Showing papers on "Master equation published in 1978"

Theory of unimolecular reactions induced by monochromatic infrared radiation

Abstract: A theory of unimolecular reactions induced by coherent, monochromatic infrared radiation (URIMIR) in the absence of collisions is presented. It is shown that the set of first order linear differential equations for the amplitudes of molecular states (Schrodinger equation) can be reduced, under specified conditions, to a much smaller set of first order linear differential equations for the coarse grained populations of levels for polyatomic molecules (master equation). Four limiting situations are identified in which such linear rate equations provide a reasonable approximation. Rate coefficients are obtained as a function of spectroscopic parameters (energy levels and transition moments). Solutions for the master equations are given as a function of time and at steady‐state. Simple limiting cases (Strong Field Limit, Weak Field Limit, Reaction Threshold Bottleneck, etc.) are identified and very simple rate expressions are obtained for these cases. A complete statistical mechanical theory of URIMIR is form...

Linear response theory revisited. I. The many‐body van Hove limit

Abstract: A critical discussion of linear response theory is given. It is argued that in the formalism as it stands no dissipation is manifest. A physical reinterpretation for the case of a system in weak interaction with a reservoir is given. Mathematically this means that the van Hove limit, as well as the large system limit, is applied to the time‐dependent Heisenberg operators of the Kubo formalism. The reduced operators can be put in a very compact form, viz.,BRα(t) =[exp(−Λdt)]Bα, where Bα is a Schrodinger operator and Λd is the Liouville space superoperator corresponding to the transition operator of the master equation. In this form the relaxation character of the transport expressions, and the approach to equilibrium is at once evident. New expressions for the generalized susceptibility and conductivity in this limit are presented. Also, the Onsager relations and other symmetry properties are confirmed.

Quantum theory of the Hanle effect: calculations of the Stokes parameters of the D 3 helium line for quiescent prominences.

Abstract: A formalism of the quantum theory of the Hanle effect is developed to obtain the Stokes parameters of the D3 line in quiescent prominences in the presence of a stationary magnetic field. The aim is to develop a method that can be used to determine the vector magnetic field in prominences. The quantum description of an ensemble of atoms or photons in terms of a density matrix is briefly recalled, and the evolution of the time-dependent density matrix of the atom is treated by means of a master equation in which the interaction between the scattering atom, the incident photons, and the magnetic field appears. The approximations leading to the master equation are discussed. The expression for the density matrix of the reemitted photons is given as a function of the atomic density matrix calculated for the steady state. The formalism has been used for explicit calculations of the D3 polarization in an earlier paper by Sahal-Brechot et al. (1977).

Subgrid-Scale Modeling of Enstrophy Transfer in Two-Dimensional Turbulence

Abstract: A method of parameterization of the energy and enstropy transfers involving subgrid scales is proposed in order to predict the time evolution of the energy spectrum at larger scales. The method is based on the strong non-localness of enstrophy transfer within the enstrophy inertial range of turbulence in two dimensions. It is applied first to a master equation derived from an eddy-damped quasi-normal Markovian model. Comparisons with reference calculations including all the scales up to the dissipation range show that the method simulates the enstrophy inertial range accurately up to the cutoff wavenumber. It is then generalized to direct simulation of the two-dimensional Navier-Stokes equation: the inertial range is again accurately simulated.

Nuclear spin–lattice relaxation in periodically irradiated systems

Abstract: A master equation for nuclear magnetic relaxation under conditions of periodic and cyclic rf irradiation is derived based on the stochastic Liouville equation. Conditions for the validity of the equation, involving both use of the motional narrowing approximation and the Magnus expansion, are discussed with particular attention given to the simultaneous presence of fluctuating and nonfluctuating interactions. The expressions derived are applied to several irradiation schemes: the Carr–Purcell sequence, where the echo decay time under translational diffusion is calculated; the cw and pulsed versions of spin locking in solids, with special emphasis on the origin and role of spin diffusion and on the exact relationship between the second moment and the prefactor in the T1ρ expression; and, finally, the four‐ and eight‐pulse sequences used for suppression of homonuclear dipolar interactions, where it is shown that the x, y, and z axes of the interaction frame are ’’principal axes’’ of relaxation. Physical int...

