Showing papers on "Master equation published in 1972"

Stochastic instability of non-linear oscillations

Abstract: CONTENTS 1. Introduction 549 2. A One-dimensional Non-linear Oscillator Acted on by a Periodic Perturbation 551 3. Fundamental Concepts of Ergodic Theory 554 4. Numerical Experiments 560 5. Stochastic Fermi Acceleration 560 6. The Stochastic Layer and the Condition of Overlapping Resonances 562 7. The Random-phase Approximation and the Fundamental Kinetic Equation (Master Equation) 564 8. Bibliography 566

Electron spin relaxation in liquids

Abstract: I. Superoperators, Time-Ordering, and Density Operators.- Superoperators.- Time-ordering.- Density operators.- II. Stochastic Processes.- Stochastic (random) variables and probability.- General remarks on stochastic (random) processes.- The relaxation function.- Gaussian and Markov processes.- The relaxation function.- Gaussian and Markov processes.- Master equation.- Fokker-Planck equation.- Functional integration technique of Kac.- III. An Introduction to the Stochastic Theory of E.S.R. Line Shapes.- The jump model.- The isotropic rotational diffusion model.- IV. Projection Operators.- "Non-Markoffian" master equation.- Application to the interaction between a spin system and a bath.- The Redfield approximation.- The two jump models.- V. Cumulant Expansion.- One-dimensional moment and cumulant expansions.- Multivariable moment and cumulant expansions.- Expansions.- Generalization.- Conclusion.- VI. Linear Response Theory and Spin Rotation.- Spin relaxation.- Derivation of the Bloch equations.- Concluding remarks.- VII. Two Approaches to the Theory of Spin Relaxation: I. The Redfield-Langevin Equation II. The Multiple Time Scale Method.- The Redfield-Langevin equation.- Motivation.- Derivation of the equation of motion for G??, (t).- The Redfield-Langevin equation - the lowest order result.- Stochastic properties of the Redfield-Langevin eq..- The Bloch-Langevin equation.- Further remarks.- Generalizations of the lowest order result.- Semiclassical treatment.- Further stochastic interpretation of the spin problem.- The multiple time scale method.- Derivation of the equations of motion.- VIII. ESR Relaxation and Line Shapes from the Generalized Cumulant and Relaxation Matrix Viewpoint.- General approach.- Relaxation matrix and spectral lineshapes.- Properties of the relaxation matrix.- Non-asymptotic solutions.- Stochastic averaging.- Gaussian processes.- Markov processes.- Diffusion models.- Internal rotations.- Anisotropic rotational diffusion.- Summation of the generalized moments for a Markoff process: stochastic Liouville equation.- IX. Spin Relaxation via Quantum Molecular Systems.- Strong collisional relaxation.- General formulation.- Applications.- Gas-phase relaxation.- Quantum effects of methyl group tunneling.- Spin relaxation via vibronic relaxation.- Non-resonant effects.- X. Electron Spin Relaxation in Liquids. Selcted Topics.- Hamiltonian: terms which determine frequencies.- Hamiltonian: terms which determine relaxation.- Complete hamiltonian and density matrix.- Time-dependent perturbation expansions.- $${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over M} }}$$(t) related to '$${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }}$$(t)$${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }}$$' pulse experiments.- Line widths: cw experiments.- Spin autocorrelation function.- Line shapes in absence of relaxation 251 Line widths and reorientation: detailed derivation.- Discussion of results.- Analysis of experiments.- Second order corrections.- Several interacting nuclei.- Breakdown of spin-hamiltonian Orbach processes.- Appendices.- XI. Spin-Rotation Interaction.- Basic theory of the interaction.- The full relaxation problem.- The dynamical problem.- Appendices.- 3j-symbols.- Spherical tensors.- Rotation matrices.- XII. Electron Spin Relaxation in 6S State Ions.- Adaption of Redfield's theory.- Relaxation via rotational modulation of the zero-field splitting.- Relaxation via the quartic terms.- Collisional fluctuations of the zero-field splitting.- Modifications demanded by a hyperfine interaction.- Symmetric linewidth variations in the spectra of manganese(II) ions.- XIII. Magnetic Resonance Line Shapes in Slowly Tumbling Molecules.- Expressions for the line shapes.- Reduction to algebraic equations.- Computational algorithms.- Reduction to an eigenvalue problem.- Reduction of band-width to tridiagonal 356 Diagonalization of a complex symmetric tridiagonal matrix.- Estimating rates of convergence.- Applications and comparison with experiments.- Diagonalization programs.- XIIIA. Appendix: Symmetry and the Slowly Tumbling Spin System.- Application of group theory.- XIV. ESR Line Shapes and Saturation in the Slow Motional Region - The Stochastic Liouville Approach.- General approach.- Free radicals of S=$${1 \over 2}$$ no saturation.- Axially symmetric secular g-tensor.- Asymmetric secular g-tensor.- g-tensor plus END-tensor including pseudosecular terms.- One nuclear spin of I = 1.- Saturation.- Rotationally invariant T1.- g-tensor (axially symmetric).- Triplets.- General solutions.- Perturbation theory.- Summary of spectra.- XV. Electron Spin Relaxation in Liquid Crystals.- Nematic liquid crystals.- The static spin hamiltonian.- The linewidth calculation.- The line shape.- XVI. Two Problems Involving ESR in Liquid Crystals.- Thin film ESR (rapid tumbling).- Slow tumbling ESR of a free radical in a bulk liquid crystal.- XVII. The ESR Line Shape of Triplet Excitons in Disordered Systems: The Anderson Theory Approach.- The static line shape.- The magneto-selection theory.- Line shape and symmetry of the triplet state.- The magneto-photo-selection.- The incoherent exciton line shape.- The experimental problem.- The single-channel transfer model.- The exciton line shape of a trimer.- The secular approximation.- The adiabatic approximation.- A line shape formula for the general case.- Simulation of the experimental data.- Excitons in thermal equilibrium.- The multi-channel transfer model.- The vibronic coupling approach of the exciton.- The coherent exciton states.- The incoherent exciton states.- The factors controlling the exciton diffusion.- XVIII. ESR Saturation and Double Resonance in Liquids.- to saturation: a simple line.- ELDOR.- ENDOR.- General approach.- Transition probabilities.- ELDOR - Generalized no saturation of observing mode.- ENDOR - Limiting enhancements.- Expressions for transition probabilities.- Heisenberg spin exchange and chemical exchange.

