Showing papers on "Master equation published in 1975"
TL;DR: In this article, a tutorial approach to cooperative phenomena in systems far from thermal equilibrium and in non-physical systems is presented, with particular emphasis on the question of how order and self-organization may arise.
Abstract: This article consists of two parts. The first part presents a tutorial approach to cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Particular emphasis is placed on the question of how order and self-organization may arise. The following example is treated among others: the ordered phase of the laser giving rise to both coherently oscillating atomic dipole moments and coherent light emission. A complete analogy of the laser light distribution function to that of the Ginzburg-Landau theory of superconductivity is found mathematically which allows us to interpret the laser threshold as a quasi-second-order phase transition with soft modes, critical slowing down, etc. Similar phenomena, again closely resembling phase transitions, are found in tunnel diodes and in the nonlinear wave interaction which occurs, for example, in nonlinear optics. Remarkable analogies between the instability of the laser and those in hydro-dynamics are elaborated. While these phenomena show pronounced analogies to phase transitions in thermal equilibrium, there are further classes of instabilities and new phases which rather resemble hard excitations known in electrical engineering. Chemical oscillations are particularly important examples. In addition, the first part of this article contains the example of the cooperative behavior of neuron networks and shows the applicability of simple physical concepts, e.g., the Ising model, to the problem of the dynamics of social groups. All these above-mentioned examples demonstrate clearly that rather complex phenomena brought about by the cooperation of many subsystems can be understood and described by a few simple concepts. One of the main concepts is the order parameter; another is the adiabatic elimination of the variables of the subsystems, which is based upon a hierarchy of time constants present in most systems. The second part of this article gives a systematic account of the mathematical tools which allow us to deal with fluctuations. It contains the master equation, the Fokker-Planck equation, the generalized Fokker-Planck equation, and the Langevin equations, and gives several general methods for deriving the stationary and, in certain cases, the nonstationary solutions of master equations and the Fokker-Planck equations. Such general classes comprise those in which detailed balance is present or in which the coupling to the reservoirs is weak. In the quantum mechanical domain, the density matrix and the projection formalism for its reduction are presented. Finally, it is shown how the principle of quantum-classical correspondence allows us to translate quantum statistical problems completely into the classical domain.
815 citations
TL;DR: In this paper, a general nonperturbative treatment of cooperative emission in systems of two-level atoms, starting from first principles and including inhomogeneous broadening, is given.
Abstract: We give a general nonperturbative treatment of cooperative emission in systems of $N$ two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to ${N}^{2}$, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length $L$ of the active volume is much larger than a suitable threshold length ${L}_{T}$ (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length $L$ is smaller or of the same order of a suitable cooperation length ${L}_{c}$ (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to $N$). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to $N$, so that superfluorescence effects disappear.
464 citations
TL;DR: In this article, a new method of treating open systems is presented, which is based on an integrodifferential equation for the statistical operator of the composite system, which naturally provides a hierarchy of equations involving higher and higher order bath-system correlations.
Abstract: A new method of treating open systems is presented. The normal treatment using the generalized master equation with the projection of Argyres and Kelley is meaningful only if the state of the reservoir never deviates appreciably from the reference state which appears in the projection. Otherwise, one must make at least a partial resummation of the perturbative expansion of the kernel of the generalized master equation. The present method avoids the introduction of a projection operator and allows us to overcome such resummation difficulties. It is based on an integrodifferential equation for the statistical operator of the composite system, which naturally provides a hierarchy of equations involving the statistical operator ϱ( t ) of the open system and suitable quantities describing higher and higher order bath-system correlations. Treating the deviations of the bath from its initial equilibrium or stationary state as expansion parameters, one gets an approximation scheme, each step of which gives a closed system of equations for ϱ( t ) and a suitable set of correlation quantities. Eliminating such quantities one obtains a closed linear integrodifferential equation for ϱ( t ). The zeroth approximation in the deviations coincides with the Born approximation of the generalized master equation which uses the projection of Argyres and Kelley. On the other hand, even the first approximation is equivalent to the resummation of infinite contribution of the Born series of such a generalised master equation. When it is suitable, the concentration of the bath can also be used as an expansion parameter to handle the hierarchy.
