Showing papers on "Master equation published in 1968"
TL;DR: In this paper, an analytic expression is given for the distribution maintained by the vibration-vibration mechanism of a simple harmonic oscillator, which reduces to the usual Boltzmann-like distribution defined by a single vibrational temperature.
Abstract: The terms in the master equation for vibrational relaxation of anharmonic oscillators are ordered according to the rates of the relaxation processes (vibrational exchange, vibrational‐energy transfer to translation). The population distributions in the master equation are expanded about their values when the vibration‐vibration mechanism is the only one present. An analytic expression is given for the distribution maintained by the vibration‐vibration mechanism. In the limiting case of the simple harmonic oscillator, this distribution reduces to the usual Boltzmann‐like distribution defined by a single vibrational temperature. The general solution also applies to a mixture of simple‐harmonic‐oscillator gases of different fundamental frequencies. For such a mixture, each gas relaxes in a Boltzmann‐like distribution, but the different gases have different (but related) vibrational temperatures at any given time. The relaxation of the first moment of the distribution function also has been investigated. Anharmonicity causes a marked departure from the Landau‐Teller model of vibrational relaxation under conditions of high vibrational energy, coupled with low translational temperature. For such conditions, the populations of the lower vibrational states can be considerably lower than those predicted by the Landau‐Teller model. Furthermore, the over‐all energy relaxation rate can be accelerated.
703 citations
137 citations
TL;DR: In this paper, an exact master equation (EME) of the markovian form for the diagonal part of the density operator is derived from quantum dynamical laws, which is valid for systems with a time-dependent Hamiltonian also.
86 citations
TL;DR: In this article, a theoretical framework for the nonequilibrium statistical mechanics of open systems is presented, which is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary system S in interaction with a large reservoir R. The dynamics of S are analyzed on the basis of a precise quantum-mechanical treatment of the microscopic equations of motion for the combined system S + R.
Abstract: A theoretical framework for the nonequilibrium statistical mechanics of open systems is constructed This is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary system S in interaction with a ``large'' reservoir R The dynamics of S are analyzed on the basis of a precise quantum‐mechanical treatment of the microscopic equations of motion for the combined system S + R On proceeding to the thermodynamical limit for R we obtain a generalized master equation for S, subject to specified conditions on the many‐particle structure of R, its initial state, and its coupling to S This master equation corresponds to a self‐contained law of motion for S, in which the R variables appear only in the forms of certain thermal averages, taken over the initial state This dynamical law is a generalization of the quantum‐mechanical Liouville equation to a form appropriate to open systems
82 citations
TL;DR: The asymptotic dependence of trapping time on N, the number of sites per trap, found to be proportional to N ln N by Pearlstein (1966) and by Robinson (1967) in square networks, has been verified and extended to the triangular case.
62 citations
TL;DR: In this paper, a Boltzmann-like master equation for a Fermi-gas system with residual two-body interaction was proposed to solve the problem of high-energy nuclear reactions.
Abstract: The usual interpretation of high-energy nuclear reactions entails a two-step mechanism: (1) a first step, which includes the emission of knock-on particles in an intranuclear cascade generated by the incident particle, followed by (2) the "evaporation" of particles from the residual excited nucleus, which is assumed to be at statistical equilibrium. If the independent-particle model with residual two-body interaction is taken as a description of the residual excited nucleus, the assumption in the second step requires that the nonequilibrium distribution of nucleons and holes that are produced in the fast step approach the equilibrium distribution before more particles leave the nucleus. This requirement has been investigated and shown to be valid by numerically solving a Boltzmann-like master equation for a Fermi-gas system.
38 citations
23 citations
TL;DR: In this article, the authors extended the analysis to the case of binary mixtures of gases and showed that large differences can exist between population factors (or vibrational temperatures in the harmonic model) for each species at steady state.
Abstract: A previous paper on vibration processes in a pure gas is extended to the case of binary mixtures of gases. Through numerical integration of the master equation for vibrational relaxation, three distinct time scales to the relaxation processes are identified. The first time scale characterizes the pure‐gas vibrational exchange reactions, the second relaxation time describes the coupling between species, while vibration–translation processes are characterized by a third time scale. Depending on the relative vibrational spacing and vibration–translation rates of the components in the mixture, vibrational distributions consistent with each of these time scales may be evident during the relaxation process. Through a steady‐state analysis to the vibrational exchange terms in the master equation, it is demonstrated that large differences can exist between population factors (or vibrational temperatures in the harmonic model) for each species at steady state. Only at high kinetic temperatures (i.e., >1000°K) or f...
21 citations
TL;DR: For weak-coupled master equation, the authors showed that the choice of H function which preserves the usual relationship with the thermodynamic entropy is convex as a function of time.
Abstract: We show, for the weak‐coupled master equation, that the choice of H function which preserves the usual relationship with the thermodynamic entropy is convex as a function of time.
14 citations
TL;DR: In this paper, the rate equation for non-Markowian behavior and the approximations that are necessary in order to obtain the conventional Markowian rate equation are investigated.
Abstract: Starting from a field theoretical representation of chemical reactions, the rate equation is developed in terms of the quantum mechanical Green's functions. The equation is valid for non‐Markowian behavior and the approximations that are necessary in order to obtain the conventional (Markowian) rate equation are investigated. The effect of finite collision times is also discussed.
10 citations
TL;DR: In this paper, a model for the production of double resonance signals in the limit of short τ seconds was presented, where the observing rf field H1 and the strong rf fields H2 are periodically pulsed so that H1 is on when H2 is off, and off when h2 is on.
