Showing papers on "Master equation published in 1983"
TL;DR: In this paper, the authors developed a theory of stochastic transport in disordered media, starting from a linear master equation with random transition rates, and employed a Green function formalism to reduce the basic equation to a form suitable for the construction of a class of effective medium approximations (EMAs).
Abstract: 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 i...
129 citations
TL;DR: In this paper, the authors considered photodesorption due to a laser resonantly coupled into an internal vibrational mode of an adsorbed molecule and calculated desorption rates as a function of temperature and laser intensity.
Abstract: Photodesorption due to a laser resonantly coupled into an internal vibrational mode of an adsorbed molecule is considered. Based on a master equation with transition probabilities calculated quantum statistically for laser-induced vibrational transitions, phonon-mediated bound-state---bound-state and bound-state---continuum transitions, and tunneling transitions from bound states degenerate with the continuum, we calculate desorption rates as a function of temperature and laser intensity. They are highly nonlinear in both, and become saturated for high intensities. The theory is applied to the C${\mathrm{H}}_{3}$F-NaCl and CO-NaCl systems.
95 citations
TL;DR: In this article, a random walk model is introduced to describe the dynamical behavior of the excitations in a domain and is used to calculate the parameter that determines the shape of the total fluorescence yield vs. pulse intensity curve in the case in which the reaction centers are all in the closed state.
95 citations
TL;DR: In this paper, the migration of a classical dynamical system between regions of configuration space can be treated as a continuous time random walk between these regions, and a short memory approximation to these memory functions is equivalent to the well-known transition state method.
Abstract: The migration of a classical dynamical system between regions of configuration space can be treated as a continuous time random walk between these regions. Derivation of a classical analog of the quantum mechanical generalized master equation provides expressions for the waiting time distribution in terms of transition memory functions. A short memory approximation to these memory functions is equivalent to the well-known transition state method. An example is discussed for which this approximation seems reasonable but is entirely wrong.
77 citations
TL;DR: In this article, the relationship between the five direct simulation methods of Nanbu, Belotserkovskiy and Yanitskiy, Koura, Deshpande, and Bird is examined by means of probability consideration.
Abstract: Interrelations between the five direct simulation methods of Nanbu, Belotserkovskiy and Yanitskiy, Koura, Deshpande, and Bird are examined by means of probability consideration. These methods are divided into two groups; one based on the Boltzmann equation and the other based on the Kac master equation. Nanbu's method belongs to the first group while all other methods belong to the second group. The methods of Koura, Deshpande, and Bird are modifications of that of Belotserkovskiy and Yanitskiy. Although the probability law for simulating molecular collisions is different from method to method, the mean rate of collision is essentially the same for all methods. This implies that there is no significant difference at least among macroscopic properties resulting from all methods.
75 citations
TL;DR: In this paper, a mean-field coupling between a system of noninteracting subsystems and the bath is investigated, and the existence and uniqueness of solutions of the corresponding nonlinear evolution equations are studied in a more abstract framework.
Abstract: The notion of a nonlinear quantum dynamical semigroup is introduced, and the existence and uniqueness of solutions of the corresponding nonlinear evolution equations are studied in a more abstract framework. The construction of nonlinear quantum dynamical semigroups is carried out for two different mean-field models. First a mean-field coupling between a system of noninteracting subsystems and the bath is investigated. As examples, a nonlinear frictional Schrodinger equation and a model for a quantum Boltzmann equation are discussed. Second, a many-body system with mean-field interaction coupled to a bath is considered. Here, again, the form of the generator is derived; however, it cannot be obtained rigorously, except for some particular examples. Finally, the quantum Ising-Weiss model is briefly studied.
73 citations
TL;DR: In this article, a reformulation and a generalization of a quantum theory of non-conservative (dissipative) systems was presented, which can be equivalently described by the Schrodinger or Heisenberg picture.