209 citations
TL;DR: In this paper, a unified microscopic statistical theory of preequilibrium and equilibrium processes of the compound nucleus, valid for mass numbers A ⪆ 40, light incident projectiles (A ′4), and for excitation energies a few MeV above neutron threshold or larger, is presented.
193 citations
TL;DR: In this paper, the steady states of a simple nonlinear chemical system kept far from equilibrium are analyzed, and a standard macroscopic analysis shows that the nonlinearity introduces an instability causing a transition analogous to a thermodynamic first-order phase transition.
Abstract: The steady states of a simple nonlinear chemical system kept far from equilibrium are analyzed. A standard macroscopic analysis shows that the nonlinearity introduces an instability causing a transition analogous to a thermodynamic first-order phase transition. Near this transition the system exhibits hysteresis between two alternative steady states. Fluctuations are introduced into this model using a stochastic master equation. The solution of this master equation is unique, preventing two alternative exactly stable states. However, a quasi-hysteresis occurs involving transitions between alternative metastable steady states on a time scale that is longer than that of the fluctuations around the mean steady state values by a factor of the formeΔφ, where Δo is the height of a generalized thermodynamic potential barrier between the two states. In the thermodynamic limit this time scale tends to infinity and we have essentially two alternative stable steady states.
110 citations
TL;DR: In this paper, general expressions for the frictional term of the Schrodinger equation were derived for both the dissipative and the non-deterministic terms, and a proof was given showing that the Frictional term causes the quantum system to lose energy.
Abstract: Frictional and dissipative terms of the Schrodinger equation are studied. A proof is given showing that the frictional term of the Schrodinger-Langevin equation causes the quantum system to lose energy. General expressions are derived for the frictional term of the Schrodinger equation.
106 citations
TL;DR: In this article, non-equilibrium quantum-statistical mechanics is applied to peripheral collisions between heavy nuclei (A≳40), where a large number of degrees of freedom are involved during the process.
Abstract: Non-equilibrium quantum-statistical mechanics is applied to peripheral collisions between heavy nuclei (A≳40) where a large number of degrees of freedom are involved during the process. By eliminating the relative motion from the explicit consideration, the transitions between different channels are determined by a Liouville equation with timedependent coupling matrix elements. The introduction of subsets of channels (coarse graining) leads to the definition of macroscopic variables which correspond to observable quantities. The equation of motion for the macroscopic variables become irreversible by assuming the values of the coupling matrix elements to be randomly distributed. The validity and possible applications of the resulting master equations are discussed.
105 citations
TL;DR: In this article, a theory of fluctuations of macrovariables in nonequilibrium systems based on a nonlinear master equation is outlined, which takes into account, via a "mean field" type of approximation, the effect of the spatial extension of fluctuations.
Abstract: A theory of fluctuations of macrovariables in nonequilibrium systems based on a nonlinear master equation is outlined. This equation takes into account, via a “mean field” type of approximation, the effect of the spatial extension of fluctuations. A comparison with the birth and death formalism reveals several unsatisfactory features of the latter.
96 citations
TL;DR: In this paper, a generalised master equation is proposed to calculate both angular distributions and spectra of preequilibrium nucleons. But the model is not suitable for the case of nuclei.
52 citations
TL;DR: Ordered operator expansions for operators forming physically important low-dimensional Lie algebras are derived in a simple unified way as mentioned in this paper, starting with the Zassenhaus formula for the disentangling of exponential operators, series expansions of both undisentangled and disentangled exponentials and comparison of the operator coefficients of equal powers of an ordering parameter alpha leads to ordered operator expansions.