Dynamics of Fluorescence Polarization in Macromolecules

Abstract: Reexamination of the theory of fluorescence time dependence owing to rotational diffusion of rigid macromolecules reveals deficiencies or hidden restrictions in each of the previous treatments. The correct master equation has five exponential decay terms, with preexponential factors that depend upon the diffusion constants and, in a completely symmetrical fashion, upon the orientations of absorbing and emitting dipoles.

Impurity Quenching of Molecular Excitons. I. Kinetic Comparison of Förster—Dexter and Slowly Quenched Frenkel Excitons in Linear Chains

Abstract: The quenching kinetics of tightly bound excitons for two different one‐dimensional models are compared. The quenching of fully incoherent, or Forster—Dexter, excitons is described by a standard master equation, and that of fully coherent, or Frenkel, excitons by an ad hoc linear differential equation whose eigenvalues are complex. Moments of the chain excitation function (probability that excitation remains at time t) are calculated on each model for finite chains with either localized or uniform initial conditions, free‐end or periodic boundary conditions, one disruptive or one nondisruptive quencher, and various quencher locations, but with only nearest‐neighbor interactions included. The ad hoc equation is treated only in the limit that the quenching is slow enough not to affect the exciton wave functions in first order. In that limit of Frenkel exciton quenching, an analytic expression is given for the mean de‐excitation time in the presence of uniform decay processes such as fluorescence. The master ...

Solutions of Master Equations and Related Random Walks on Quenched Linear Chains

Abstract: Four separate but related contributions to the theory of quenched stochastic processes in one dimension are presented. First, Green's functions are derived for Laplace transformed master equations (here described in the language of, but not restricted to, hopping excitons) on finite chains with either periodic or free‐end boundary conditions, and with either a disruptive (substitutional impurity) or a nondisruptive quencher. We solve these problems in spectral form for short‐range quenching with arbitrary quencher location and quenching rate parameter Qo. Second, the analogous random walk situations are treated. The solution of the generating function (finite‐difference analog of the Laplace transform) equation is identical to that of the Laplace‐transformed master equation with a disruptive quencher, but not with a nondisruptive quencher. Unlike the master equation case, slowly damped oscillations of the random walk chain excitation function can exist. Other differences also exist and are discussed; thes...