Abstract: We present here in a systematic way a reformulation and a generalization of a quantum theory of nonconservative (dissipative and antidissipative) systems already outlined by us many years ago. In particular, following a procedure first introduced by Levi Civita we give a detailed formulation of the corresponding classical Lagrangian and Hamiltonian treatments and consequently we show that quantum nonconservative systems can be equivalently described by the Schrodinger or Heisenberg picture. Furthermore, a detailed discussion of uncertainty rules for nonconservative systems is developed. By means of such a formulation it is possible to overcome easily criticisms raised against the so-called Caldirola-Kanai equation. Finally the connection between the Schrodinger equation for nonservative systems and the master equation is shortly discussed and some new possible developments of the theory are suggested.
70 citations
TL;DR: In this article, a stochastic description of an exothermic reaction leading to adiabatic explosion is set up, and the numerical solution of the master equation reveals the appearance of a long tail and of multiple humps of the probability distribution, which subsist for a certain period of time.
Abstract: A stochastic description of an exothermic reaction leading to adiabatic explosion is set up. The numerical solution of the master equation reveals the appearance of a long tail and of multiple humps of the probability distribution, which subsist for a certain period of time. During this interval the system displays a markedly chaotic behavior, reflecting the random character of the ignition process. An analytical description of this transient evolution is developed, using a piecewise linear approximation of the transition rates. A comparison with other transient phenomena observed in stochastic theory is carried out.
66 citations
TL;DR: In this article, the Newman-Penrose formalism is used to work with gravitational, electromagnetic, and Dirac field perturbations of the Kerr-de Sitter space.
Abstract: The Newman-Penrose formalism is used to work with gravitational, electromagnetic, and Dirac field perturbations of the Kerr---de Sitter space. It is shown that the resulting equations are separable, and the radial parts (for the massless fields) combine into a master equation resembling that of Teukolsky. This master equation includes the Teukolsky equation and the equation arising from the de Sitter-Schwarzschild universe, and can be reduced to these cases under appropriate limiting conditions. Finally, the radial parts of the electromagnetic and neutrino fields are transformed to the form of the one-dimensional barrier-penetration equation.
61 citations
TL;DR: In this paper, a general theory to describe equilibrium as well as nonequilibrium transport properties of systems in which the carriers perform an incoherent motion that can be described by means of a set of master equations.
Abstract: We present a general theory to describe equilibrium as well as nonequilibrium transport properties of systems in which the carriers perform an incoherent motion that can be described by means of a set of master equations. This includes hopping as well as trapping in the localized energy region of amorphous or perturbed crystalline semiconductors. Employing the mathematical analogy between the master equations and the tight binding problem we develop approximation schemes using methods of many-particle physics to derive expressions for the averaged propagator of the carriers and the conductivity tensor. The calculated conductivity and Hall conductivity of hopping systems compare extremely well to computer simulations over the whole range of frequency, density, and temperature. We are able to derive expressions for dispersive transport in hopping as well as trapping systems that contain the results of earlier theories of Scher, Montroll and Noolandi, Schmidlin as special cases and establish criteria for the occurrence of dispersive transport in such systems. We find that in principle hopping can lead to dispersive transport if the times and densities are very low, but actual experimental data are more easily explained in terms of multiple trapping.
57 citations
TL;DR: In this article, the authors derived the kinetic equations for a one-dimensional model of phonon-induced desorption, and discussed approximations that reduce it to a master equation of the Pauli type.
Abstract: We derive the kinetic equations for a one‐dimensional model of phonon‐induced desorption. We start with a generalized master equation and discuss approximations that reduce it to a master equation of the Pauli type. This describes desorption as a stochastic process in which the energy of the chemisorptive bond has random variations caused by the thermal fluctuations of the lattice atoms. Desorption occurs when the bond energy exceeds a certain value, placing the adsorbed particle in a continuum state of the surface‐particle potential. The calculation of the rates of energy transfer includes multiphonon processes and allows transitions between all the bound levels of the adsorbed particle. Numerical calculations are presented in a subsequent paper.
TL;DR: In this paper, an exact algebraic (master) equation for the euclidean master field of any large-N matrix theory, including quantum chromodynamics, is derived, which is the quenched Langevin equation, a translationally covariant function of (uniform) random momenta and (gaussian) random noise.
