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Showing papers on "Master equation published in 1969"


Journal ArticleDOI
TL;DR: In this article, the theory of stochastic motion is formulated from a new point of view, and it is shown that the fundamental equations of the theory reduce to Schrodinger's equation for specific values of certain parameters.
Abstract: The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrodinger's equation for specific values of certain parameters. A generalized Fokker‐Planck‐Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrodinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum‐mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non‐Markoffian terms are briefly studied.

91 citations


Journal ArticleDOI
TL;DR: In this article, the Schrodinger equation of motion for the density operator is derived from the phase-space equivalent to the Schroff equation of Motion for the energy and the time-correlation functions satisfy an integral equation of Volterra type.
Abstract: Using the recently discussed quantum dynamics in phase space, we derive a master equation, starting from the phase-space equivalent to the Schrodinger equation of motion for the density operator. Use is made of Zwanzig's projection-operator techniques and some explicit realizations of the projection operators are given. The master equation is then applied to show that the time-correlation functions, as defined in the text, satisfy an integral equation of the Volterra type. Next, a master equation for a system interacting with a large system is derived. As an illustration, we determine the lowest-order Born approximation and carry out a short-memory-approximation calculation for an oscillator coupled to a reservoir and for a two-level system interacting with an oscillator heat bath; we obtain equations of the Fokker-Planck type. Some physical implications of these equations are also discussed.

63 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered a microscopic system coupled to a bath and established a non-Markoffian master equation for the reduced statistical operator of the system, valid in the Born approximation.
Abstract: AbstractWe consider a microscopic system $$\\mathfrak{S}$$ coupled to a bath $$\\mathfrak{B}$$ and establish a non-Markoffian master equation for the reduced statistical operator of $$\\mathfrak{S}$$ , valid in theBorn approximation. Discussing in detail theBorn approximation we find as a general condition for its validity that a certain “strength function” should not degenerate to one or more extremly sharp and high lines.

40 citations



Journal ArticleDOI
J. Byrne1
TL;DR: In this paper, the statistical distribution of the number of electrons in the avalanche generated in a proportional counter by a single initial electron is analyzed by dividing the electrons into two groups differentiated by whether they have or have not sufficient energy to produce additional electrons by ionization.

34 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the total entropy is a completely monotonic function of the time and that the phenomenological rate constants for both dissociation and recombination remain virtually independent of time and conform accurately to the rate-quotient law.
Abstract: : The master equation, as described in part I of this series, is solved numerically for the dissociation of H2 diluted in He, and also for the reverse reaction, the recombination of H atoms, using an assumed set of transition probabilities. In both processes, it is found that the total entropy is a completely monotonic function of the time. Once the transient period is over, the phenomenological rate constants for both dissociation and recombination remain virtually independent of time and conform accurately to the rate-quotient law; there are, however, some conditions attached to this statement if integrated rate constants are used. (Author)

33 citations


Journal ArticleDOI
TL;DR: In this paper, the atomic coordinates are eliminated exactly from the laser master equation with the help of a projector technique, and the resulting integrodifferential equation for the field statistical operator with the kernel given correctly up to fourth order in the coupling constant is shown to be equivalent to a fourth order differential equation with respect to time.
Abstract: The atomic coordinates are eliminated exactly from the laser master equation with the help of a projector technique. The resulting integrodifferential equation for the field statistical operator with the kernel given correctly up to fourth order in the coupling constant is shown to be equivalent to a fourth order differential equation with respect to time. All theories footing on an “adiabatic” elimination procedure, especially the wellknown Fokker Planck equation treatment and a recently published theory containing a second order time derivative are shown to be successive approximations to our treatment. Non-Markoffian effects are discussed.

32 citations


Journal ArticleDOI
TL;DR: In this article, the solution of the generalized master equation for a macroscopic insulated system is studied in the limit t → + ∞ using the theory of holomorphic operators in finite-dimensional linear spaces, the spectral properties of the kernel of the G.M.E. are analyzed.

29 citations


Journal ArticleDOI
TL;DR: In this paper, the authors apply the non-Markoffian master equation discussed in a previous paper to the test case of an exactly solvable model and study the theory of the damped harmonic oscillator.
Abstract: Applying the non-Markoffian master equation discussed in a previous paper to the test case of an exactly solvable model we study here the theory of the damped harmonic oscillator. The results confirm the criterion for the applicability of the Born approximation in which our master equation is valid. On the other hand, the present paper generalizes Markoffian theories of the damped oscillator.

29 citations


Journal ArticleDOI
TL;DR: In this paper, a generalization of the projection methods developed in the previous paper is used to obtain kinetic equations for systems in strong fields, with arbitrary time dependence, and it is demonstrated that a linear approximation for small (but time dependent) outside fields can not be expected to be valid for arbitrary long times.