Abstract: Periodically pulsed nuclear magnetic double resonance is described by the Bloch equations for a single nucleus of spin 12, and by the density matrix master equation for AmXn spin systems. In this experiment, the observing rf field H1 and the strong rf field H2 are periodically pulsed so that H1 is on when H2 is off, and off when H2 is on. The H1 and H2 pulse intervals are each equal to τ seconds. The Bloch‐equation description illustrates the basic features of the experiment in terms of the macroscopic moment. This description provides a physical model for the production of double‐resonance signals in the limit of short τ. The density‐matrix analysis separates into two parts: the production of population changes by H2; and the production of detected signals by H1. The formalism is illustrated by the AX2 system, 1, 1, 2‐trichloroethane. Theoretical and experimental frequency‐sweep spectra are compared for values of τ ranging from 1.25 to 0.0315 sec. The spectra are dynamic in the sense that the detector ou...
TL;DR: The formal solution of the Liouville equation for systems in an external inhomogeneous, nonstationary, electromagnetic field based on perturbation theory is found in this paper.
TL;DR: The generalized master equations for composite particles were derived in this article for diagonal and off-diagonal matrix elements by a generalization of the projection technique of Zwanzig, and the form of the equations was exactly the same as that obtained by Prigogine for structureless particles.
TL;DR: In this article, the exact master equation of Prigogine and Resibois is derived for inhomogeneous as well as homogeneous systems from the following postulate: the singularities of the analytically continued Liouville resolvent nearest the real axis are isolated simple poles arising from irreducible vacuum-to-vacuum transitions.
Abstract: The exact master equation of Prigogine and Resibois is derived for inhomogeneous as well as homogeneous systems. The asymptotic master equation is obtained from the following postulate: The singularities of the analytically continued Liouville resolvent nearest the real axis are isolated simple poles arising from “irreducible vacuum‐to‐vacuum” transitions. The asymptotic distribution function found in this way differs from the one obtained previously. If the asymptotic distribution function at time t1, fa(t1) is taken as a new initial value in the exact master equation, and the subsequent asymptotic solution is calculated for a time t2 later, the result is fa(t1 + t2).
TL;DR: An exact markovian master equation for the smoothed classical distribution function f = Mf is derived using the existence of the operator [1 + M(−1 + exp (-it L))]−1.
Abstract: An exact markovian master equation for the smoothed classical distribution function f = Mf is derived using the existence of the operator [1 + M(−1 + exp (-it L))]−1. It is shown that according to the information theory f0 = 0 (“initial random phase approximation”) should be taken. Then in the first order of a perturbation approach the master equation given by POMPE and VOSS can be derived in the long time approximation.
TL;DR: In this article, the radial SCHRODINGER equation is considered in a more general form and is then transformed into another equation to permit straightforward application of the perturbation formalism used previously by the author.
Abstract: The radial SCHRODINGER equation is considered in a more general form and is then transformed into another equation to permit straightforward application of the perturbation formalism used previously by the author. The potential considered is assumed to be expandable in ascending even powers of r. The potential (sinh r)−2 emerges as a special case and is considered separately. For the general case, expansions are obtained for the eigenenergies and eigenfunctions valid in regions of small angular momenta and coupling constants. A simple analytic expression for the S-matrix is obtained. Finally several consequences are discussed.
TL;DR: In this paper, a thermodynamically consistent microscopic model is used to derive a Fokker-Planck equation governing voltage fluctuations across a diode in the limit of small electron charge.
Abstract: A thermodynamically consistent microscopic model is used to derive a Fokker-Planck equation governing voltage fluctuations across a diode in the limit of small electron charge. In going to this limit, strong nonlinearities in the diode current characteristic are allowed to persist for voltages in the range of the thermal fluctuations. This requires greater freedom in the choice of the microscopic transition probabilities than exists in other diode models that have been studied. It is pointed out that this added freedom is essential for the achievement of a regime in which the thermal fluctuations are strongly nonlinear while at the same time the charge may legitimately be treated as a continuous variable. The transition probabilities appearing in the master equation are required to be consistent with the thermodynamically correct equilibrium voltage distribution and to satisfy the condition of time reversibility. It is shown that the resulting Fokker-Planck equation is compatible with an equivalent circuit that includes a fluctuating current source having the form of white noise multiplied by a function of the diode voltage. This equivalent noise source has the form of a mathematical representation developed by Hurwitz and Kac for a related stochastic problem.
TL;DR: In this article, approximate solutions of the Schrodinger equation with arbitrary local (or non-local) potential, are obtained in analytic form, which can be applied to a wide class of integro-differential equations and of systems of such equations.
TL;DR: In this paper, it was shown that the Boltzmann relaxation equation (BRE) is often a direct consequence of the master equation in the steady state and that the BRE can be justified in those cases for which the BRE is justified.
TL;DR: In this paper, it was shown that Fulinski's master equation without time convolution is very closely related to master equation with time convolutions and can indeed be reduced to the latter in a few simple steps.
TL;DR: The kinetic equations for quantum distribution functions are usually in integral form as discussed by the authors, which is consistent with the kinetic equation for the quantum distribution function introduced by Blokhintsev [1,2].
Abstract: The kinetic equations for quantum distribution functions are usually in integral form. For example, the quantum distribution function introduced by Blokhintsev [1,2], l 1 x p F N (x, p, t) = ( 2 ~ h ~ 2 ~x~ "e h , (1) where N is the number of particles'in the system; x{x~,. . . ,XN}, P{P~' ' 9 ] ,PN} are the sets of particle coordinates and momenta; ~ is the Planck's constant; t is time; and p is the density matrix in the x, p representation, is consistent with the kinetic equation