Showing papers on "Master equation published in 1970"
TL;DR: In this article, the authors derived a Fokker-planck equation to describe the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system.
Abstract: Spontaneous emission from a system of $N$ identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state $|{z}_{0}〉$.
149 citations
TL;DR: In this paper, it was shown that the behavior at very low and very high frequencies can be described in terms of apparent relaxation times, which differ by approximately 50% from the temperature dependence predicted by Parker, but the magnitudes are larger by a factor of 2.
Abstract: In his well‐known theory, Parker assumes that the rotational relaxation of diatomic gases may be described by a single relaxation time which is calculated for the special case of initially nonrotating molecules. In the present theory the evolution of the rotational distribution function is described by a diffusion‐equation approximation to the master equation. This equation is linearized and solved for the case of acoustic waves. The results indicate that the absorption and dispersion of acoustic waves cannot be described by a single relaxation time. However, the behavior at very low and very high frequencies can be described in terms of “apparent relaxation times” which differ by approximately 50%. The temperature dependence of the apparent relaxation times is similar to that predicted by Parker, but the magnitudes are larger by approximately a factor of 2.
98 citations
TL;DR: In this paper, a modified master equation was proposed for the case of two coupled boson fields, which leads to additional terms in the usual master equation arising from correlations between the heat baths introduced by the coupling.
Abstract: The usual derivations of the master equation for coupled systems such as the laser make an assumption of weak coupling both for the coupling of the components to their heat baths and for the internal coupling between the components. It is this second condition that we wish to relax. In the usual derivation of the irreversible part of the master equation the approximation is made that the density operator for the coupled system factorizes into a product of the density operators for the two components. However when strong coupling is present, such as in high intensity lasers this approximation is no longer valid. To illustrate how the irreversible part of the master equation may be derived without making the factorization ansatz we consider the case of two coupled boson fields. Our derivation leads to additional terms in the usual master equation arising from correlations between the heat baths introduced by the coupling. This modified master equation yields the correct stationary solution for the density operator of the coupled system, whereas the usual master equation leads to a stationary solution for the density operator correct for the free components only.
50 citations
TL;DR: In this paper, the master equation for the dissociation of a dilute diatomic gas is extended to the non-dilute case by the inclusion of V-V transitions.
Abstract: : A previous formulation of the master equation for the dissociation of a dilute diatomic gas is extended to the nondilute case by the inclusion of V-V transitions. Using existing experimental and theoretical data for other molecules, estimates are made for the V-V transition probabilities in H2. The master equation is solved for the dissociation of a series of M-H2 mixtures at 2000K, and the results are compared with those previously obtained for the same reaction at infinite dilution. The present solution is achieved by brute-force integration, continued through the transient period and out into the pseudosteady regime. (Author)
17 citations
TL;DR: In this article, the derivation of the quantum mechanical Boltzmann equation for an atomic gas is presented based on the corresponding generalized master equation, where the atoms are treated as composite particles and the degeneracy of their excited levels is taken into account.
16 citations
15 citations
12 citations
TL;DR: The part of the statistical operator which is relevant for the description of macroscopic dynamics is treated in this paper, where new mathematical properties of the solution of Zwanzig's generalized master equation which are important for the deduction of a markoffian master equation for Φ 0(t) are pointed out.
11 citations
TL;DR: In this article, a model was studied in which it was possible to obtain an expression for the exact solution of the master equation for the problem of spontaneous emission in a finite system, and the model chosen for discussion is that of the Wigner-Weisskopf atom in interaction with a massless boson field, here 1-dimensional.
Abstract: In this paper, a model is studied in which it was possible to obtain an expression for the exact solution of the master equation for the problem of spontaneous emission in a finite system. The model chosen for discussion is that of the Wigner‐Weisskopf atom in interaction with a massless boson field, here 1‐dimensional. The solution takes the form of a constant term plus a time‐dependent one, expressed as the sum of residues at a series of poles along the real axis of a Laplace transform variable. Numerical calculations were performed on various aspects of the solution, and although these were too delicate to be quite certain around sensitive values of the time, the general picture is clear: After an initial decay to a value near zero, the series gave rise at fairly regular intervals to rapid and large fluctuations, the size of which never quite attains the initial value, but may nonetheless be large even after very long times. This result seems to be in agreement with the observation made by Zwanzig, namely, that for finite systems the master equation might demonstrate properties associated with the finite size of the system which would become important over certain long‐time scales and would be ignored by the use of the customary ``thermodynamic limit'' of statistical mechanics. The relationship between this work and that of Montroll and Mazur and of Rubin is also discussed.
