Showing papers on "Master equation published in 1980"

Generalised P-representations in quantum optics

Abstract: A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.

Derivation of the Continuous-Time Random-Walk Equation

Abstract: The transport of electrons or excitations on a lattice randomly occupied by guests is considered. The equation governing the transport in any configuration is assumed to be the master equation. A projection operator technique is used to derive the exact equation governing the transport averaged over all configurations, which can be written as either a generalized master equation or the continuous-time random-walk equations (CTRW), establishing the correctness of the CTRW for these problems.

On electronic energy transfer in disordered systems

Abstract: A generalized master equation (GME) describing the incoherent motion of an excitation in a disordered system is developed. The connection of the GME to the semi‐Markovian theory of Scher and Lax, the generalized continuous random walk, and the self‐energy approaches to the temporal properties of the transport is discussed. The theory is used in a model calculation to compute the mean square displacement and the probability of the excitation to remain at the origin as one dimensional systems, in which only nearest neighbor interactions are included.

Infrared multiphoton decomposition: A comparison of approximate models and exact solutions of the energy-grained master equation

Abstract: The approximate ’’thermal’’ model and ’’continuum’’ model have been compared to exact calculations, and neither gives satisfactory results. In particular, the thermal model is based on an inaccurate and unphysical approximation to the correct molecular density of states, as well as a restrictive expression for the absorption cross sections. As an alternative to the approximate models, the exact stochastic method gives exact results and can be implemented using small computers or programmable pocket calculators. The results obtained from this method are exact and precision depends only upon the number of ’’trajectories’’ calculated. It was discovered that the yield versus fluence results from exact numerical integration of the master equation approximately conform to a cumulative log–normal distribution function in decomposition time or fluence. Thus, to a good degree of approximation, computed yield versus fluence results can be expressed simply in terms of a mean and a dispersion parameter. This suggests that the use of probability graph paper and the log–normal distribution function may prove to be a useful means for the presentation and analysis of experimental and computed data. Moreover, deviation from the log–normal distribution may indicate that the simple master equation is not a sufficient description of the experimental results.

Theory of rate processes in condensed media

Abstract: I Basic Quantum Mechanical Equipment.- 1.1 Basic concepts of quantum theory.- 1.2 Two simple quantum-mechanical systems.- 1.3 Transition probability per unit time. Master equations. General properties of irreversible motion.- 1.4 The Born-Oppenheimer approximation. Phonons. Phonon-phonon interaction and relaxation.- II Basic Processes and Models.- 2.1 Adiabatic and non-adiabatic transitions.- 2.2 Adiabatic transitions. Tunneling..- 2.3 Non-adiabatic transitions. Landau-Zener transition.- 2.4 Electron transfer.- 2.5 Energy transfer.- III The Equation of Motion.- 3.1 Derivation of master equations from von Neumann equation.- 3.2 Equations for density matrix of dynamic subsystem.- 3.3 Master equations for dynamic subsystem.- IV Calculation of Rate Coefficients in Various Temperature Regions.- 4.1 Equation of motion describing electron transfer.- 4.2 Calculation of rate coefficients by the saddle-point method.- 4.3 Integral equation for non-radiative transition rates.- 4.4 Symmetric case of zero energy gap. Temperature dependence.- V Equations of Motion at Zero and Low Temperature Range.- 5.1 Equation of motion in the case when the interaction energy V is not a small parameter.- 5.2 Validity conditions of perturbative approach.- VI Adiabatic Rate Processes.- 6.1 Tunneling in condensed media.- 6.2 Overcoming the potential barrier.- 6.3 General features of adiabatic rate processes.- VII Competition Between Electronic and Vibrational Relaxations.- 7.1 Master equations describing competition between electronic and vibrational relaxations.- 7.2 Master equation approach to non-adiabatic rate processes.- VIII Concluding Remarks.- 8.1 General quantum mechanical analysis.- 8.2 Hierarchy of equations of motion.- 8.3 Models.- References.

