Linear Master Equation Research Articles

Two-Dimensional Model for Polymer-Based Photovoltaic Cells: Numerical Simulations of Morphology Effects

In this contribution, we develop and investigate a general 2D hopping model for the photovoltaic action in polymer-based thin films. The model takes a microscopic origin and accounts for the molecular photonic and electronic processes by a simple kinetic scheme that eventually leads a linearized master equation for the time evolution of the photovoltaic system. With an emphasis on the topology of blends of donor/acceptor functionalized polymers, we attempt to characterize the dependence of the short-circuit current, internal quantum efficiency, IV characteristics, and fill factors on the morphology of the blend. Several different morphologies for the polymer film are considered, and they show quite different transport and efficiency behavior (e.g., for so-called double cable structures, nearly quantitative conversion efficiencies are computed, and for other structures similar efficiencies may be found, but with short-circuit currents orders of magnitude lower). The model neglects effects such as exciton migration, the built-in potential, and interaction in the third dimension. Nontheless, significant conclusions can be drawn: in particular, we demonstrate that a viable photovoltaic system driven only by concentration gradients of charge carriers (no built-in field) is possible.

Geometric Investigation of Association/Dissociation Kinetics with an Application to the Master Equation for CH3 + CH3 ↔ C2H6

The dynamics of association/dissociation kinetics is studied, with an application to the titled reaction. The focus is the geometry of the phase space of the phenomenological rate law, a Lindemann mechanism, and the master equation of this reversible reaction, all of which are nonlinear. It is shown that all three systems possess similar phase space structure, including a 1-D manifold. This 1-D manifold describes asymptotic motion for either dissociation or recombination and is the analogue of the corresponding eigenvector for linear master equations describing dissociation without recombination. The 1-D manifold allows for the separation of asymptotic motion from transient behavior and together with other manifolds in phase space allows a better understanding of the dissociation and recombination processes. The 1-D manifold also allows us to test various approximations that have been used in the past to calculate association rate constants from the master equation and Lindemann mechanism and develop new methods for calculating association rate constants and generating rate laws.

Detection statistics in the micromaser

We present a general method for the derivation of various statistical quantities describing the detection of a beam of atoms emerging from a micromaser. The use of non-normalized conditioned density operators and a linear master equation for the dynamics between detection events is discussed as are the counting statistics, sequence statistics, and waiting time statistics. In particular, we derive expressions for the mean number of successive detections of atoms in one of any two orthogonal states of the two-level atom. We also derive expressions for the mean waiting times between detections. We show that the mean waiting times between detections of atoms in like states are equivalent to the mean waiting times calculated from the uncorrelated steady state detection rates, although like atoms are indeed correlated. The mean waiting times between detections of atoms in unlike states exhibit correlations. We evaluate the expressions for various detector efficiencies using numerical integration, reporting results for the standard micromaser arrangement in which the cavity is pumped by excited atoms and the excitation levels of the emerging atoms are measured. In addition, the atomic inversion and the Fano-Mandel function for the detection of deexcited atoms are calculated for comparison to the experimental results of Weidinger et al. [Phys. Rev. Lett. 82, 3795 (1999)], who report the observation of trapping states.

Observation and analysis of nonlinear vibrational relaxation of large molecules in shock waves

We report the observation of highly nonlinear vibrational relaxation for a number of large molecules in shock waves, together with an attempt at a master-equation modeling of this phenomenon. In all these molecules laser-schlieren measurements show a clear and often well-resolved local maximum in the density gradient, indicating a similar maximum in the rate of energy transfer. This unexpected phenomenon is seen in the relaxation of benzene (C6H6), cubane (C8H8), cyclopropane (C3H6), furan (C4H4O), norbornadiene (C7H8), oxirane (C2H4O), and quadricyclane (C7H8). It has also been detected in cyclopentadiene (C5H6) and pyrazine (C4N2H4), as well as CF3Br and CF3Cl but in these was not well resolved. The phenomenon thus seems nearly ubiquitous; of the “large” molecules where relaxation could be resolved, only norbornene (at C7H10 the largest such molecule) exhibits a fully linear relaxation. The gradients are clearly and solely from vibrational relaxation; integrated gradients are in good agreement with thermodynamic calculations of total density change, and near-equilibrium relaxation times in pure cyclopropane and oxirane are fully consistent with overlapping ultrasonic results. It appears we are seeing a delay in the development of series coupling in these experiments. An attempt is made to model the process using a linear master equation with exponential gap probabilities having an α(∼〈ΔE〉down) linear in energy. Although this does introduce sufficient nonlinearity through the rate coefficients to produce maxima, these and the overall gradients are consistently too small. It is suggested that inclusion of a true nonlinearity through VV transfer will be needed to explain the observations, and a possible mechanism for this is proposed.

