Showing papers on "Master equation published in 1981"

Stochastic processes in physics and chemistry

Abstract: Preface to the first edition. Preface to the second edition. Abbreviated references. I. Stochastic variables. II. Random events. III. Stochastic processes. IV. Markov processes. V. The master equation. VI. One-step processes. VII. Chemical reactions. VIII. The Fokker-Planck equation. IX. The Langevin approach. X. The expansion of the master equation. XI. The diffusion type. XII. First-passage problems. XIII. Unstable systems. XIV. Fluctuations in continuous systems. XV. The statistics of jump events. XVI. Stochastic differential equations. XVII. Stochastic behavior of quantum systems.

Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions

Abstract: The relationship between the atomic coherent-state representation of Arecchi et al. [Phys. Rev. A 6, 2211 (1972)] and the state multipoles is established. The state multipoles are used to develop a theory of generalized phase-space distributions for angular momentum (collective atomic) systems. The general theory for angular momentum systems is shown to have many features in common with the general theory for boson systems [Phys. Rev. D 2, 2161 (1970)]. These generalized phase-space distributions contain as a special case the coherent-state representation of Arecchi et al. The applications of the generalized phase-space distributions and state multipoles to the dynamical problems and to the calculation of multitime correlations are given. State-multipole techniques are used to give a brief discussion of the master equation describing cooperative resonance fluorescence.

On the theory of hopping conductivity in disordered systems

Abstract: Starting with the linearised master equation, the authors present a first-principles theory of conductivity for disordered systems. The theory is valid for all situations to which the master equation description applies. The path summation and configurational average are evaluated by generalising relations obtained from exactly soluble models. Both symmetric and asymmetric energy-dependent transition frequencies are considered. In the latter case they are able to define an energy-dependent conductivity from which it is possible to evaluate the thermopower. All electronic transport properties, including the frequency-dependent conductivity can be evaluated self-consistently, the only input parameter being the density of states. Numerical results for the DC conductivity and thermopower are presented using several model density-of-states functions. For random statistics, the results are in complete agreement with percolation theory for low densities (temperature). The theory is exact in the high-density (temperature) limit.

Milne's differential equation and numerical solutions of the Schrodinger equation. I. Bound-state energies for single- and double-minimum potentials

Abstract: Milne's approach to the numerical solution of the Schrodinger equation via a non-linear differential equation an the quantisation of a quantum action is investigated in detail. An accurate and efficient computational method is presented which allows a rapid second-order convergence onto a desired eigenenergy En. Numerical sample calculations demonstrate the efficiency of the method, which has special advantages for accurate calculations of high quantum states. The present method can be easily extended to the calculation of quasi-bound levels at resonance (complex-values) energies.

Master equation derivation of quantum regression theorem

Abstract: The master equation approach is used to relate the calculation of correlation functions to the calculation of single-time expectation values. The quantum regression theorem is shown to result on neglecting a certain term. The properties of this neglected term are briefly discussed.

Quantum kinetic equations in systems with bound states

Abstract: Starting from the quantum mechanical BBGKY-hierarchy kinetic equations in systems with two particles bound states are given in this paper. With this equation it is possible to consider transport properties in nonideal gases with three particles reactions.

Nonequilibrium fluctuations in μ space

Abstract: The properties of fluctuations inμ space in or outside thermal equilibrium are obtained by solving hierarchies of equations derived either from the Liouville or the Master equation. In particular we study the one-, two-, etc., time correlation functions that describe the spatial and temporal behavior of the fluctuations inμ space. Explicit solutions are obtained for a dilute gas. The Langevin approach is briefly discussed. Our results are compared with those obtained in the extensive literature, which is reviewed in some detail.

Brownian motion of harmonic systems with fluctuating parameters: III Scaling and moment instabilities

Abstract: We examine the stability properties of equilibrium moments of all orders for the damped mechanical oscillator with a delta correlated fluctuating frequency. A Markovian master equation is derived startimg from a frequency fluctuation process with finite correlation time τc and the limit τc→0 is taken. To approach this limit systematically, the oscillator and frequency fluctuation parameters are expressed in terms of a dimensionless scaling parameter. We derive exact integer moment transport equations in the limit of vanishing correlation time. These equations, and hence the moments, depend only on the second cumulant of the frequency fluctuations and not on the higher cumulants. The conjecture of Bourret et al.1) that for given frequency fluctuations, however weak, all moments beyond a certain order diverge is proved. We therefore conclude that the equilibrium distribution of the oscillator displacement and momentum cannot be Gaussian. A simple algebraic relation is established between the order of the lowest unstable moments and the system parameters.