Abstract: Ordered operator expansions for operators forming physically important low-dimensional Lie algebras are derived in a simple unified way. Starting with the Zassenhaus formula for the disentangling of exponential operators, series expansions of both undisentangled and disentangled exponentials and comparison of the operator coefficients of equal powers of an ordering parameter alpha leads to ordered operator expansions. This 'comparison method' gives an alternative simple derivation of some already known formulae and a number of new formulae in the physical and chemical applications of the harmonic oscillator and for master equation problems with nearest-neighbour transition probabilities. The 'comparison method' cannot be applied to the angular momentum algebra directly. By a slight modification it can be used to derive from one matrix element or trace of JxkJylJzn all possible combinations k, l, n by simply comparing powers of ordering parameters.
51 citations
TL;DR: In this article, a quantum-mechanical master equation approach to the spontaneous emission from a two-level atom in the presence of a strong pump field is reported, and access is available to both a one-photon treatment and a treatment incorporating photon cascades.
Abstract: A quantum-mechanical master equation approach to the spontaneous emission from a two-level atom in the presence of a strong pump field is reported. Access is available to both a one-photon treatment and a treatment incorporating photon cascades. A peak height ratio of 3:1 for the complete many photon approach is reduced to 2:1 in a one-photon approximation. Broadening of the sidebands is also lost through this approximation.
TL;DR: A theory of microscopic fluctuations, which is based on generalized fluctuation-dissipation assumptions, is used to describe concentration fluctuations in a uniform system undergoing chemical reactions as discussed by the authors, and the theory is shown to agree with the linear Langevin-type theory of concentration fluctuations near equilibrium, and provides a method for calculating the time evolution of an arbitrary initial probability distribution for a system far from equilibrium.
Abstract: A theory of microscopic fluctuations, which is based on generalized fluctuation–dissipation assumptions, is used to describe concentration fluctuations in a uniform system undergoing chemical reactions. The theory is shown to agree with the linear Langevin‐type theory of concentration fluctuations near equilibrium, and provides a method for calculating the time evolution of an arbitrary initial probability distribution for a system far from equilibrium. These results are easily compared to the birth and death (master equation) theory of chemical reactions, and recent work on the asymptotic solutions of these master equations indicates that in the limit of a macroscopic system, the two approaches are identical.
TL;DR: In this paper, two approximate solutions are developed that permit removing one of the dimensions from the integrations, i.e., the rate of rotational dissociation at a given vibrational energy, averaged over an appropriate non-equilibrium vibrational distribution function.
TL;DR: In this article, the authors derived a master equation for the reduced atomic density operator, which gives rise both to a damping and to a time-dependent frequency shift in the dynamics of collective modes.
Abstract: We discuss from first principles the cooperative decay of a system of two-level atoms, initially prepared in an uncorrelated excited state with population inversion $N$, and we give the conditions under which the superfluorescence effect occurs. Describing the atomic system in terms of collective variables, we derive a master equation for the reduced atomic density operator, which gives rise both to a damping and to a time-dependent frequency shift in the dynamics of collective modes. The coupled equations of motion are solved with a self-consistent approach. It is found that the system goes through a nonexponential decay if the maximum length of the active volume is smaller than a "cooperation range" and larger than a "threshold length," in agreement with the one-mode theory. The radiation burst has a time width proportional to ${N}^{\ensuremath{-}1}$, and its intensity is proportional to ${N}^{2}$. Specializing to a pencil-shaped volume, we find that only two atomic modes need to be considered; in this case, the average emitted radiation is all condensed in the two diffraction patterns of the opposite axial modes.
TL;DR: In this article, the authors derived a Markoffian master equation for a spin system which interacts with a phonon heat bath, and obtained the transition rate on a time scale of one microscopic event (one spin flip).
