Showing papers on "Master equation published in 1971"
TL;DR: In this article, it was shown that large quantum fluctuations occur only if the initial total occupancy of the excited state differs from the total number of atoms at most by a number of order unity.
Abstract: We discuss the solution of the "superradiance master equation" derived in a preceding paper. During the first few photon transient times the cooperative atomic decay goes through a non-adiabatic oscillatory regime. For later times the decay takes place monotonically in time with the electromagnetic field following it adiabatically. The emitted light pulse has different statistical properties for an incoherently and a coherently prepared "superradiant" atomic initial state. The former case is characterized by large quantum fluctuations and strong atom-atom and atom-field correlations. In the latter case quantum fluctuations are small and the system behaves essentially classically. By also solving for a class of coherently prepared intermediate initial states we show that large quantum fluctuations occur only if the initial total occupancy of the excited state differs from the total number of atoms at most by a number of order unity.
362 citations
TL;DR: In this paper, a set of master equations describing the equilibration process and utilizing Williams' transition-rate expressions are presented to study the occupation probabilities and the instantaneous and time-integrated particle energy spectra as a function of the time elapsed from the initial target-projectile interaction.
211 citations
TL;DR: The theory of Brownian motion of a quantum oscillator is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir as discussed by the authors.
Abstract: The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
172 citations
TL;DR: In this paper, it was shown that the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima.
Abstract: It is shown that there is a simple relation between master equation and random walk solutions. We assume that the random walker takes steps at random times, with the time between steps governed by a probability density ψ(Δt). Then, if the random walk transition probability matrix M and the master equation transition rate matrix A are related by A = (M − 1)/τ1, where τ1 is the first moment of Ψ(t) and thus the average time between steps, the solutions of the random walk and the master equation approach each other at long times and are essentially equal for times much larger than the maximum of (τn/n!)1/n, where τn is the nth moment of ψ(t). For a Poisson probability density ψ(t), the solutions are shown to be identical at all times. For the case where A ≠ (M − 1)/τ1, the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima...
133 citations
TL;DR: The exact solution of the master equation proposed by Glauber to describe spin relaxation of the one-dimensional Ising-model in interaction with a heat bath was given in this paper, where the methods of solution are similar to those employed in the calculation of the partition function for the two-dimensional ising model.
104 citations
TL;DR: In this article, the terms corresponding to the frequency shifts were properly treated and the appropriate master equation was given, and the explicit form of the frequency-shift terms was also given.
Abstract: This is an addendum to the author's previous paper [Phys. Rev. A 4, 1778 (1971)], where master equations describing the spontaneous emission from a collection of identical two-level atoms and oscillators were derived without the use of rotating-wave approximation. However, the terms corresponding to the frequency shifts were not adequately included. Here such terms are properly treated and the appropriate master equation is given. Explicit form of the frequency-shift terms is also given.
58 citations
TL;DR: In this paper, the authors studied the kinetics of the spin 4 Ising chain for a class of master equations describing transitions in the spin system owing to interactions with a heat bath.
43 citations
Book•
01 Jan 1971
TL;DR: The laws of thermodynamics statistical theory of information and of ensembles statistical mechanics of ideal systems ideal classical gases of polyatomic molecules ideal relativistic classical and quantum gases non-electrolyte liquids and solutions phase stability, co-existence, and criticality surfaces the imperfect classical gas the imperfect quantum gas phase transitions cooperative phenomena green function methods the plasma negative temperatures and population inversion recombination rate theory in semiconductors transport in gases transport in metals transport in semiconductorors fluctuations of energy and number of particles fluctuations of general classical mechanical variables fluctuations of thermodynamic variables time dependence of
Abstract: The laws of thermodynamics statistical theory of information and of ensembles statistical mechanics of ideal systems ideal classical gases of polyatomic molecules ideal relativistic classical and quantum gases non-electrolyte liquids and solutions phase stability, co-existence, and criticality surfaces the imperfect classical gas the imperfect quantum gas phase transitions cooperative phenomena green function methods the plasma negative temperatures and population inversion recombination rate theory in semiconductors transport in gases transport in metals transport in semiconductors fluctuations of energy and number of particles fluctuations of general classical mechanical variables fluctuations of thermodynamics variables time dependence of fluctuations - correlation functions, power spectra, Wiener-Khintchine relations Nyquist's theorem and its generalizations Onsager relations stochastic methods - master equation and Fokker-Planck equation ergodic theory, H-theorems, recurrence problems variational principles and minimum entropy production
35 citations
TL;DR: In this paper, a theory of the irreversible statistical mechanics of flexible polymer chains is developed on the basis of new ideas, where the Brownian motion of polymer chains are assumed to be a Markoff random transition among their rotational isomeric states.