TL;DR: In this article, a new approach to describe fluctuations of reversible chemical reactions in closed systems is proposed, where deterministic rate laws are cast into the form of nonlinear Onsager type closed laws, and a Fokker-Planck equation describing the stochastic process of concentration fluctuations is obtained.
Abstract: The paper proposes a new approach to describe fluctuations of reversible chemical reactions in closed systems. The deterministic rate laws are cast into the form of nonlinear Onsager type closed laws. By means of nonlinear transport theory a Fokker-Planck equation describing the stochastic process of concentration fluctuations is obtained. It is shown that the stochastic formulation reduces to the correct deterministic rate laws in the thermodynamic limit V → ∞ with the concentrations kept fixed. Concrete examples of reactions in ideal mixtures are given and the results of the presented approach are compared with those of the usual approach by means of a birth and death type master equation. It is shown that both approaches lead to the same stationary probability and exhibit the same natural boundaries reflecting the fact of a restricted state space. The proposed Fokker-Planck equation is different from the Fokker-Planck equation obtained from the master equation by truncating its Kramers-Moyal expansion. However, the two equations are shown to have identical Fokker-Planck coefficients in the vicinity of the deterministic equilibrium state. Compared with the usual master equation approach the proposed stochastic modeling of chemical reactions has the advantage of allowing for a straightforward extension to reactions in non-ideal mixtures.
TL;DR: In this paper, the exciton model of nuclear reactions with the distinguishability of protons and neutrons explicitly included is discussed and the Pauli master equation and transition rates in this two-component formulation are given.
Abstract: The exciton model of nuclear reactions with the distinguishability of protons and neutrons explicitly included is discussed. The Pauli master equation and transition rates in this two-component formulation are given. Both the simple considerations and actual calculations show that the two-component model exhibits new features and modifies conclusions of the one-component formulation. The ratio of the proton to neutron preequilibrium emissions in proton induced reactions decreases twice to three times when compared with the usual one-component model.
TL;DR: In this article, a new master equation describing the irreversible process of a quantum mechanical Brownian particle is proposed, which obeys the symmetry of detailed balance leading to a quantum analog of the reciprocity relations.
Abstract: A new master equation describing the irreversible process of a quantum mechanical Brownian particle is proposed. The master equation is shown to obey the symmetry of detailed balance leading to a quantum analog of the reciprocity relations, and the fluctuation-dissipation theorem is obtained. The method is applied to the damped harmonic oscillator. The relation to previous approaches is discussed.
TL;DR: In this article, the stochastic solution method of two integro-differential equations for probability density is described, one is the master equation appearing in the theory of Stochastic processes and the other is the Kac model of the Boltzmann equation, and the basic idea of the method is in that a set of mutually independent random variables sampled from the probability density can take the place of the role of a probability density function.
Abstract: Here is described the stochastic solution method of two integro-differential equations for probability density; one is the master equation appearing in the theory of stochastic processes and the other is the Kac model of the Boltzmann equation. The basic idea of the method is in that a set of mutually independent random variables sampled from the probability density can take the place of the role of the probability density function. A few examples are calculated. When the exact solution is known, it is ascertained that the solution obtained from the stochastic solution method agrees with the exact solution.
TL;DR: A generalized quantum Liouville equation of the form iℏ(∂ρ/∂t)=Huρ−ρHl for the density operator ǫ(t) is introduced in this paper.
Abstract: A generalized quantum Liouville equation of the form iℏ(∂ρ/∂t)=Huρ−ρHl for the density operator ρ(t) is introduced; the quantum mechanical Hamiltonians Hu and Hl are, in general, different operators. Such an equation is of interest for a variety of problems involving systems at finite temperatures, including the calculation of vibrational and electronic spectra of molecules that are initially distributed over a range of eigenstates. It is shown that the generalized Liouville equation can be solved by exploiting its equivalence to a time‐dependent Schrodinger equation in the coordinate space representation. In particular, this equivalence makes it possible to utilize techniques of Schrodinger wave packet propagation to compute the time evolution of the desired operator. Application of Gaussian wave packet dynamics and its extensions is considered and shown to be justified when the coordinate representation of ρ(t) remains essentially Gaussian throughout the course of the relevant dynamics. As an...