On fluctuations in μ-space

Abstract: A master equation is derived microscopically to describe the fluctuating motion of the particle density in μ. space. This equation accounts for the drift motion of particles and is valid for any inhomogeneous gas. The Boltzmann equation is obtained from the first moment of this equation by neglecting the second cumulant (the pair correlation function). The successive moments form coarse-grained BBGKY-like hierarchy equations, in which small spatial regions with rij < the force range are smeared out. These hierarchy equations are convenient for investigating the nonequilibrium long-range pair correlation function, which arises mainly from sequences of isolated binary collisions and gives rise to the much-discussed long-time tail and the logarithmic term in the density expansion of transport coefficients. It is shown to have a spatial long tail, like the Coulombic potential, in a steady laminar flow. The stochastic nature of the nonlinear Boltzmann-Langevin equation is also investigated; the random source term is found to be expressed as a linear superposition of Poisson random variables and to become Gaussian in special cases.

Lattice diffusion and the Heisenberg ferromagnet

Abstract: It is shown that the master equation for a general diffusion problem with exclusion and symmetric binary transfer rates can be mapped exactly on the Schr\"odinger equation for an equivalent Heisenberg ferromagnet. Quantities of physical interest, e.g., the site occupation probability, are related to the lowest eignstates of the ferromagnet which play no thermodynamic role. The thermodynamics is only reflected in unobservable quantities such as the joint occupation probability of all sites. An additional result, obtained by elementary considerations, is the exact equation for the time evolution of the site-occupation probabilities. For symmetric transfer rates the equation reduces to a linear form in which exclusion effects are no longer present.

Generalized master equations under delocalized initial conditions

Abstract: The initial condition term that must be appended to the generalized master equation (GME) when the density matrix is not initially diagonal in the representation chosen is studied and explicit expressions are obtained for several cases. The term is shown to vanish for initial occupation of a Bloch state of arbitrary wave vector if the system is a crystal and the representation is that of site states, despite the violation of the initial diagonality condition. It is pointed out how one is to use the expressions for the initial term in transport calculations.

Deterministic and stochastic theory of sustained oscillations in autocatalytic reaction systems

Abstract: A special type of reaction in a uniform system with two reactants is considered which is a generalization of the first Lotka scheme. Using the Poincare-Bendixson theorem the analytical conditions for the existence of limit cycles are derived and some examples are treated numerically in the deterministic picture. The onset of limit cycles is considered as a second order phase transition. The master equations are formulated and a general analysis of limit cycle reactions in the stochastic picture is given. The fluctuations of phase and amplitude and the correlation functions are discussed. Finally, Monte-Carlo solutions of the master equation are presented. The relations between the deterministic and the stochastic picture are discussed.

Master equation approach to the second harmonic and subharmonic generation

Abstract: The statistical properties of the second harmonic and subharmonic are investigated on the basis of the generalized Fokker-Planck equation making use of the coherent state technique. An iterative procedure of solving the Fokker-Planck equation is adopted making it possible to obtain some recursion equations, particularly two iterations are explicitly performed providing approximate solutions for the normal and antinormal characteristic functions. Higher-order corrections to the superposition of coherent and chaotic fields are found in the subharmonic mode or if the coupling of modes is taken into account. Earlier results for the anticorrelation effect are rederived, the reservoir effect is obtained and the related problems of the existence of the Glauber-Sudarshan quasidistribution are discussed.

Stochastic theory of nonlinear rate processes with multiple stationary states. II. Relaxation time from a metastable state

Abstract: We have developed a methodology for obtaining a Fokker-Planck equation for nonlinear systems with multiple stationary states that yields the correct system size dependence, i.e., exponential growth with system size of the relaxation time from a metastable state. We show that this relaxation time depends strongly on the barrier heightU(x) between the metastable and stable states of the system. For a Fokker-Planck (FP) equation to yield the correct result for the relaxation time from a metastable state, it is therefore essential that the free energy functionU(x) of the FP equation not only correctly locate the extrema of U(x), but also have the correct magnitudeU at these extrema. This is accomplished by so choosing the coefficients of the FP equation that its stationary solution is identical to that of the master equation that defines the nonlinear system.