Random Matrix Theory and the Master Equation for Finite Systems

Abstract: In this paper we consider the justification for the use of a stationary Markov assumption to describe vibrational relaxation and dissociation in isolated, highly excited, polyatomic molecules. We discuss first a model in which only a single oscillator is allowed to break, the remaining vibrations serving as a heat bath which interacts weakly with the reactive degree of freedom. A random matrix hypothesis is used to derive a time‐independent form for the rates of internal energy exchange. In the limit of very ``hot'' molecules these transition probabilities are shown to have the expected linear and quadratic dependences on the individual vibrational quantum numbers. The constants of proportionality are simple functions of the size of the molecule and the interaction strengths coupling the nuclear motions. Finally, we return to the full dynamical theory of unimolecular reactions which was introduced in an earlier paper [J. Chem. Phys. 52, 5718 (1970)] as an alternative to the conventional statistical (e.g., transition‐state) approaches. We use the Zwanzig projection operator formalism to derive the sufficient conditions for which Liouville's equation for internal energy exchange assumes the Markov form appropriate to an irreversible relaxation process. The familiar Green's function (resolvent operator) techniques are generalized to the case of tetradic propagators and a stochastic description is shown to be justified for time scales which correspond to experimental observation of the vibrational dissociation.

Diffusion for weakly coupled quantum oscillators

Abstract: We construct a simple model which exhibits some of the properties discussed by van Hove in his study of the Pauli master equation. The model consists of an infinite chain of quantum oscillators which are coupled so that the interaction Hamiltonian is quadratic. We suppose the chain is in equilibrium at an inverse temperature β and study the return to equilibrium when a chosen oscillator is given an arbitrary perturbation. We show that in the limit as the interaction becomes weaker and of longer range, the evolution of the chosen oscillator becomes a diffusion equation. Moreover we give an explicit example where the evolution of the chosen oscillator has the Markov property and where the Pauli master equation is exactly satisfied.

Rotational Relaxation of Molecules in Fluids for Reorientations of Arbitrary Angle

Abstract: A theory of rotational motion of molecules in liquids is presented. The traditional Debye diffusion model is generalized so as to include arbitrary reorientation angles by means of a nonlocal master equation. Random walk models are a special case of this generalization. We have obtained the conditional probabilities for the orientation of a body and of a vector fixed in the body for arbitrary body geometry. These results are then specialized to specific symmetries. From these conditional probabilities we calculate correlation functions which describe a large number of spectroscopic experiments such as NMR, ESR, NQR, dielectric relaxation, and infrared and Raman spectroscopy. We find that the form of these results for the correlation functions depend only on medium isotropy and body geometry. The same number of decaying exponentials are found as in the diffusion model regardless of the details of the reorientation mechanism. Only the amplitudes and relaxation times depend on these details.

Laser-Induced Rate Processes in Gases: Dynamics, Unimolecular Decay, and Scattered Field

Abstract: We have recast a linear quantum-mechanical Boltzmann equation, describing the dynamics of a gas of $N$-level atoms absorbing light between an arbitrary pair of levels, into the form of a generalized master equation. The atoms are permitted to undergo both phase-changing and energy-changing collisions with a heat bath of inert molecules. The possibility of unimolecular decay is incorporated into the Boltzmann equation. The solutions of the generalized master equation give an accurate description of the system for short as well as long times and for arbitrary light-field intensity. Eigenvalues obtained from the solution of the $N$-level master equation are shown to have particular significance with regard to the scattered electromagnetic field, with the imaginary parts of the complex eigenvalues providing the location and the real parts of the eigenvalues giving the widths of scattered radiation bands. The dynamics and scattered field for a two-level system is discussed in detail. Two collision parameters giving the frequency of phase-changing and energy-changing collisions are important. For special values of these parameters, our results reduce to scattered field spectra calculated previously. When unimolecular decay is included, one identifies the smallest eigenvalue of the generalized master equation with the macroscopic rate constant. Observation of the scattered field for different incident field intensities would permit separation of radiative and thermal effects in an enhanced reaction rate, without requiring an accurate temperature measurement.