TL;DR: In this paper, the authors carried out a systematic adiabatic elimination of the atomic degrees of freedom from the quantum-mechanical master equation for the single-mode laser.
Abstract: We carry out a systematic adiabatic elimination of the atomic degrees of freedom from the quantum-mechanical master equation for the single-mode laser We represent the reduced density operator of the field mode by various quasiprobabilities and construct the respective equations of motion The generators of infinitesimal time translations are obtained as series in powers of two parameters, the smallness of which defines the adiabatic limit Our ''adiabatic'' expansion treats that part of the atom-field interaction as a zeroth-order effect which describes the action of the field on the atoms, while the reaction of the atoms is treated perturbatively The adiabatic equilibrium thus assigned to the atomic variables at all times is a conditional one, contingent on the current state of the field mode As a result, saturation effects in the atoms are fully accounted for in low orders of our expansion In second order, especially, we obtain a Fokker-Planck equation for the Wigner function of the field mode which is valid for arbitrary pump strengths below, near, and above threshold We compare our results with those of previous theories
01 Feb 1983
TL;DR: In this article, a finite-basis set method of solving the weak-collision master equation of thermal unimolecular reactions in the general pressure regime is presented, where successive sections of the equilibrium probability density are used as basis functions.
Abstract: A new finite-basis-set method of solving the weak-collision master equation of thermal unimolecular reactions in the general pressure regime is presented. Consecutive sections of the equilibrium probability density are used as basis functions. Significant advantages in terms of efficiency and applicability are obtained. Representative calculations are performed to illustrate the method's convergence properties and storage requirements. Calculations of collisional efficiency factors in the low-pressure limit β 0 and weak-collision broadening factors F W C (ω) are performed to offer a simple concise representation of weak-collision effects. It is noted that weak-collision effects can be incorporated into fall-off curves with an accuracy of within an order of magnitude by simple scaling of the strong-collision fall-off curves. However, for accurate representation of weak-collision effects the weak-collision broadening must be taken into account.
TL;DR: In this article, a master equation with nearest-neighbor transfer rates (symmetric or asymmetric) was proposed to describe the one-dimensional lattice and the transfer rates associated with bonds were assumed to be independent, equally distributed random variables.
Abstract: Diffusion on the one-dimensional lattice ℤ is described by a master equation with nearest-neighbor transfer rates (symmetric or asymmetric). The transfer rates associated with bonds are assumed to be independent, equally distributed random variables. Under various conditions on their common distribution the large time behavior of averaged site probabilities and/or related quantities is exhibited.
TL;DR: In this article, the authors show that the occupation probability of a compound system decays in time like a superposition of exponentials, with decay rates equal to the energy autocorrelation widths of the n eigenclasses of the system.
01 May 1983
TL;DR: In this paper, a critical review of reaction-rate theory and experiment is given from the engineers' point of view, where rates of heavy-particle, collision-induced reaction in gas phase are formulated in terms of the cross sections and activation energies for reaction.
Abstract: Reaction-rate theory and experiment are given a critical review from the engineers' point of view. Rates of heavy-particle, collision-induced reaction in gas phase are formulated in terms of the cross sections and activation energies for reaction. The effect of cross section function shape and of excited state contributions to reaction both cause the slope of Arrhenius plots to differ from the true activation energy, except at low temperature. The master equations for chemically reacting gases are introduced, and dissociation and ionization reactions are shown to proceed primarily from excited states about kT from the dissociation or ionization limit. Collision-induced vibration, vibration-rotation, and pure rotation transitions are treated, including three-dimensional effects and conservation of energy, which have usually been ignored. The quantum theory of transitions at potential surface crossing is derived, and results are found to be in fair agreement with experiment in spite of some questionable approximations involved.
TL;DR: Solutions to the Fokker-Planck equation for a damped harmonic oscillator corresponding to uniquely quantum-mechanical initial states with no classical correspondence are given in this article.
Abstract: Solutions to the Fokker–Planck equation for a damped harmonic oscillator corresponding to uniquely quantum‐mechanical initial states with no classical correspondence are given.