Diffusion of small clusters on metal (100) surfaces: Exact master-equation analysis for lattice-gas models

Exact results are presented for the surface diffusion of small two-dimensional clusters, the constituent atoms of which are commensurate with a square lattice of adsorption sites. Cluster motion is due to the hopping of atoms along the cluster perimeter with various rates. We apply the formalism of Titulaer and Deutch [J. Chem. Phys. {bold 77}, 472 (1982)], which describes evolution in reciprocal space via a linear master equation with dimension equal to the number of cluster configurations. We focus on the regime of rapid hopping of atoms along straight close-packed edges, where certain subsets of configurations cycle rapidly between each other. Each such subset is treated as a single quasiconfiguration, thereby reducing the dimension of the evolution equation, simplifying the analysis, and elucidating limiting behavior. We also discuss the influence of concerted atom motions on the diffusion of tetramers and larger clusters. {copyright} {ital 1999} {ital The American Physical Society}

Stochastic Wave Function Approach to Generalized Master Equations

A generalization of the stochastic wave function method is presented that allows the unraveling of arbitrary linear quantum master equations that are not necessarily in Lindblad form and, moreover, the explicit treatment of memory effects by employing the time-convolutionless projection operator technique. The crucial point of this construction is the description of the open system in a doubled Hilbert space, which has already been successfully used for the computation of multitime correlation functions.

A NONLINEAR MODEL OF THE COLLECTIVE SPONTANEOUS EMISSION BY A SPIN SYSTEM

We propose a nonlinear quantum master equation, which results in the nonlinear Bloch-like equations, similar to those which are used usually to describe the collective spontaneous emission by a system of two-level atoms or a spin system in a magnetic field. On the one hand, the new equation is the unique simplest homogeneous and nonsingular generalization of the linear master equations for quantum systems in the finite-dimensional Hilbert space. On the other hand, it corresponds to the mean-field approximation, and in this form it can be used for a larger class of systems. Our equations describe both triggered and pure spin emission processes (when the population is completely inverted), and their solutions show the occurrence of secondary emission peaks, as observed in experiments.

Quantum mechanics with event dynamics

Event generating algorithm corresponding to a linear master equation of Lindblad's type is described and illustrated on two examples: that of a particle detector and of a fuzzy clock. Relation to other approaches to the foundations of quantum theory and to the description of quantum measurements is briefly discussed.

One-atom maser: Statistics of detector clicks.

We present a general theory for the calculation of various characteristic properties of the beam of atoms emerging from a resonator in one-atom-maser experiments. The beam is described in terms of the statistics of the detector clicks. The evolution of the state of the maser photons between clicks is governed by a nonlinear master equation. The nonlinearity originates in the necessity to account for the atoms that escape detection. Despite the permanent reductions of the photon state, resulting from the detections, the steady state of the conventional linear master equation determines the statistics of the detector clicks. The whole process is ergodic, in the sense that a single run of the experiment contains all reproducible statistical data, provided the duration of the run is much longer than all relevant correlation times. The nonlinear master equation is used to calculate the distribution of waiting times between detector clicks. Other statistical properties of the clicks that are derived include correlation functions and variances of the counting statistics. The formalism is applied to standard one-atom-maser experiments and to parity measurements on both unpumped and pumped cavities. We find that, for the standard one-atom-maser operation, none of the said statistical properties is a simple immediate indicator for a sub-Poissonian variance of the photon number. For example, the detector clicks may be antibunched although the photon distribution has a super-Poissonian variance.

Quantum theory of correlated-emission lasers: Vacuum state for the mode of the relative phase and the relative amplitude.

We present a comprehensive quantum theory of correlated-emission lasers (CEL's) in a combination-mode approach, which is valid for both the quantum-beat laser and the CEL with injected atomic coherence. We first show that, under certain conditions, the linear master equation of the CEL's breaks down into two uncoupled equations in the combination modes, which are linear combinations of two CEL modes

A theoretical prediction of hydrogen molecule dissociation-recombination rates including an accurate treatment of internal state nonequilibrium effects

This paper presents the results of a detailed study of dissociation and recombination of H2 over the temperature range 1000 to 5000 K. The chemical processes are modeled by solving the master equation for the concentrations of the full set of rovibration states. All of the state-to-state energy transfer rate coefficients required by the master equation are evaluated by the quasiclassical trajectory method using a potential energy surface which is a fit to ab initio electronic structure calculations. The analysis of the results of the master equation to obtain the phenomenological rate coefficients was carried out using several techniques. This is required because there is a mixture of third bodies and their concentrations change as the reaction proceeds. The methods based on one-way fluxes are not reliable, while the methods based upon an eigenvalue solution of the linearized master equation are in reasonable agreement with our preferred method, which is based on fitting the concentrations from a two component master equation. The results from various methods of determining the phenomenological rate coefficient for H2 as a third body are in good agreement with experimental estimates. The recombination rate coefficients are most sensitive to the collision induced dissociation rates of initial states with moderate values of vibrational and rotational quantum numbers. A comparison of the orbiting resonance theory to the accurate results shows that this simple model is not valid under the conditions of the present study.