Abstract: A theory of the irreversible statistical mechanics of flexible polymer chains is developed on the basis of new ideas. The Brownian motion of polymer chains is assumed to be a Markoff random transition among their rotational isomeric states. The theory is described for ring polymer chains, for which the “normal coordinates” can be determined by the consideration of their symmetry alone. First, we derive the master equation which describes the discrete Brownian motion of a ring polymer chain. The master equation is averaged over all the configurations, fixed several normal coordinates to certain values. This averaging process is called “coarse graining.” By Taylor expansion of the coarse‐grained master equation, we get a Fokker–Planck diffusion equation which is specified by two kinds of molecular constants, the diffusion constant Dα and the expansion parameter γα, both of which depend on the suffix α of the normal coordinates. For slow relaxation phenomena, the diffusion equation is reduced to that of the ...
30 citations
TL;DR: In this paper, an exact time-irreversible non-markovian master equation is derived, which is due to the special (retarded or advanced) form of the boundary conditions imposed on the non-equilibrium statistical operator (NSO) of the system.
27 citations
TL;DR: In this article, a general theorem is deduced which illuminates the problem of extracting reduced dynamics from the general dynamics of a system, and a possible generalization of quantum mechanics is indicated, by which an independent dynamics of macroscopic observables holds exactly.
TL;DR: In this paper, a general theory for spontaneous emission from a system of N identical atoms or molecules was developed and the normally ordered correlation functions for the oscillator system were calculated and these were then used to calculate the radiation-field correlation functions in the far zone.
Abstract: In a previous investigation, a general theory for spontaneous emission from a system of N identical atoms or molecules was developed. This theory was based on the master equation recently derived in another paper. In that paper, the master equation relating to spontaneous emission from a system of harmonic oscillators was also derived. In the present investigation, the normally ordered correlation functions for the oscillator system are calculated and these are then used to calculate the radiation-field correlation functions in the far zone. These correlation functions are compared for two different modes of excitation, viz., (i) when each of the oscillators is excited initially to a Fock state, and (ii) when each of the oscillators is excited to some coherent state. It is found that the even-order (2n) correlation functions for the second mode of excitation (superradiant excitation) are of order N 2n higher than those for the first mode of excitation. It is also shown that the photoelectron counting distribution for the superradiant excitation is Poissonian. Finally, the non-Markoffian effects in the spontaneous emission are studied in detail and their connection with exact results is described.
TL;DR: In this article, the Schrodinger equation for a two-state quantum-mechanical system with sinusoidal perturbation is numerically integrated with respect to time, and a general formula for the induced transition probability is extracted.
Abstract: The Schr\"odinger equation for a two-state quantum-mechanical system with sinusoidal perturbation is numerically integrated with respect to time. From these results a general formula for the induced transition probability (as a function of time, perturbation frequency, and perturbation strength) is extracted.
TL;DR: In this article, the weak-coupling solution of the generalized Prigogine-Resibois master equation for the model of the Wigner-Weisskopf atom in a one-dimensional radiation field is studied.
Abstract: A study is undertaken to cast light on difficulties, which arose in the first two papers of this series, pertaining to the occurrence of negative probabilities in the weak‐coupling solution of the generalized Prigogine‐Resibois master equation for the model of the Wigner‐Weisskopf atom in a one‐dimensional radiation field. The Schrodinger equation is solved exactly for the model with the initial condition for spontaneous emission, and then the weak‐coupling approximations to the solution, both for an infinite and for a finite system, are derived as inverse Laplace transform integrals. An extensive analysis, theoretical and numerical, of these is undertaken, and comparison is made with the corresponding results based on the master equation. In particular, quantitative estimates of the Poincare recurrence times for finite systems are made. It is found that considerable differences exist between the statistical‐mechanical and quantum‐mechanical results, but that both manifest nonanalyticity in the coupling p...
TL;DR: In this article, a general theory for the treatment of spontaneous emission from a system of N identical two-level atoms or molecules is developed and the master equation for the reduced density operator of the atomic system alone is obtained and is used to derive an expression for the radiation rate.
TL;DR: In this article, a transport equation for dilute gases with internal degrees of freedom, derived earlier from the BBGKY hierarchy, is obtained from the generalized master equation, and the Zwanzig projection operator which has been used previously by various authors to derive the Waldmann-Snider equation, is not applicable in case generalizations of this equation are required.
TL;DR: In this article, the decay time for the kernel of a generalized master equation and the evolution of macroscopic quantities of the measuring device were investigated. And they were found to justify the possibility of a macro-scopic description of the measurement apparatus, and a relevant hypothesis was found to be the separation of two characteristic times.
Abstract: A contribution is given to the attempts towards a solution of the measurement problem in quantum mechanics, in the spirit of some previous papers. In order to justify the possibility of a macroscopic description of the measuring apparatus, a relevant hypothesis is found to be the separation of two characteristic times: the decay time for the kernel of a generalized master equation and the characteristic time for the evolution of macroscopic quantities of the measuring device.
TL;DR: In this paper, the Glauber model is used to describe a time-dependent Ising spin system in interaction with a heat bath, and the fluctuation dissipation theorem expressing the linear response in terms of fluctuation properties of the system in equilibrium is derived.