10 citations
TL;DR: Using projection operator techniques and the Liouville formalism, a derivation of an exact evolution equation for the internal degrees of freedom of a molecule in a temperature bath is presented in this article.
Abstract: Using projection operator techniques and the Liouville formalism, a derivation of an exact evolution equation for the internal degrees of freedom of a molecule in a temperature bath is presented. This non‐Markovian “master equation” gives the time evolution of the diagonal part of the reduced density matrix of the internal degrees of freedom. Since the collision term in this equation depends explicitly on the intensive variables of the reservoir, the simultaneous limits of low reservoir density and long time may be used to reduce the exact equation to a Markovian master equation. Using an identity which connects the Liouville formalism to scattering theory, it is shown that the collisional transition probabilities which occur in this master equation can be written in terms of scattering cross sections. Finally, it is demonstrated that the transition probabilities satisfy the condition of detailed balance and that the master equation agrees with that obtained by the usual physical derivations.
10 citations
TL;DR: In this article, a generalized master equation for a system described in a phase space of generalized coordinates w and momenta J was derived, which gives the time evolution of the reduced density distribution function ρ(t, w, J) for the momenta.
Abstract: For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced‐density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non‐Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time‐evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.
TL;DR: In this paper, a new perspective of nonequilibrium statistical mechanics is given: the basic equation is formally very close to the Fokker-Planck equation and may readily be modified to a universal master equation by a slight change.
Abstract: By Fourier‐transforming the author's recently proposed state functional formalism for the BBGKY hierarchy, a new perspective of nonequilibrium statistical mechanics is given: the basic equation is formally very close to the Fokker‐Planck equation and may readily be modified to a universal master equation (with irreversibility) by a slight change. Hence, the problem reduces to one of a generalized random‐walk such that the stochastic quantity to be considered is the particle‐number density in the 1‐body phase space. A general solution is formulated for the weak interaction case.
TL;DR: The generalized Kramers-Fokker-Planck equation for a rather general quantum system described by a set of gross variables is derived by the method of the non-equilibrium statistical operator developed by one of authors as mentioned in this paper.
TL;DR: In this article, the authors derived quantum-mechanical kinetic equations for the matter density and radiation density matrix, which describe optical pumping phenomena and obtained the present theories of optical pumping with appropriate linearizations.
Abstract: We derive quantum-mechanical kinetic equations for the matter density $\ensuremath{\rho}$ and radiation density matrix $R$, which describe optical pumping phenomena. The resultant kinetic equations are a set of coupled nonlinear equations for $\ensuremath{\rho}$ and $R$. With appropriate linearizations, we can obtain the present theories of optical pumping. The nonlinear equations describe multiple scattering and line narrowing due to imprisonment of resonant radiation. We show that the coupled equations for $\ensuremath{\rho}$ and $R$ are equivalent to coupled equations for $\ensuremath{\rho}$ and a generalized polarization matrix II, whose matrix elements are the second moments of $R$. The polarization matrix II constitutes a complete description of linear phenomena in the same manner as present theories describe optical pumping phenomena by using the matter density matrix $\ensuremath{\rho}$ alone. As a consequence of our nonphenomenological treatment of radiation, we can provide a completely microscopic treatment of the externally modulated light-beam experiment of Bell and Bloom. We show that the atom absorbs the modulation envelope directly from the external thermal light beam in the same way that the atom absorbs transverse polarization directly from the light beam in optical pumping experiments.
TL;DR: Several dynamical models of the Slater K.D.P. model of a ferroelectric are studied and, following Glauber, master equations are derived for the time development of these models.
Abstract: Several dynamical models of the Slater K.D.P. model of a ferroelectric are studied and, following Glauber, master equations are derived for the time development of these models. A computer simulation of the models is used to study the solutions to these equations, and to compare the equilibrium results with Lieb's exact solution of the equilibrium case. Excellent agreement is obtained in one case.
TL;DR: In this article, exact master equations of the Fulinski-kramarczyk type were derived in a completely rigorous way, and the exact master equation was shown to be equivalent to
TL;DR: In this article, a method based on the approximation of the time evolution operator by a finite symmetric matrix is found to yield a simple, rapid, and quite accurate algorithm for integrating the time-dependent Schrodinger equation; this method is also applicable to other partial differential equations of physical interest.