Kinematics of the forced and overdamped sine-Gordon soliton gas

Abstract: Motion of a driven and heavily damped sine-Gordon chain with a low density of kinks and tight coupling between particles is controlled by the nucleation and subsequent annihilation of pairs of kinks and antikinks. We show that in the steady state there are no spatial correlations between kinks or between kinks and antikinks. For a given number of kinks and antikinks all geometrical distributions are equally alike, as in equilibrium. A master equation for the probability distribution for the number of kinks on a finite chain is solved, and substantiates the physical reasoning in previous work. The probability distribution characterizing the spread along the direction of particle motion of a finite chain in equilibrium as well as in the driven overdamped case is derived by simple combinatorial considerations. The spatial spread of a driven chain in the thermodynamic limit does not approach a steady state; a given particle followed in time deviates as t1/2 from its average forced motion. This result follows from the hydrodynamic equations for the dilute kink gas. Comparison is made with other recent results.

First passage time problems for one-dimensional random walks

Abstract: This note contains a development of the theory of first passage times for one-dimensional lattice random walks with steps to nearest neighbor only. The starting point is a recursion relation for the densities of first passage times from the set of lattice points. When these densities are unrestricted, the formalism allows us to discuss first passage times of continuous time random walks. When they are negative exponential densities we show that the resulting equation is the adjoint of the master equation. This is the lattice analog of a correspondence well known for systems describable by a Fokker-Planck equation. Finally we discuss first passage problems for persistent random walks in which at each step the random walker continues in the same direction as the preceding step with probability a or reverses direction with probability 1−α

Theory of quantum beats: Master equation approach

Abstract: Based on the master equation approach, an expression for the time dependent fluorescence intensity has been derived taking into account both collision effects resulting from the elastic and inelastic scattering from the heat bath and the statistical properties of the incident light. The degree of coherence for the molecular eigenstates is defined. The pure dephasing constant between the excited states attenuates the degree of coherence for the molecular eigenstates during and after the light pulse duration; on the other hand, the pure dephasing between the ground and excited states attenuates it during the pulse duration. A separable nonstationary correlation function for the incident radiation field is introduced to investigate the effects of the field coherence on the time dependent fluorescence. Model calculations of the quantum beats in the fluorescence from molecules with two levels in the excited state in the weak radiation field have been performed to demonstrate the effects of the dephasing consta...

Precompound decay in heavy-ion reactions

Abstract: The Boltzmann master equation approach of Harp, Miller, and Berne has been modified to treat the coalescence and equilibration sequence which might be expected in heavy-ion reactions. Assumptions made to accomplish these ends are discussed. Results are presented for sample reactions $^{109}\mathrm{Ag}(^{40}\mathrm{Ar},X)$ at 240 and 320 MeV projectile energy, and $^{197}\mathrm{Au}(^{16}\mathrm{O},X)$ at 214 and 320 MeV. For the latter case, theoretical proton spectra (angle integrated) are compared with experimental yields measured by Symons et al. Very good agreement was found for the shapes of the calculated and experimental precompound spectra. Types of experimental data necessary to rigorously test this model approach are discussed.NUCLEAR REACTIONS Precompound model for heavy-ion reactions, Boltzmann master equation approach for coalescence plus equilibration phases. Predict $\frac{d\ensuremath{\sigma}}{d\ensuremath{\epsilon}}$ for $n$ and $p$ emission for $^{197}\mathrm{Au}(^{16}\mathrm{O},X)$ and $^{109}\mathrm{Ag}(^{40}\mathrm{Ar},X)$ reactions each at two bombarding energies.

IR photochemistry: A unified approach for single‐channel reactions. I. Theory and computational examples

Abstract: A model for infrared multiphoton decomposition of polyatomic molecules based on an energy‐grained master equation (EGME) shows remarkably uniform behavior over a wide range of physically reasonable model parameters. These parameters encompass the size of the molecule, the reaction threshold energy, the variation of the decomposition rate constant above the threshold, and the functional dependence of the infrared absorption process on the internal energy of the molecule. The results are shown to be uniformly well represented by a cumulative log–normal distribution function for reaction yield versus time or fluence. A complementary approach based on the distribution of first passage times for a Markovian stochastic process in discrete state space and continuous time shows that the yield behavior of the EGME can be conveniently calculated without solution of the full set of equations. The uniform behavior of the model results yields a powerful method for presenting and analyzing experimental data.

A comment on the dynamics of excitation trapping in molecular crystals

Abstract: In this paper we consider the long‐time trapping rate, by a deep sink, of an excitation which migrates incoherently in a one‐dimensional molecular crystal. The approaches based on the first passage time version of the continuous time random walk and on the master equation formalism are compared and put in a unified framework. Time domains are predicted for the different behaviors of the trapping rates for both one‐ and two‐dimensional systems.

Non-Markov processes: The problem of the mean first passage time

Abstract: The theory of the mean first passage time is developed for a general discrete non-Markov process whose time evolution is governed by a generalized master equation. The mean first passage time is determined by an adjoint matrixΩ + in a form analogous to the Fokker Planck case. The theory is illustrated by two examples: A one-dimensional unit step non-Markov process and a non-Markov process with two-step transitions. Explicit expressions for the mean first passage time are derived.

On the equivalence of some exact master equations

Abstract: In this note we show the equivalence of two procedures for obtaining exact, convolutionless master equations.