Abstract: Linear response theory is developed for systems whose time dependence is described by a master equation. The fluctuation dissipation theorem expressing the linear response of the system in terms of fluctuation properties of the system in equilibrium is derived. The time-dependent Ising spin system in interaction with a heat bath, the Glauber model, is discussed as a particular case of the formalism.
TL;DR: In this article, the entropy of the neutral kaon system interacting with a (low temperature) boson gas is derived and the equilibrium state and the total entropy change during the approach to it are the same in both time schemes.
TL;DR: By generalizing the nonperturbative method to the time-independent case, a generalized Balescu master equation was derived in this article, which is a generalization of the non-perturbation method.
TL;DR: In this paper, the spectral density of conduction electrons is calculated taking into account the photon correlations using a system of random points on the time axis using the formalism of Stratonovich.
TL;DR: In this article, the validity of the collisional master equation for the relaxation of molecular internal degrees of freedom in a high-density reservoir of inert particles is examined using the generalized master equation.
Abstract: The validity of the collisional master equation for the relaxation of molecular internal degrees of freedom in a high‐density reservoir of inert particles is examined using the generalized master equation for this system. When the hypotheses of the “Markovian approximation” to this equation are examined, it is found that they are invalidated by correlated collision processes which occur at elevated reservoir densities. At densities where the mean free time is long compared to a characteristic frequency of the internal states and the duration of a collision, the usual multiple collision theory master equation is shown to hold.
TL;DR: In this paper, steady state solutions of the master equation of the density matrix of the spin system in the rotating reference frame have been obtained in the limit of weak spin-lattice interaction and high temperature.
Abstract: By using the method of projection operators introduced by Zwanzig, steady-state solutions of the master equation of the density matrix of the spin system in the rotating reference frame have been obtained in the limit of weak spin-lattice interaction and high temperature. With the aid of these solutions, the saturation behaviors of the spin system in solids have been investigated in the two limiting cases, weak- H 1 and strong- H 1 , where the spin system is not uniform and some sorts of cross relaxation take place. Furthermore, the remaining terms of the steady-state solution obtained under Redfield's assumption of a single spin temperature have been derived directly starting from the master equation.
TL;DR: The non-Markoffian kinetic equation for the single-particle distribution function of a classical system is derived by means of introducing in the Liouville equation the infinitesimal sources as mentioned in this paper.
TL;DR: A generalized time irreversible master equation for a relevant part of the density matrix is derived using the special form of the initial or boundary condition, which selects the retarded solutions of the Liouville equation as mentioned in this paper.
TL;DR: In this paper, the authors showed that the position of the bottleneck does not necessarily occur at that level above which direct collisional dissociation can take place rapidly, and that there is no simple correspondence between the position and the temperature coefficient for dissociation.
Abstract: Our previous vibration–dissociation coupling calculations for H2 have been repeated for D2. Closer spacing of the vibrational energy levels in D2 leads to increased translation–vibration transition probabilities, and the effect of this is to increase the rate of recombination by the vibrational mechanism at all temperatures. These numerical experiments also clarify two other issues: (i) that the position of the bottleneck does not necessarily occur at that level above which direct collisional dissociation can take place rapidly, and (ii) that there is no simple correspondence between the position of the bottleneck and the Arrhenius temperature coefficient for dissociation.
TL;DR: In this article, the authors present a master equation for amplified spontaneous emission and show the nonlinear saturation effects are dominated by parametric terms which are the analogue of mode-locking terms in laser oscillator theory.
TL;DR: In this paper, a comparative study of spontaneous emission and spin-lattice relaxation at zero temperature is presented, in particular, the time evolution of the density matrix for two simple models as determined from an analysis of the Prigogine-Resibois master equation.
Abstract: In this paper we present a comparative study of spontaneous emission and spin-lattice relaxation at zero temperature. In particular, we study the time evolution of the density matrix for two simple models as determined from an analysis of the Prigogine-Resibois master equation. The first model treated is that of the Wigner-Weisskopf atom in a three-dimensional radiation field; the second model is that of a single, effective spin in interaction with the phonon modes of a three-dimensional lattice. The divergence which arises in the solution of the master equation for the first model is avoided using a frequency cutoff. A frequency cutoff in the second model is imposed by the upper bound of the spectrum of modes in the crystal, and this fact manifests itself when one integrates over the first Brillouin zone only. From a detailed numerical study of the analytic results obtained in solving the master equation, we find that for both models the relaxation to equilibrium is characterized, in part, by a sequence of slowly damped oscillations. This result seems to be in agreement with the observation made by Zwanzig, namely, that exponential decay in time seems not to be universal, and may, in fact, be hidden behind some other kind of time dependence. The numerical study also reveals, however, that the nonexponential modes of decay can be quantitatively different in magnitude and qualitatively different in structure for atomic versus spin systems. Finally, based on the solution obtained for the spin problem, an estimate is made of the relaxation time for cerium ethyl sulfate, and this estimate is found to be consistent with experiment. © 1971 The American Physical Society.