Abstract: Phenomenological master equations are useful for discussions of various aspects of the kinetics of phase transitions The transition rate in such master equations is of a form proposed by Langer; the rate is an exponential of the change of the free energy of the system due to a change in a thermodynamic variable We present a derivation of this form from a microscopic theory for an Ising model which exhibits first and second order phase transition Using Zwanzig’s formalism we derive a master equation for a spin system which interacts with a phonon heat bath Thus, we obtain the transition rate on a time scale of one microscopic event (one spin flip) This form differs from the phenomenological one in that it depends on details of the dynamics of the system, the heat bath, and the interaction between them These details are removed by a further averaging (coarse‐graining) If a Markoffian master equation is assumed to exist on a phenomenological time scale large compared to the microscopic time scale, yet small compared to the relaxation time of the system, then we show that the transition rate in that master equation has the form proposed by Langer
TL;DR: In this article, the authors investigated properties of the cooperative emission of radiation from identical two-level atoms by using, for the collective atomic operators, the Holstein-Primakoff transformation to boson operators.
Abstract: We investigate properties of the cooperative emission of radiation from $N$ identical two-level atoms by using, for the collective atomic operators, the Holstein-Primakoff transformation to boson operators. In the basis of the Glauber states, a weight function is associated with the atomic density operator. The equation of motion for the weight function is derived from the master equation and approximate solutions are given in the two limit cases of superradiance and diffusion. The moments of the radiation intensity are calculated and are found to be in close agreement with previous works.
TL;DR: In this paper, the evolution of a cluster by a postulated stochastic master equation in which the transition probabilities are given by a formula proposed by Langer is described. And they derive from these assumptions, by a path integral method, an equation for the most probable evolution of the size of the cluster and show thereby that under these circumstances the two definitions are equivalent.
Abstract: A critical cluster in homogeneous nucleation can be defined in two ways. Thermodynamically, the size of the critical cluster corresponds to a maximum of the free energy of the system; kinetically the critical cluster is defined such that all supercritical clusters grow and all others decay. We describe the evolution of a cluster by a postulated stochastic master equation in which the transition probabilities are given by a formula proposed by Langer. We derive from these assumptions, by a path integral method, an equation for the most probable evolution of the size of the cluster and show thereby that under these circumstances the two definitions are equivalent.
TL;DR: In this article, the effects on the energy absorbed and the distribution of energy resulting from a phased response of a molecule to a driving laser field were investigated, comparing the dynamic and steady-state solutions to our generalized master equation with solutions to an ordinary master equation, for truncated (finite level) and nontruncated (infinite-level) harmonic oscillators.
Abstract: We consider the effects on the energy absorbed and the distribution of energy resulting from a phased response of a molecule to a driving laser field. Contrasting a Landau–Teller collision model with a strong collision model we compare the dynamic and steady‐state solutions to our generalized master equation with solutions to an ordinary master equation, for truncated (finite‐level) and nontruncated (infinite‐level) harmonic oscillators. We find significant coherent enhancement in upper energy level populations which is most pronounced when an intense but below saturation laser field is present; these coherent effects, predicted for truncated oscillator level populations, are contrasted with the infinite level cases. We show that at steady state an ordinary master equation predicts the average energy of the harmonic oscillator correctly but fails to predict the correct population distribution of energy states.
TL;DR: In this paper, the generalized master equation (GME) was analyzed in the weak radiation field limit for a two-level absorber and it was shown that the GME reduces to an ordinary master equation when the absorber response is perturbed by collisions, by phase interruptions in the light field, or by a combination of both.
Abstract: The generalized master equation (GME) describing the time evolution of energy level populations for a laser‐pumped molecular gas, suffering phase‐changing and thermalizing collisions, is analyzed in the weak radiation field limit for a two‐level absorber. We show that the GME reduces to an ordinary master equation (OME) when the absorber response is perturbed by collisions, by phase interruptions in the light field, or by a combination of both. The GME to OME reduction is possible only in the low intensity and weak collision limits thereby placing a stringent limitation on the use of ordinary master equations to obtain accurate estimates of the evolution of energy level populations.
TL;DR: In this article, a new formalism of the theory of magnetic relaxation is given for systems where the random molecular motions are describable as a Markoff process, and the master equation for the density matrix is derived form the stochastic Liouville equation.