On the relations between Markovian master equations and stochastic differential equations

Abstract: We have investigated the relationship between Markovian master equations (m.e.) and the corresponding stochastic differential equations (s.d.e.) for closed systems, i.e., systems not subjected to external pumping. We show that the form of the fluctuations in the s.d.e., i.e., additive or multiplicative, depends upon the properties of the kernel of the m.e. and the range of the state space of the stochastic variable(s), i.e., bounded or unbounded. The knowledge of these two properties of the m.e. permits the determination of the way in which the fluctuations enter into the s.d.e. (i.e., additive or multiplicative) and the calculation of their statistics. Several examples are presented to illustrate the general theory.

Master equations for photochemistry with intense infrared light (IV). A unified treatment of case B and case C including nonlinear effects

Abstract: A unified master equation for unimolecular reactions induced by monochromatic infrared radiation (URIMIR) is presented. Its effective rate coefficient matrix covers both case B (Pauli equation) and case C, properly including the nonlinearity of the latter. Exact quantum mechanical model solutions are compared with results from the approximate unified master equation. The exact analytical solutions of the master equation are presented for the URIMIR of some realistic molecular models. The important new properties of the transition range between case B and case C are quantitatively discussed with respect to time dependent and steady state level populations, time dependent and steady state rate coefficients and their nonlinear intensity dependence, and with respect to the influence of molecular properties. The role of case C for the interpretation of static field effects and its importance for efficient isotope separation are pointed out.

Statistical-dynamical theory of nonlinear stochastic processes: II. Time-convolutionless projector method in nonequilibrium open systems

Abstract: A new statistical-dynamical theory of nonlinear stochastic processes in nonequilibrium open systems is presented. It is shown by means of the time-convolutionless projector method that multiplicative type stochastic equations of motion for relevant variables Ai(t) can always be transformed exactly into additive type (Langevin type) stochastic equations of motion for Ai(t) and the corresponding master equation for the probability distribution. The Langevin type equation consists of a drift term and a fluctuating force. The fluctuating force is shown to give a new kind of additive stochastic process which has a quite different feature from the ordinary additive processes. The importance of Langevin equations of this type in the multiplicative stochastic process is pointed out from the statistical-dynamical viewpoint. A new cumulant expansion of the master equation in powers of stochastic forces is also found. The application of the dynamic renormalization group method to steady states far from thermal equilibrium is discussed from the statistical-dynamical standpoint developed in the present paper. The present theory is applied to renormalize nonlinear stochastic equations by eliminating the short-wavelength modes and thus to derive nonlinear Langevin equations with renormalized kinetic coefficients and the corresponding master equation. The renormalized coefficients are then used not only to find recursion relations but also to calculate explicitly a cutoff dependence of the kinetic coefficients. As an actual example, fluctuations in the generalized time-dependent Ginzburg-Landau model are investigated near its nonequilibrium tricritical point.

Two-component equilibration in the exciton model of nuclear reactions

Abstract: A generalized two-components master-equation approach for a nuclear system towards equilibration is described, two-components being the proton and the neutron components. An approximate closed form solution of the two-component equation is discussed. Further an effective one-component master equation is derived from it. Explicit expressions for effective transition and emission rates are derived under a binomial model. The mean free path of an exciton has been found to be reduced in accounting for the two-component equilibration.

Connection of the velocity autocorrelation function to the mean-square-displacement and to the memory function of generalized master equations

Abstract: General connections between the velocity autocorrelation function and the mean-square-displacement for a special initial condition are established and shown to reduce in the high-temperature limit to earlier such results obtained by Scher and Lax. A simple and useful relation, which is valid in the high temperature limit, between the velocity autocorrelation function and memory functions in generalized master equations is given.

Kinetic equations for desorption

Abstract: Starting from a set of rate equations for the bound-state occupation functions for gas-solid systems in which the surface potential has many physisorbed bound states, we derive a master equation; its kernel is explicitly calculated for phonon-mediated adsorption and desorption in a Morse potential. We give the equivalent Smoluchowski-Chapman-Kolmogorov equation for which we find the Kramers-Moyal expansion. Identifying van Kampen's large parameter $\ensuremath{\Omega}$ for such gas-solid systems, we establish explicit criteria for the validity of a Fokker-Planck equation.

Desorption times from rate equations, the master equation, and the Fokker-Planck equation

Abstract: A set of rate equations for the bound-state occupation functions with transition probabilities calculated microscopically serves as a basis for a detailed study of various approximations available for the calculation of desorption times in gas-solid systems exhibiting physisorption. The exact time evolution of the adsorbate during the desorption process shows that quasiequilibrium is only maintained at low temperatures, where perturbation theory of the master equation yields a simple analytic expression for the desorption time in weakly coupled gas-solid systems. At intermediate temperatures we derive another simple expression from the Fokker-Planck equation. Classical and phenomenological equilibrium theories of desorption are critically assessed. Lower limits for the preexponential factor in the desorption time of the order of ${10}^{\ensuremath{-}16}$ sec proportional to the inverse of the heat of adsorption are derived.