Comparative study of various quasiprobability distributions in different models of correlated-emission lasers.

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'

Time-dependent solution of multidimensional Fokker-Planck equations in the weak noise limit

The time-dependent problem of multidimensional Fokker-Planck equations (FPE), satisfying potential conditions, is solved in the weak noise limit. For the relaxation from an intrinsically unstable state to the metastable state, the author applies the theory of the Omega expansion of the Green function and reveals the possibility of dimensional reduction. From a metastable state to the stationary state, the time-dependent problem of the FPE is reduced to linear master equations. The first passage times are given explicitly.

Linear master equation in the generalized coordinate representation

Linear master equation in the generalized coordinate representation

New techniques for the study of non-equilibrium effects in non-first-order systems

We demonstrate that the phenomenological rate constants for a second-order system far from equilibrium may be calculated by eigenanalysis of a linearized master equation. We also calculate some of these rate constants by a simple iterative method. The results are in good agreement with numerical integration of the non-linear master equation. Furthermore the eigenanalysis provides additional insights into the mechanism of non-equilibrium effects.

Stochastic transport in disordered systems

We develop a theory of stochastic transport in disordered media, starting from a linear master equation with random transition rates. A Green function formalism is employed to reduce the basic equation to a form suitable for the construction of a class of effective medium approximations (EMAs). The lowest order EMA, developed in detail here, corresponds to recent approximations proposed by Odagaki and Lax [Phys. Rev. B 24, 5284 (1981], Summerfield [Solid State Commun. 39, 401 (1981)], and Webman [Phys. Rev. Lett. 47, 1496 (1981)]. It yields an effective transition rate Wm which can be identified as the memory kernel of a generalized master equation, and used to define an associated continuous-time random walk on a uniform lattice. The long-time behavior of the mean-squared displacement arising from an initially localized state can be found from Wm, as can diffusion constants in any case where the long-time behavior of the system is diffusive. Detailed calculations are presented for seven lattice systems in one, two, and three dimensions, and for a variety of probability density functions f(w) for the transitions rates. For percolation-type densities, i.e., those with only a fraction p<1 of the bonds transmitting, the EMA predicts three distinct kinds of behavior: localization, ‘‘fractal’’ transport with slower than linear growth of the mean-squared displacement, and diffusion in the cases p<pc, p=pc, p≳pc, respectively, where pc is the bond percolation threshold of the lattice. Depending on the form of f(w) near w=0, critical exponents may take values independent of f(w) (‘‘universality’’) or heavily dependent on f(w) (‘‘nonuniversality’’).

Effect of laser-frequency fluctuations on optical bistability

We derive, solve, and analyze the linearized master equation and evaluate the spectrum for absorptive optical bistability in the high-Q cavity when the incident laser has jitter or frequency fluctuation. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. With our results we can explain the qualitative results of machine simulations and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters.

THE STOCHASTIC MODELS FOR NON-EQUILIBRIUM SYSTEMS AND FORMULATION OF MASTER EQUATIONS

In this paper we present clearly a set of postulates on probabilities which are the basis of the nonlinear master equations and multivariate linear master equations appeared in non-equilibibrium statistical physics. By using the methods of probability theory, we deduced these master equations from our general postulates. We further discussed the conditions under which the master equations are consistent with the macroscopic kinetic equations. Finally, we discussed the fluctuation in Brusselator model using multivariate linear master equation approach and made a preliminary explanation on the appearance of dissipative structure through fluctuation.

Non-gaussian, non-markov processes

A non-Gaussian treatment of the Mori theory is presented with the intention of explaining the results of recent computer simulations. The Mori equation is replaced by a multi-dimensional Markov process represented by a set of interrelated matrix equations. These are related straight-forwardly to a linear master equation and its Kramers–Moyal expansion. The non-Gaussian nature of the molecular dynamics results may then be represented by truncating the latter at third order, involving an extra operator Γ(1)L. This may be expanded in a matrix over the basis set of Hermite polynomials in terms of which may be represented eigenstates of the usual Fokker–Planck operator Γ(0)L.

Stochastic theory of ligand migration in biomolecules.

When ligand binding to proteins involves the presence of more than one ligand inside a given biomolecule, linear deterministic rate equations become useless. A stochastic approach, however, permits a treatment of the migration and binding of small molecules to proteins even at high ligand concentrations. An appropriate linear master equation and its analytic solution are given. As an example, the binding of carbon monoxide to myoglobin at partial pressures from 1 to 10(3) bars (0.1 to 100 MPa) is treated.