15 citations


Journal ArticleDOI
TL;DR: In this article, the so-called phase equation is established in a general way for one-dimensional, linear second-order differential equations and for 2×2-linear differential systems.
Abstract: The so-called phase equation is established in a general way for one-dimensional, linear second-order differential equations and for 2×2-linear differential systems. Systematic derivations give phase equations for particular quantum mechanical problems. New results are obtained using elementary mathematical tools, and some problems particularly well adapted for the phase method are suggested.

Journal ArticleDOI
TL;DR: In this paper, the master equation representing the dissociation-recombination kinetics of a dilute diatomic gas can be reduced to a system of inhomogeneous nonlinear Volterra integral equations of the second kind.
Abstract: : The master equation representing the dissociation-recombination kinetics of a dilute diatomic gas can be reduced to a system of inhomogeneous nonlinear Volterra integral equations of the second kind. Accurate solutions of these equations can be obtained using the self-consistent matrix iteration procedure proposed by Rush and Pritchard, under certain relatively unrestrictive conditions. (Author)

Journal ArticleDOI
TL;DR: In this paper, the generalized master equation of Zwanzig for a macroscopic system is approximated by the solution of a markoffian master equation, and the reliability of this approximation is studied at arbitrary times.

Journal ArticleDOI
TL;DR: In this article, a generalized master equation for time-dependent Hamiltonians is proposed, which appears to be a generalization of Zwanzig's generalized master equations, and is formally solved.



Journal ArticleDOI
TL;DR: In this paper, a general stochastic theory is formulated for these time-defined processes, in this case, undergoing vibronic relaxation and fluorescence emission, and the solution is given in a form suitable for direct computation.
Abstract: One method for the absolute measurement of rate constants for defined quantum states is to study their response to sharply defined pumping. Such an experiment is the observation of the relaxation process after excitation of selected vibronic levels with amplitude modulated light. The observation could either involve secondary absorption or radiative emission. A general stochastic theory is formulated here for these time‐defined processes, in this case, undergoing vibronic relaxation and fluorescence emission. From the general transport matrix describing the relaxation problem via the master equation, the observables are an unambiguous consequence. The solution is given in a form suitable for direct computation. For this particular case, the theory predicts a phase angle in terms of a many‐shot expansion, which is not restricted to exponential fluorescence decay.

Journal ArticleDOI
TL;DR: In this article, a simple theoretical treatment of dipolar relaxation is presented, essentially based on the autocorrelation formalism and derived from the general theory developed by Boon and Rice for the auto-correlation function of a dynamical variable.
Abstract: In this paper, we present a simple theoretical treatment of dipolar relaxation, essentially based on the autocorrelation formalism and derived from the general theory developed by Boon and Rice for the autocorrelation function of a dynamical variable. A simple model is introduced to derive analytical expressions for the dipolar autocorrelation function, from which other quantities, like the response function and the complex dielectric function, are calculated. These theoretical results are tested against the experimental data obtained from a molecular dynamics study by Bellemans, Kohler, and Gancberg on a two‐dimensional system of electric dipoles on a rigid lattice, subject to dipole–dipole interactions. We also append some results concerning the memory function governing the evolution of the autocorrelation function as described by the master equation.

Journal ArticleDOI
TL;DR: In this article, an extensive study is made on the dynamic behavior of an Ising system within the framework of the Bethe or constant coupling theory, and a set of master equations for the spin average and for the nearest-neighbor-correlation average is constructed.

Journal ArticleDOI
TL;DR: In this article, the probability of occurrence of fluctuations around nonequilibrium steady states is discussed from a kinetic viewpoint, and it is shown that in a large class of continuous media it is possible to extend the thermodynamic theory of fluctuations, provided one uses suitable steady-state parameters rather than equilibrium quantities.
Abstract: The probability of occurrence of fluctuations around nonequilibrium steady states is discussed from a kinetic viewpoint. It is shown that in a large class of continuous media it is possible to extend the thermodynamic theory of fluctuations, provided one uses suitable steady-state parameters rather than equilibrium quantities.