TL;DR: In this article, the authors derived a generalized master equation for precompound processes with the Hauser-Feshbach formula as a special case, based on the statistics of the nuclear matrix elements.
TL;DR: In this article, it was shown that the Volterra-lotka model does not have a steady state, by considering the Fokker-Planck-type equation for small fluctuations.
Abstract: Nicolis and Prigogine have shown that the Volterra-Lotka model taken as a Markovian stochastic process does not have a steady state, by considering the Fokker-Planck-type equation for small fluctuations. Here we use the exact master equation to show that the only steady state in the model is the trivial one.
Journal Article•
TL;DR: In this paper, the authors generalized the Waldmann-Snider kinetic equation to a chemically reacting system and derived the master equations for the reacting system from the derived kinetic equations and the rate coefficients are identified.
Abstract: The Waldmann–Snider kinetic equations are generalized to a chemically reacting system AB+C ⇄ AC+B ⇄ A+BC ⇄ AB+C. In the formulation Kirkwood’s time averaging is introduced, which provides in a natural and concise way a route to exploit time‐independent scattering theoretic technique. The master equations for the reacting system are obtained from the derived kinetic equations and the rate coefficients are identified.
TL;DR: In this paper, the influence of nonlinear velocity fluctuations on the velocity correlation function Π (t) is studied for the Rayleigh model of a massive particle in an ideal gas as an example of a master equation system.
Abstract: The influence of nonlinear velocity fluctuations on the velocity correlation function Π (t) is studied for the Rayleigh model of a massive particle in an ideal gas as an example of a master equation system. It is shown that the Mori kernel K (t), which determines the decay of Π (t), has a slow mass‐dependent decay on the time scale of the decay of Π (t) and has no well‐behaved expansion in the mass ratio. Both features are contrary to standard assumption. The origins of the slow decay are traced to nonlinear fluctuations and the relationship to previous work on requisite conditions for exact exponential decay is discussed. The slow decay of Π (t) is shown to lead to divergent ’’Burnett’’ coefficients in macroscopic friction laws and the resolution of this difficulty is discussed. The relationship of the microscopic ’’bare’’ friction constant to the macroscopic friction constant is considered. Explicit expressions for Π (t) and K (t) for small mass ratio are obtained by mode–mode coupling analysis and perturbation methods. The influence of nonlinear fluctuation effects is found to be numerically negligible despite their long lifetime. The remaining deviation from standard Brownian motion results is examined numerically. The validity of some standard assumptions in mode–mode coupling theory is also examined.
TL;DR: In this paper, two methods for asymptotic expansion of kinetic equation, namely, the system size expansion of master equation and the reductive perturbation for macroscopic equation of motion, are incorporated to calculate the fluctuation of macrovariables in systems far from thermal equilibrium.
Abstract: Two methods for asymptotic expansion of kinetic equation, namely, the system size expansion of master equation and the reductive perturbation for macroscopic equation of motion, are incorporated to calculate the fluctuation of macrovariables in systems far from thermal equilibrium. As an illustration, analytical expressions for the moments of fluctua tion in an open chemical system slightly beyond the instability point are obtained.
TL;DR: In this paper, a master equation for the coupled coherent and incoherent motion of Frenkel excitons is derived, which describes the energy transfer in materials with non-periodic structure.
Abstract: Starting from the Haken-Strobl-model for the coupled coherent and incoherent motion of Frenkel excitons, a master equation is derived describing the quasi-incoherent motion of these excitations. In contrast to previous derivations no use is made of special symmetries. Therefore this equation may also describe the energy transfer in materials with non-periodic structure, which play a role in biological systems. The master equation is solved for crystals with one and two molecules in the unit cell, and explicit expressions are given for the case of nearest neighbour interaction too. Furtheron, asymptotic forms of the solutions are discussed.
TL;DR: In this article, the authors derived exact formulae for correlations in a certain non-equilibrium chemical reaction using coupled chemical and diffusion master equations, and showed that the probability distribution is poissonian in small volumes, and non-polynomial in large volumes.