Electronic energy transport in substitutionally disordered molecular crystals

Abstract: In this paper we present a theoretical study of incoherent electronic energy transfer (EET) in an impurity band of substitutionally disordered, mixed, molecular crystals. Dispersive diffusion of the electronic excitation was treated by an Average Dyson Equation Approximation (ADEA) to the master equation for EET. The ADEA rests on expressing the Green’s function (GF) for the probability of site‐excitations in a single fixed spatial configuration in terms of a Dyson equation with a normalized vertex function and subsequently performing the configurational average of the GF, invoking a decoupling approximation which omits many‐site contributions. Explicit expressions were derived for the initial site occupation probability, Po(t), the mean square displacement, Σ2(t), and the time dependent diffusion coefficient D(t). We have explored the relation between the ADEA and the stochastic continuous time random walk (CTRW) model applied by us for EET. We have demonstrated that the ADEA and the CTRW results for Σ2(...

Kinetic theory of hopping transport

Abstract: Localized states are characterized kinetically (via rate constants) and separated into transport states and traps according to whether the states actually enable transport (mobilize a particle) or disable transport (or act as temporary immobilizers). A technique is developed for effecting the separation uniquely. It is based on the principle that the mobility of a particle should be maximum under trap-free conditions. An alternative percolation technique is also described which is believed to produce the same result. Either technique is general and can be applied to any array of states linked by a set of transition probabilities, such as those which define the Pauli master equation (PME). The separated states and certain approximations then make it possible to transform the PME into a generalized form of trapping equations, which reduce to the conventional trapping equations for extended-state transport in the limit in which the mean hopping time between transport states tends to zero. The genera...

The dependence of rate coefficients and product yields upon fluence, intensity, and time in unimolecular reactions induced by monochromatic infrared radiation

Abstract: The influence of the three parameters (with two degrees of freedom) fluence, intensity, and time on rate coefficients and product yields in collisionless Unimolecular Reactions Induced by Monochromatic Infrared Radiation (URIMIR) is discussed in some detail in terms of the recently proposed logarithmic reactant fluence plots. Model calculations for several archetypes of such plots are presented, based on solutions of the Pauli master equation and solutions of the quantum mechanical equations of motion for spectra involving many states at each level of excitation. Linear diagrams, turnups, and turnovers are found and are discussed systematically. Experimental examples re‐evaluated from the literature and new measurements on the laser induced decomposition of CF2HCl are reported which nicely illustrate the various theoretical possibilities. Steady state rate coefficients for six molecules are evaluated and summarized. In some situations the intrinsic nonlinear intensity dependence of the steady state rate coefficients and deviations from simple fluence dependence of the product yields both before and at steady state are shown to be important theoretically and experimentally. The role of the reducibility of the rate coefficient matrix is discussed in connection with turnovers and with the strong influence of initial temperature that is found in the laser induced decomposition of CF2HCl.

Analytical and numerical results for the steady state in cooperative resonance fluorescence

Abstract: The cooperative resonance fluorescence steady state is discussed within the context of an operator master equation which conserves total pseudospin. Emphasis throughout is on quantum fluctuations and their significance in relation to a background of factorised dynamics. Atom-atom correlations are shown to play a fundamental role for systems driven beyond the linear regime. Use of the atomic coherent state representation yields a Fokker-Planck description closely allied to the dynamics for a classical angular momentum oscillator. For intense incident fields the quantum-mechanical steady state is understood in terms of diffusion both around and between classical trajectories on the Bloch sphere. In the limit of infinite systems simple closed-form expressions for steady-state features are derived. Coherent and incoherent fluorescent intensities are obtained together with the second-order correlation function for fluorescent light. Specific features are illustrated by numerical results for systems of from two to fifty atoms.