On the Microscopic Conditions for Linear Macroscopic Laws

Abstract: We have investigated the conditions which must be imposed on the microscopic equations of motion to obtain exact linear laws for macroscopic (phase averaged) variables. The starting point in this study has been the lowest order master equation (Pauli equation) which is a linear microscopic equation in the state probabilities with a time‐independent transition matrix. Discrete and continuous variable master equations as well as their multivariate generalizations have been considered. In the case of continuum state variables, we have used various Fokker‐Planck equations and their corresponding Langevin equations as our starting microscopic equation of motion. In each case the conditions which must be imposed to obtain linear macroscopic transport equations have been derived and discussed. The problem of the derivation of linear macroscopic laws from microscopic laws which are nonlinear in the dynamical variables has been discussed in the context of our results. We find that exact linear macroscopic laws can...

On the solutions and the steady states of a master equation

Abstract: A complete characterization of the time behavior of the means and variance of a stochastic process which is generated by a finite number of independent systems is presented based on the master equation for the conditional probability. It is found that the means and variance relax to a steady state and that the steady state will be independent of the initial state if and only if a matrix related to the transition matrix is nonsingular. Finally, the result that the variance approaches its steady-state form at twice the rate of the means is shown to depend on the nonsingularity of the same matrix.

Vibrational Relaxation in Gases—Some Constraints on the Use of Master Equations

Abstract: The master equation for vibrational relaxation of diatomic dilute gases is derived. The derivation begins with the quantum Liouville equation and uses projection operator techniques; it can be extended to dense fluids under certain conditions. A careful discussion of the master equation's range of validity reveals restrictions on the experimental initial conditions and the relevant scattering cross sections.

Monte Carlo Trajectory Calculations of the Dissociation of HCl in Ar

Abstract: The modified phase‐space theory of reaction rates is applied to the dissociation of HCl in a heat bath of argon atoms. Excellent agreement is obtained between the theoretical predictions and the shock‐tube measurements of the dissociation rate coefficient over the temperature range 2500–5000°K. The recrossing correction factor and nonequilibrium correction factor are obtained from Monte Carlo trajectory calculations for states near the dissociation limit. The trajectories were sampled within the reaction zone, with a weight proportional to the equilibrium reaction rate, and numerically integrated in both timewise directions to determine the complete histories of the collisions. A simple separable function for the equilibrium transition rate R(ei, ef) from initial states ei to final states ef was obtained to fit the numerical data with sufficient accuracy and was used to solve the steady‐state master equation. Important features of collisions of highly asymmetric diatomic molecules are discussed, and sever...

Models in nonequilibrium quantum statistical mechanics

Abstract: In this paper, quantum versions of statistical models are constructed. All aspects of the systems can be explicitly solved. It is possible to give magnetic realizations of these models. The most interesting conclusions are: (1) the state for time going to infinity is approached in an oscillatory manner in the quantum case; (2) in both classical and quantum cases, the exact description gives limiting states which remember the initial specifications; and (3) in these models, the time evolution generally cannot be described. even approximately, by a master equation.