On the Absorbing Zero Boundary Problem in Birth and Death Processes

Abstract: The probability of absorption as well as the mean absorption time and its variance, have been calculated for a model which does not satisfy detailed balance. The explicit calculation of the variance is helpful in understanding the different predictions between the deterministic and the stochastic descriptions. These results can be obtained directly from the master equation by Laplace transform techniques.

Theory of the interplay of luminescence and vibrational relaxation: A master-equation approach

Abstract: A theory is developed to describe the interplay of vibrational relaxation and luminescence occurring simultaneously in a molecule, in terms of a master equation involving true sinks of probability. Specifically, the basic equation is the Montroll-Shuler equation augmented by the addition of sink terms which can be nonlinear as well as linear in the vibrational energy. These terms describe radiative and nonradiative decay and expressions for the former are derived explicitly in terms of Franck-Condon factors. Exact solutions in terms of known functions are obtained in the linear case for several physically motivated initial vibrational distributions, viz., $\ensuremath{\delta}$-function, Boltzmann, Poisson, and Laguerre. Two perturbation schemes are developed to analyze the nonlinear case, one of which is useful for small nonlinearities and the other for arbitrarily large nonlinearities but for low temperatures. Illustrative applications of the theory include the exact calculation of time- and frequency-resolved emission spectra (linear decay calculation) and the perturbation analysis of quantum yields in the presence of strongly energy-dependent nonradiative transitions (nonlinear decay calculation).

Dynamical correlations near instabilities in nonlinear chemical reaction systems

Abstract: Projection operator techniques are applied to stochastic master equations for homogeneous chemical reaction systems to derive a continued fraction representation for dynamical correlations of the particle numbers. The formalism is applied to two simple nonlinear chemical reactions which exhibit first and second order phase transition analogies, respectively. Numerical results are obtained and various approximations are investigated to describe memory effects arising at the instability points. The method presented here provides a systematic way of investigating the dynamics of nonlinear chemically reacting systems showing unstable behaviour and enhanced fluctuations far from thermal equilibrium.

A master equation approach to the hyper-Raman effect

Abstract: The hyper-Raman effect is treated theoretically from the quantum statistical point of view. For this purpose a master equation, based on a microscopically correct Hamiltonian, is derived and then solved analytically. The solution obtained is quite general and enables computation of the complete joint probability distribution for arbitrary time and any initial conditions.

Dynamics of the generalized Glauber-Ising chain in a magnetic field

Abstract: A one-dimensional kinetic Ising model with nearest neighbor interactionJ and magnetic fieldH ⩾ 0 is treated in both linear and nonlinear response, using the most general single spin-flip transition probabilities that depend on nearest neighbor states only. The dynamics is reformulated in terms of kinetic equations for the concentration nl +(t) [@#@ nl(t) of clusters containingl up- [or down-] spins, which is exact in the homogeneous case. The initial relaxation time τ* of the magnetization is obtained rigorously for arbitraryJ, H, and temperatureT. The relaxation function is found by numerical integration forJ/T < 2. It is shown that “coagulation” of minus-clusters becomes negligible for bothJ/T andH/T large, and the resulting set of equations is solved exactly in terms of an eigenvalue problem. A perturbation theory is developed to take into account the neglected coagulation terms. The relaxation function is found to be non-Lorentzian in general, in contrast to the Glauber results atH = 0, which are recovered as a special case. In addition, nonlinear and linear relaxation functions differ forH ≠ 0. Consequences for the application to biopolymers are briefly mentioned.

Stochastic theory of phase transition in multiple steady state chemical system: Nucleation

Abstract: We study on a simple reaction–diffusion system the role of stochastic fluctuation in first order type transition between two stable steady states. We focus especially on the large fluctuations in a small spatial region that give rise to nucleation. The theory of nucleation developed is based on a birth‐and‐death type master equation. The result is different from the Langevin approach, and the physical basis of the difference is discussed.