Langevin description of markovian integro-differential master equations

Abstract: For a given master equation of a discontinuous irreversible Markov process, we present the derivation of stochastically equivalent Langevin equations in which the noise is either multiplicative white generalized Poisson noise or a spectrum of multiplicative white Poisson noise. In order to achieve this goal, we introduce two new stochastic integrals of the Ito type, which provide the corresponding interpretation of the Langevin equations. The relationship with other definitions for stochastic integrals is discussed. The results are elucidated by two examples of integro-master equations describing nonlinear relaxation.

Impulsive stochastic models of molecular relaxation and isomerization reactions

Abstract: We discuss several aspects of the problem of modeling the effect of collisions on isomerization reactions. We show how to generate stochastic dynamics for systems which suffer more than one type of collision, and derive the appropriate master equation for the phase space distribution function. We discuss a variety of different collision models and sketch how one might deduce the form and parameters of the model from experiment and molecular dynamics. Finally, we examine a one‐dimensional model of an isomerization reaction in the presence of two types of collisions, finding that the reaction rate varies nonlinearly with the various collision rates.

Systematic analysis of the multivariate master equation for a reaction-diffusion system

Abstract: The multivariate master equation for a reaction-diffusion system is analyzed using a singular perturbation approach. It is shown that in the vicinity of a bifurcation leading to two simultaneously stable steady states, the steady-state probability distribution reduces asymptotically to the exponential of the Landau-Ginzburg functional. On the other hand, for a system displaying quadratic nonlinearities and an absorbing state, critical behavior is ruled out.

Process with delta-correlated cumulants

Abstract: In a recent paper1) a differential equation was studied which involves a stochastic process having the property that all its cumulants are delta-correlated. It is here shown that such processes consist of a random sequence of delta functions with random coefficients. As a consequence the solutions of the differential equation are Markov processes, whose master equation can be constructed. From it closed equations for the successive moments may be obtained, and the auto-correlation is determined, in agreement with the results of reference 1. Some generalizations are given in Appendices B and C.

The frequency of cyclic processes in biological multistate systems.

Abstract: Cyclic processes in stochastic models of macromolecular biological systems are considered. The diagram solution of the model equations (master equation) gives rise to special functions of the rate constants, called the circuit (or one-way cycle) fluxes. As Hill has shown, these functions are the fundamental theoretical components of the operational fluxes, i.e., of the rates of reaction, of transport, of energy conversion, etc. Evidence recently has been found by Monte Carlo simulations that the circuit fluxes can be interpreted as the frequencies of circuit completions. Making use of the theory of graphs, we prove that this physical interpretation of the circuit fluxes is generally valid.

Asymptotic solution of the master equation near a nonequilibrium transition: The stationary solutions

Abstract: The time evolution for the logarithm of the probability as derived from the master equation, is expanded in terms of the inverse of the size of the system. The first term of the expansion yields the Hamilton-Jacobi equation studied by Kubo et al. 7 ). The second term, which plays a fundamental role in the description of transition phenomena, is retained. In this way, the exact form of the stationary solution for a Schlogl type model is recovered. Taylor expansions around extrema yield approximate stationary solutions both at the transition point as well as above it.

Approximating the master equation by Fokker–Planck‐type equations for single‐variable chemical systems

Abstract: The stochastic evolution of a well stirred chemically reacting system containing a single time‐varying species X is accurately described by the master equation, in which the total number x of X molecules is an integer variable. We investigate here the legitimacy of approximating the master equation by a Fokker–Planck type partial differential equation in which x is treated as a real variable. Taking the position that any partial differential equation may be regarded as a legitimate approximation to the master equation if and only if it reduces to the master equation when subjected to a proper discretization procedure, we deduce the following: For the special case in which the various chemical reactions can alter the X molecule population by no more than one molecule at a time, a second order (two term) Fokker–Planck equation suffices. However, for the more general case in which reactions are allowed that alter the X molecule population by two molecules at a time, it is necessary to use at least a fourth o...