Sufficient Conditions for the Arrhenius Rate Law

Abstract: Kinetics are discussed for a system of identical degrees of freedom each of which is subject to a potential containing a barrier separating minima and is coupled weakly to other degrees of freedom serving as a heat reservoir. Temperature is defined via the assumption that the reservoir does not depart significantly from internal equilibrium. Use is made of an especially detailed form of master equation in which, for each transition in the system‐cum‐reservoir, account is taken of events in the reservoir as well as in the system. This permits application of the laws of microscopic reversibility and energy conservation to each transition. A generalized form of Arrhenius expression is obtained for the temperature dependence of the net flux between the set of quantum levels essentially contained within either potential well (initial state or final state) and the set of (higher energy) uncontained levels (``activated'' or ``transition'' state). Application of a steady state assumption to transition state concentration (as contrasted with the familiar equilibrium assumption for this) eliminates the latter and provides an equation for the overall net flux over the barrier. The forward and backward components of this flux depend upon ``free energies of activation'' in the familiar exponential fashion. On the other hand the formal expressions obtained for the ``frequency factors'' associated with these forward and backward components are not necessarily identical except under special conditions in which the quantum levels of initial state are equilibrated among themselves and in which similar statements apply (separately) to activated state and final state. Even these special conditions do not require an equilibrium between initial state (or final state) and activated state. Nevertheless they suffice to reduce the general result to one identical, except for a possibly important difference in the form of the ``frequency factor,'' to that of the transition state theory of Eyring.

Interference of coherent and incoherent interactions in the Master Equation for open systems

Abstract: The limits of validity of a well known type of Master Equation for open systems are discussed. Using Zwanzig's projector formalism we derive a “Generalized Master Equation” which describes interference processes between coherent and incoherent interactions. The conditions under which this equation reduces to the known ones is given and discussed for laser systems.

Quantum Theory of a Damped Two-Level Atom

Abstract: The interaction of a two-level atom with the radiation field in a cavity is discussed in quantum mechanical terms that are closely analogous to those used in discussing the classical Langevin equation; the quantum Langevin equation for this case, however, is nonlinear. A technique is developed for finding the measurable properties of the solution of the equations exactly; the properties of the solution are compared with those for the harmonic oscillator.

Some approximate solutions to the discrete master equation

Abstract: Recent mathematical developments on approximate diffusionlike solutions to the master equation are summarized. The technique is applied to two master equations of physical interest-one that describes the phenomenon of superradiance and a second that characterizes generation-recombination noise in semiconductors. For this second case, some previously obtained equilibrium results are found and the extension of these results to finite times is given.

Time-dependent statistics of binary linear lattices

Abstract: The time-dependent statistics of binary linear lattices is investigated on the basis of a master equation at the microscopic level. It is assumed that the kinetics may be formulated as transformations of specified sequences of clusters ofA units andB units into other specified sequences. On the basis of aStosszahlansatz, a master equation at the macroscopic level is derived. In the limit of a large system, the densities of clusters of all types satisfy rate equations similar to the equations of chemical kinetics. AnH-theorem is proven and the nonequilibrium thermodynamics of the system is studied. The theory has application to the kinetics of the helix-coil phase transition in biopolymers.

Stochastic Master Equations: A Perturbative Approach

Abstract: Linear stochastic master equations for wave propagation in a continuous random medium are derived along the lines of the resolvent theory used in nonequilibrium statistical mechanics. Equations for the mean and fluctuating fields are subsequently obtained by operating directly on the stochastic master equations with statistical projection operators. The findings are compared with the results determined using the method of renormalization and the method of smoothing.

The Master Equation for the Dissociation of a Dilute Diatomic Gas. VI. Solution of Coupled Sets of Master Equations

Abstract: Three coupled sets of master equations, representing the equilibration of via atoms only, have been solved by the normal-mode technique; the set of 57 simultaneous differential equations describing...

Nonequilibrium statistical mechanics of finite classical systems-II

Abstract: The work of the previous paper is applied to the study of weakly interacting systems. Either by “quasilinear” techniques or by analyzing the perturbation series for the smoothed probability density, it is possible to derive a master equation equivalent to that of Brout and Prigogine without requiring the size of the system to become infinite. The properties of this equation are discussed. The equation is self-consistent provided the interactions are weak enough; however, examination of higher terms in the perturbation series shows that their effect might make the master equation invalid for times longer than that taken by a typical particle to cross the containing vessel. In many physical cases, the relaxation time will be shorter than this; also, further studies may show the higher terms to be less important than they seem.

Time-correlation function for macroscopic observables in a classical gas

Abstract: The state of a gas is characterized by occupation numbers of cells inμ-space. The mean values and fluctuations of these numbers are studied with the help of a master equation. The results are discussed within the framework of the theory of random forces. An equation of motion of the time-correlation function (TCF) is derived and it is shown that the temporal development of the TCF can be described by a linearized Boltzmann equation.