to be due to a decrease in an apparent diffusion coefficient for phase fluctuation

Dynamics of Macrovariables for Nonuniform Systems --Scaling Theory--

Abstract: A scale transformation of the nonequilibrium macroscopic system to larger similar sys­ tems is introduced to find kinetic equations for the evolution and fluctuation of the macro­ variables. In the scale transformation, we postulate that the probability distribution for the fluctuation of the macroscopic degrees of freedom and the quantities determined by the micro­ scopic degrees of freedom per unit volume are invariant. The characteristic length of the macroscopic state l, the macroscopic state variables Yand their fluctuation variables z. are transformed by h=Ll, (L):-1), YL=L-"yand z.L=L-Pz., respectively. The probability dis­ . tribution then takes the form P( {z;,lP}, {ql}, SJjl", t/l'), where q, SJ, d and t denote the wave vectors, volume, dimensionality and time, respectively. If a being the frequency. § I. Introduction Macroscopic systems have characteristic properties which do not appear in systems of small numbers of degrees of freedom. The central limit theorem and the phase transitions are outstanding examples. In a previous paper/> we have proposed a general type of kinetic equations from the statistical-mechanical point of view. In this paper we shall explore the most dominant features of macro­ scopic systems by introducing a new method of asymptotic evaluation for large systems and deriving the asymptotic form of the kinetic equations. A similar attempt has been jD-ade by van Kampen2> and by Kubo, Kitahara and Matsuo 3> for uniform systems by the use of the Kramers-Moyal expansion of the master. equation. They extended the central limit theorem in the form of the system-size expansion of the master equation, and showed that, for large values of the system size !2, the master equation is reduced to a linear Fokker-Planck equation and the probability distribution of the macrovariables is normal or Gaus­ sian around their mean evolution. These works, however, are limited to the uni­ form disordered systems which are described by a small number of macrovariables.

Kinetic modeling of CW CO electric-discharge lasers

Abstract: A rate equation analysis of convective CO electric-discharge lasers (EDL's) is presented. The spatial distribution of CO vibrational levels is governed by a kinetic master equation which contains terms representing electron impact pumping, vibration-vibration (VV) and vibration-translation (VT) energy transfer, and spontaneous and stimulated emission. Electronic pumping rates are derived from the solution to the electron Boltzmann equation. Spectral distributions of small-signal gain and optical flux are calculated in terms of the cross sections for these processes. Good agreement is obtained between predicted and experimentally reported gain and power spectra. For coincident discharge/cavity configurations, the quasi-steady small-signal gain is predicted to have a strong dependence on translational temperature for representative operating conditions. The predicted power output for a wide variety of bulk gas and plasma properties is found to be correlated by a simple energy-time parameter. The performance of other devices which involve separate discharge and cavity regions is also examined. The importance of the threshold excitation energy in understanding a variety of configurations is discussed.

Kinetics of Order-Disorder

Abstract: Various theoretical models of order-disorder kinetics are presented from a unified point of view The time evolution of both single-site and pair-site probabilities are derived from a single master equation for the time dependence of configuration probabilities in binary solid solutions Linearized diffusion equations are solved in the Fourier representation and theoretical predictions are compared to experimental results of disordering kinetics in binary and ternary solid solutions A nonlinear equation for long-range order kinetics is also derived from the master equation, and compared to the classical theories of Dienes and of Vineyard, and to available experimental data The phenomenon of critical slowing down and the kinetics of short-range order are briefly covered

Fluctuations in chemical reactions around the steady states

Abstract: Birth‐and‐death type master equations for linear chemical reactions in closed and open systems are solved to discuss the properties of fluctuations. For nonlinear chemical reactions the generating functions of the probability of number fluctuation are usually second order differential equations, and a method for obtaining the moments of fluctuations is presented with applications to simple examples. The deterministic kinetic equation is valid for large systems and the second moments are the same as those of Poisson distributions in open systems treated here, while for the higher moments this is not always the case.

On the relaxation to quantum‐statistical equilibrium of the Wigner‐Weisskopf atom in a one‐dimensional radiation field. VI. Influence of the coupling function on the dynamics

Abstract: An investigation is made of the effect on the dynamics of spontaneous emission of the Wigner‐Weisskopf atom in an infinite‐system limit of various choices of the coupling function or form factor describing the atom's interaction with the spectrum of the radiation field This is carried out both for the exact solution to the problem of spontaneous emission, obtained in the earlier papers in the series, and for some approximate solutions, also previously considered, in particular one based on the Schrodinger equation of the problem and one based on the weak‐coupling Prigogine‐Resibois master equation The details of the form factor are found, by numerical computation of the solutions, to be critical in determining the nonexponential parts of the solutions, and these parts are seen to be capable in some cases of dominating the exponential parts, which are given only by the values of the form factor near the resonance energy The approximate solutions discussed are found to vary widely in their worth, and one, which yields the exact solution for the Wigner‐Weisskopf problem, is singled out as being of probable use in the statistical‐mechanical description of more complicated systems