Abstract: Numerical integration techniques are used to investigate the quantum dynamical behavior of the Morse, Fues, and truncated square‐well oscillators. These procedures, unlike the methods involving expansion in terms of a finite basis set of functions, readily permit the study of dissociating systems. Stability problems arise when the time‐dependent Schrodinger equation is solved by difference equation methods; an extension of a difference equation scheme reported by Mazur and Rubin is found to yield a conditionally stable system. A method based on the approximation of the time evolution operator by a finite symmetric matrix is found to yield a simple, rapid, and quite accurate algorithm for integrating the time‐dependent Schrodinger equation; this method is also applicable to other partial differential equations of physical interest. The agreement between the results of numerical integration methods and those of expansion in a basis set of functions is found to be excellent for systems in which bound states ...
TL;DR: In this paper, the authors derived time correlation formulas for phenomenological coefficients of solid state diffusion and derived convergent summations for ionic conducting crystals whose defect composition is arbitrary except that there are no macroscopic amounts.
Abstract: Previously derived time correlation formulas for phenomenological coefficients of solid‐state diffusion are studied. The model is that the system evolves by the master equation of a Markoff process and is characterized by known defect‐configuration dependent jump probabilities. To study the effect of defect interactions on matter transport the cluster and diagram expansion methods of equilibrium theory are used to convert the expressions to series expansions in powers of defect concentrations, the migration mechanisms being vacancy and interstitial. The coefficients of the expansion are cluster summations dependent on concentration through equilibrium correlation functions. To obtain convergent summations for ionic crystals a further chain summation step like that of Mayer ionic solution theory, as well as an additional bond summation of a different kind, are necessary. Formal results are given for ionic conducting crystals whose defect composition is arbitrary except that there are no macroscopic amounts...
TL;DR: In this paper, a straightforward extension of time-dependent quantum-mechanical perturbation theory is made to obtain the equivalent formulae for a statistical mixture of states.
Abstract: A straightforward extension of time-dependent quantum-mechanical perturbation theory is made to obtain the equivalent formulae for a statistical mixture of states. The 'master equation' resulting from this procedure is, in its lowest order, the analogue in quantum statistics of the 'Golden Rule' for transition rates and reduces exactly to this form in the limit of pure initial states. The authors give results to fourth order and illustrate their use by applying them to the problem of the resonant interaction of an electromagnetic field and a two-level atom.
TL;DR: In this paper, a universal method was found to express the real amplitudeA of the complex wave-functionψ=Aei S/ħ as the real function, ofS (action) from the continuity equation of wave mechanics.
Abstract: A universal method is found to express the real amplitudeA of the complex wave-functionψ=Aei S/ħ as the real function, ofS (action) from the continuity equation of wave mechanics. In this way, quantum potential may be given as a function ofS. Thus the wave-mechanical eikonal equation contains the functionS only. Considering the resulting equation as the wave-mechanical generalization of the Jacobian point dynamical equation, the wavemechanical generalization of the Newtonian equation of motion might be given. Moreover, as a consequence, it is pointed out that theψψ * =A 2 really means density of particles or the probability of their finding even in this interpretation.
TL;DR: In this paper, the generalized master equation (GME) was applied to the linear approximation of a system subject to a small mechanical disturbance, and it was shown that the approach to such an asymptotic state is well described at a macroscopic level by a markoffian equation.
TL;DR: In this article, three different switching-on processes are studied: Q-switching, switching off the detuning, and switching on the pump power, and the transient behaviour of the laser field is divided into a rapidly and in a slowly time-dependent part.
Abstract: Three different switching-on processes are studied —Q-switching, switching off the detuning, switching on the pump power. The transient behaviour of the laser field is different for these switching-on processes, and may generally be divided into a rapidly and in a slowly time-dependent part. The latter one corresponds to the quasi stationary behaviour of the laser field.
TL;DR: In this article, the Peierls equation for the time rate of change of the mean number of polaritons in a dielectric crystal was derived using coherent states as a basis.
01 May 1970
TL;DR: In this paper, the Schrodinger equation is reduced to a partial differential equation which is equivalent to the classical Liouville equation, to the approximation of nullifying h squared in the former.
Abstract: : The theme of this report is a problem of correlation, as is illustrated in section 1 by considering the two-electron problem according to the Schrodinger equation. In section 2, the Schrodinger equations is reduced to a partial differential equation which is equivalent to the classical Liouville equation, to the approximation of nullifying h squared in the former.
IBM1
TL;DR: In this article, the numerical formulation for the nonlinear master equation for the dissociation and recombination of a diatomic molecule is presented, making the necessary variable and similarity transformation, the equation becomes more tractable from a computational viewpoint.