Fluctuations in laser theories

Abstract: The influence of fluctuations on two models relevant in solid-state laser theory is discussed. One model is the usual solid-state laser model, the other describes a laser with an unstable saturable absorber. For the usual solid-state laser it is shown that fluctuations act as singular perturbations of a bifurcation. Also derived is a simplified master equation whose stationary solution is in near-perfect agreement with the known exact stationary properties of the laser. For the laser with absorber the fluctuations are shown to act as singular perturbations for some solutions and singular destruction for other solutions. A simplified master equation is also derived for this model, and its stationary solution is compared with the known exact stationary distribution. Here too very good agreement is found whenever comparison is possible.

Critical Dynamics: The Expansion of the Master Equation Including a Critical Point

Abstract: In this thesis it is shown how to solve the master equation for a Markov process including a critical point by means of successive approximations in terms of a small parameter. A critical point occurs if, by adjusting an externally controlled quantity, the system shows a transition from normal monostable to bistable behaviour. Examples of the external quantity (the pump parameter) are temperature, electric discharge current, chemical concentrations and mechanical force. The appropriate small parameter may be either the diffusion coefficient or the inverse size of the system. The latter is usually given by the volume or by the total number of constituents such as spins, photons or molecules. The fundamental idea of the theory is to separate the master equation into its proper irreducible part and a corrective remainder. The irreducibleor zeroth order stochastic approximation will be a relatively simple Fokker-Planck equation that contains the essential features of the process. Once the solution of this irreducible equation is known, the higher order corrections in the original master equation can be incorporated in a systematic manner. In part I of this thesis we consider the problem of diffusion in an externally applied potential showing a monostable to bistable transition. The appendix of part I presents a discussion of the irreducible solutions. In part II we examine the general Markov process. The appendix of part II is devoted to an example, namely the magnetic mean field Ising model.

Exact analytical solution for the change of the photon statistics due tok-photon absorption

Abstract: For unsaturatedk-photon absorption the exact solution of the master equation for the diagonal elements is presented. This solution is found by Laplace-transformation and recursion. It has the form of a double sum over the initial values. Commuting the order of summation we then show, that fork≧2 the photon distribution does not depend on the initial distribution if the initial mean photon number is sufficiently high and the square of the resulting photon number is small against the initial photon number.

Analytic theory of extrema. III. Results for master equations with application to unimolecular decomposition

Abstract: We obtain exact expressions for the mean first passage time to reach a given state in systems described by a master equation. Our results are simple and lend themselves to direct physical interpretation and to a straightforward asymptotic analysis. An asymptotic series is obtained for the mean first passage time that is rapidly convergent in many cases and that is easy to implement. We apply our formulas to the problem of unimolecular dissociation and calculate the deviation of the dissociation rate from that obtained using statistical theories.

Exact derivation and solution of the Nakajima-Zwanzig generalized master equation and discussion of approximate treatments for the coupled coherent and incoherent exciton motion

Abstract: Starting from the stochastic Liouville equation of the full Haken-Strobl model, describing the coupled coherent and incoherent motion of excitons in molecular crystals, the Nakajima-Zwanzig generalized master equation (GME) for the probability of finding an exciton at a specific lattice site is derived by an exact straightforward evaluation of its memory function. Various recently derived generalized master equations describing the excition motion are obtained as limiting cases and the Born approximation is discussed. It is shown that, even in the case of nearest-neighbor interaction in the stochastic Liouville equation, in the GME generalized time-dependent transition rates evolve between non-nearest neighbors and that their time behavior shows damped oscillations. Applying the Born approximation to the GME, the range of the generalized transition rates reduces to that of the interaction in the stochastic Liouville equation. Furthermore in this approximation the transition rates show a purely exponential decay with increasing time. Taking into account the interaction with an arbitrary number of neighbors, the mean square displacement of the exciton motion is calculated exactly from the GME. Finally the GME is solved exactly in the general case and several limiting expressions are discussed.

Solutions of the master equation for multiphoton dissociation

Abstract: Iterative analytical solutions of a master equation for multiphoton dissociation are derived and compared with numerical results. Simple expressions for the incubation time and for the steady‐state rate constant are obtained identifying the relevant molecular parameters.