Journal ArticleDOI
TL;DR: In this paper, a real probability measure is introduced for weighting trajectories and this measure is expressed as the boundary case of a measure containing one real parameter for the finite values of the real parameter, and an equation of motion of the Fokker-Planck type is used in the zero boundary case.
Abstract: The paper is concerned with the generalization ofFeynman’s method A real “probability” measure is introduced for weighting trajectories and this is expressed as the boundary case of a measure containing one real parameter For the finite values of the real parameter an equation of motion of the Fokker-Planck type is used In the zero boundary case a Schrodinger equation is derived A simple deduction is given for theWigner phase space distribution by using the differentiability of quantum trajectories, which is also proved It is suggested that the equations of motion obtained for the finite values of the real parameter included in the theory describe real processes

Journal ArticleDOI
TL;DR: In this article, the authors applied the master equation to an Ising model and obtained equations of motion that predict a self-consistency relationship among the internal field, temperature, and exchange energy.
Abstract: Applying the master equation to an Ising model, equations of motion are obtained for the average of a single spin and for the average of a pair of spins. These equations relax to equilibrium values that predict a self‐consistency relationship among the internal field, temperature, and exchange energy. The critical temperature obtained is that of the Bethe approximation, and is extended to include a static external field in the magnetization direction. With this, the relaxation of an Ising model near the Curie temperature has been investigated in a self‐consistent manner. The results are similar to those of Suzuki and Kubo, but in the Bethe approximation instead of the Bragg‐Williams approximation, and hence predict the relaxation of the correlation as well. In particular we studied a nearly Ising spin system that is a system with relatively small isotropic transverse coupling. An oscillating, linear, and external magnetic field applied transverse to the strongly coupled direction of the spins leads to low...

Journal ArticleDOI
TL;DR: In this paper, four analytical variants are given, and their range of applicability is discussed, and one of these variants is used in the determination of the Channel Coupling radial wave functions, the Optical Model wave functions and the regular Coulomb function.


Journal ArticleDOI
TL;DR: In this paper, the differential equation for the quasidistribution function of a laser field with non-markoffian character was derived and its solution was compared with the Fokker-Planck equation given by Risken1 andHempstead andLax2.
Abstract: Starting from the masterequation for the density matrix the differential equation for the quasidistribution function of a laser field with non-markoffian character is derived. This equation and its solution may be compared with the Fokker-Planck equation given byRisken 1 andHempstead andLax 2, where the laser field has markoffian character. Both solutions approximately coincide only in the threshold region.

Journal ArticleDOI
TL;DR: In this paper, the authors deal with two kinds of infinities occurring in Prigogine's master equation when it is applied to a system containing charged particles (one electron and N ions) and radiation.
Abstract: The paper deals with two kinds of infinities occurring in Prigogine’s master equation when it is applied to a system containing charged particles (one electron andN ions) and radiation The first kind is caused by the presence of the radiation field since the usual Hamiltonian for an electron and a radiation field necessarily contains also the self-field of the electron, which is infinite on the electron’s world line We then analyse how precisely this infinity enters into the formalism The analysis leads to a prescription for rewriting the Liouville operator which allows relativistic and radiation reaction effects to be approximately included and which yields a theory without divergences due to the presence of a radiation field Another kind of infinity is shown to be present It occurs upon integrating over the particle co-ordinates, as is necessary for many applications The immediate cause of this infinity is shown to be an asymptotic time integration employed to derive the master equation However, reasons are given to show that the asymptotic time integration cannot simply be replaced by some other formalism leading to finite results; the convergence of the series for the momentum distribution function is indeed poor for small relative velocities between electron and ions, so that only the sum of a large number of terms in the series can provide a correct (and finite) result for small velocities

Journal ArticleDOI
TL;DR: In this article, a transport equation for the transverse field polarization matrix is established by a perturbation and diagram resummation method, which is then transformed into an equation for Stokes parameters of the radiation.
Abstract: By a perturbation and diagram resummation method, a transport equation for the transverse field polarization matrix is established. This equation is then transformed into an equation for the Stokes parameters of the radiation. The equation takes the usual form of a transfer equation; the absorption and emission coefficients are matrix, the elements of which are given as a function of the dissipative part of the microcurrent correlation tensor and conductivity tensor. Finally this equation is expressed as a system for the intensities of the proper modes. The equations of the system are usually coupled.

Journal ArticleDOI
TL;DR: In this paper, the solutions of finite-dimensional time-invariant transport equations are considered and with the aid of an auxiliary equation the given equation is shown to be equivalent to a pair of coupled linear equations.
Abstract: The solutions of finite-dimensional time-invariant transport equations are considered. With the aid of an auxiliary equation the given equation is shown to be equivalent to a pair of coupled linear equations. Further, any four distinct solutions of a transport equation are related to the solution of the auxiliary equation. For stationary processes with dimension one, this result reduces to that on the classic Riccati's equation. Results presented here are being generalized by R.M. Redheffer and the writer in an abstract space. However, this note points out some essential properties and some of the underlying methods without the abstract setting. To accomplish this goal we assume the system is finite dimensional and time-invariant. A one-dimensional example is given to illustrate the results.

Journal ArticleDOI
TL;DR: In this paper, the classical classical plasma quasi-linear equations are derived from the Prigogine-Resibois master equation, which is derived from a class of classes of quasilinear equations.