TL;DR: In this article, the authors analyzed one-dimensional random walks with static disorder using real space renormalization group procedure and showed that the presence of disorder leads to a non-Markovian description of the macroscopic behavior of the random walk.
Abstract: One-dimensional random walks with static disorder are analyzed using a real space renormalization group procedure The presence of disorder leads to a non-Markovian description of the macroscopic behavior of the random walk We consider random walks with nearest-neighbor hopping described by a master equation with both on-site and site-to-site disorder in the transition matrix Site-to-site disorder leads to a generalized diffusion coefficient with at−3/2 long time tail whereas on-site disorder leads to a generalized Burnett coefficient with at−1/2 long time tail
TL;DR: In this article, the first passage time problems for a class of one-dimensional master equations with separable kernels are discussed. But the boundary conditions for higher moments differ slightly from those appropriate to simple diffusion.
Abstract: We discuss first passage time problems for a class of one-dimensional master equations with separable kernels. For this class of master equations the integral equation for first passage time moments can be transformed exactly into ordinary differential equations. When the separable kernel has only a single term the equation for the mean first passage time obtained is exactly that for simple diffusion. The boundary conditions, however, differ from those appropriate to simple diffusion. The equations for higher moments differ slightly from those for simple diffusion. Analysis is presented, of a generalization of a model of a random walk with long-range jumps first investigated by Lindenberg and Shuler. Since the equations can be solved exactly one can study the behavior of boundary conditions in the continuum limit. The generalization to a larger number of terms in the separable kernel leads to higher order equations for the first passage time moments. In each case, boundary conditions can be found directly from the original master equation.
TL;DR: A variety of considerations from different points of view including non-Markovian stochastic processes, basic quantum mechanics, and a mechanism based on condensed matter physics, all lead to the fractional exponential decay at long times in relaxation processes as mentioned in this paper.
Abstract: A variety of considerations from different points of view including non-Markovian stochastic processes, basic quantum mechanics, and a mechanism based on condensed matter physics, all lead to the fractional exponential decay at long times in relaxation processes. Implications of this decay law and its verifiable predictions in a broad range of phenomena in condensed matter physics are pointed out.
TL;DR: In this paper, the authors derive equations which relate coherent medium results for bond and site averaging and show how these reduce to the two-body solution results of Gochanour, Andersen, and Fayer.
Abstract: Average-T-matrix and coherent medium theories are used to study the motion of localized excitations on substitutionally disordered lattices. We derive equations which relate coherent medium results for bond and site averaging and show how these reduce to the two-body solution results of Gochanour, Andersen, and Fayer. Numerical results for Po(t), the probability of remaining at the origin for two-dimensional nearest-neig~abor lattices are presented.
TL;DR: In this paper, a Langevin equation for the Gross-Kitazawa constrained gluon is presented, and total space-time quenching is demonstrated to all orders for this field, resulting in an algebraic master equation for a properly constrained quenched master field for continuum QCD.
TL;DR: The possible existence of squeezed states in two-atom resonance fluorescence is discussed in Lehmberg's master equation approach in this article, and it is shown that squeezing strongly depends on interatomic separations r12.
TL;DR: In this article, the existence of straight-line reaction paths for various reaction systems was deduced and the general conditions for their existence in isomerization systems, dissociation-recombination reactions, and bimolecular reactions were discussed.
Abstract: We have deduced the existence of straight‐line reaction paths for various reaction systems. The general conditions for their existence in isomerization systems, dissociation–recombination reactions, and bimolecular reactions are discussed. We have also derived the invariant quantity that characterizes the internal‐state distributions of either reactants or products when the phenomenological rate constant is steady. An invariant vector is obtained for first‐order reactions, and an invariant tensor is obtained for second‐order reactions. Analytic expressions for these invariant quantities are given for several cases. The application and usefulness of an invariant vector is illustrated for an example reaction by using it to calculate the internal‐state population distributions at any time during the forward or reverse reactions from the internal‐state distributions at any one time during either. These calculations check satisfactorily against the distributions obtained by numerical solution of the master equation.