Generalised multiphoton rate equations

Abstract: Starting from the multi-level master equation a non-Markoffian rate equation is derived. Explicit expressions for the transition rates between each pair of atomic levels in this equation are derived at it is shown that these may be interpreted as the sum of one-photon, two-photon,..., etc. transition rates connecting the given pair of levels. The non-Markoffian master equation is exact and so can describe long- or short-time behaviour under monochromatic or broad-band excitation. The exact steady-state solutions are obtained by solving a Markoffian rate equation.

On the theory of fluctuations around non-equilibrium steady states: A generalized time-convolutionless projector formalism

Abstract: A new method of finding nonlinear Langevin type equations of motion for relevant macrovariables and the corresponding master equation for systems far from thermal equilibrium is presented by generalizing the time-convolutionless formalism proposed previously for equilibrium hamiltoian systems by Tokuyama and Mori. The Langevin type equation consists of a fluctuating force, and the nonlinear drift coefficients which are always identical to those of the master equation. A simple formula which relates the drift coefficients to the time correlation of the fluctuating forces is derived. This is a generalization of the fluctuation-dissipation theorem of the second kind in equilibrium systems and is valid not only for transport phenomena due to internal fluctuations but also for transport phenomena due to externally-driven fluctuations. A new cumulant expansion of the master equation is also obtained. The conditions under which a Langevin and a Fokker-Planck equation of a generalized type for non-equilibrium open systems can be derived are clarified. The theory is illustrated by studying hydrodynamic fluctuations near the Rayleigh-Benard instability. The effects of two kinds of fluctuations, internal fluctuations of irrelevant macrovariables and external (thermal) noises, on the convective instability are investigated. A stochastic Ginzburg-Landau type equation for the order parameter and the corresponding nonlinear Fokker-Planck equation are derived.

Chapman-Enskog development of the multivariate master equation

Abstract: A systematic development for a Multivariate Master Equation (MME), describing reaction diffusion systems, is presented along the same lines as the Chapman-Enskog development of kinetic gas theory. Diffusion, which occurs at a very fast rate, brings the system near a state of diffusional equilibrium, the analogue of local equilibrium for gases. In diffusional equilibrium, the global Master Equation (ME) is shown to be an exact consequence of the MME. For finite systems, corrections to the global ME, resulting from the finiteness of the diffusion times, are calculated. The development is verified on an exactly solvable model and illustrated on the Schlogl model. The difficulties encountered in the thermodynamic limit are discussed, and possible outcomes suggested.

Stochastic hydrodynamic theory for one-component systems

Abstract: A nonlinear diffusion approximation for a previously derived master equation describing an inhomogeneous Boltzmann gas in a lumped phase space is proposed. A fluctuating kinetic equation is obtained which differs from the usual Langevin equations in three essential properties: the drift and random force are nonlinear, the random noise obeys a generalized fluctuation-dissipation theorem, and there is no reference to equilibrium. Relations with other approaches to hydrodynamic fluctuations are discussed. © 1980 The American Physical Society.

Correction to the laser master equation

Abstract: The effects of the interaction between atoms and a laser field on the damping phenomena of the laser system are examined by the perturbational treatment of the rigorous damping theory. Correction terms to the conventional master equation of the laser system are obtained and those to the equations of motion for moments are also calculated. The significance of the correction terms is explicitly shown. The modified conditions of threshold and diffusion constant are derived by eliminating the atom variables near the threshold in terms of the boson coherent-state representation. Remarks on the conventional Langevin-equation approach are also made.

Kinetic equation theory of multi‐component crystal growth. I. Master equations in the lattice model

Abstract: A general approach to describe the growth kinetics of multi-component crystals is worked out on the basis of the lattice model. A system of kinetic equations is obtained for binary alloys in the two-partition approximation of the distribution function, taking into account diffusion processes in the solid and the liquid phases. The relaxation properties of kinetic equations are studied. [Russian Text Ignored].