Showing papers on "Master equation published in 1984"

Basic governing equations for the flight regimes of aeroassisted orbital transfer vehicles

Abstract: The basic governing equations for the low-density, high-enthalpy flow regimes expected in the shock layers over the heat shields of the proposed aeroassisted orbital transfer vehicles are derived by combining and extending existing theories. The conservation equations are derived from gas kinetic principles for a four-component ionized gas consisting of neutral molecules, neutral atoms, singly ionized ions, and electrons, assuming a continuum flow. The differences among translational-rotational, vibrational, and electron temperatures are accounted for, as well as chemical nonequilibrium and electric-charge separation. Expressions for convective and viscous fluxes, transport properties, and the terms representing interactions among various energy modes are explicitly given. The expressions for the rate of electron-vibration energy transfer, which violates the Landau-Teller conditions, are derived by solving the system of master equations accounting for the multiple-level transitions.

Bistable systems: Master equation versus Fokker-Planck modeling

Abstract: Relaxation and fluctuations of nonlinear macroscopic systems, which are frequently described by means of Fokker-Planck or Langevin equations, are studied on the basis of a master equation. The problem of an approximate Fokker-Planck modeling of the dynamics is investigated. A new Fokker-Planck modeling is presented which is superior to the conventional method based on the truncated Kramers-Moyal expansion. The new approach is shown to give the correct transition rates between deterministically stable states, while the conventional method overestimates these rates. An application to the Schl\"ogl models for first- and second-order nonequilibrium phase transitions is given.

Theory of relaxation in viscous liquids and glasses

Abstract: A theory of relaxation (the time‐dependent change of macroscopic properties) of nonpolymeric viscous liquids and glasses is presented. The fluid is described by a set of quasiequilibrium structures, and a master equation gives the transitions among these structures. Any structural change is presumed to require a cooperative rearrangement involving many atoms, and this rearrangement entails a fluctuation to a high‐energy transition state. The structure of the fluid varies from point to point, and the rate of this transformation depends crucially on the local structure. The resulting kinetic equation describes very well the main features of observed relaxation—namely, the broad distribution of relaxation times and the nonlinearity (in ΔT) of relaxation following a temperature jump ΔT, where the apparent activation energy for relaxation depends on time. The kinetic equation is solved exactly, and the resulting solution is exhibited for a particular set of the parameters. The resulting relaxation function for energy relaxation goes as e −t 1 / 4 , except at the very shortest times. (The precise exponent— 1/4 in this case—will depend on the parameters used in the particular calculation.) A detailed comparison with other theories is made, and suggestions of how the theory can be used to develop a phenomenological model of relaxation of glass are given.

Continuous measurement: Watchdog effect versus golden rule

Abstract: Conditions which may lead to a freezing of the motion of a system under continuous observation (the so-called "Zeno paradox" or "watchdog effect") are examined. The measurement process is treated phenomenologically by the usual wave-packet reduction as well as in a more realistic way by including the measuring apparatus. For this purpose a model for an ideal measurement process is employed, following an example given by von Neumann. The resulting behavior varies between complete freezing and a mere suppression of interference terms and constant transition rates as represented by a master equation (rate equation). The most familiar example of the latter is Fermi's golden rule, with integration leading to exponential decay. Reviewing and extending the derivation of the Pauli master equation, the conditions leading to constant transition rates are discussed. The importance of the interaction with the natural environment for establishing a master equation is emphasized. Some consequences for the derivation of macroscopic equations of motion and for the physical foundations of superselection rules are pointed out.

Noise in strong laser-atom interactions: Phase telegraph noise

Abstract: We discuss strong laser-atom interactions that are subjected to jump-type (random telegraph) random-phase noise. Physically, the jumps may arise from laser fluctuations, from collisions of various kinds, or from other external forces. Our discussion is carried out in two stages. First, direct and partially heuristic calculations determine the laser spectrum and also give a third-order differential equation for the average inversion of a two-level atom on resonance. At this stage a number of general features of the interaction are able to be studied easily. The optical analog of motional narrowing, for example, is clearly predicted. Second, we show that the theory of generalized Poisson processes allows laser-atom interactions in the presence of random telegraph noise of all kinds (not only phase noise) to be treated systematically, by means of a master equation first used in the context of quantum optics by Burshtein. We use the Burshtein equation to obtain an exact expression for the two-level atom's steady-state resonance fluorescence spectrum, when the exciting laser exhibits phase telegraph noise. Some comparisons are made with results obtained from other noise models. Detailed treatments of the effects ofmly jumps, or as a model of finite laser bandwidth effects, in which the laser frequencymore » exhibits random jumps. We show that these two types of frequency noise can be distinguished in light-scattering spectra. We also discuss examples which demonstrate both temporal and spectral motional narrowing, nonexponential correlations, and non-Lorentzian spectra. Its exact solubility in finite terms makes the frequency-telegraph noise model an attractive alternative to the white-noise Ornstein-Uhlenbeck frequency noise model which has been previously applied to laser-atom interactions.« less

Onset of spatial correlations in nonequilibrium systems: A master-equation description

Abstract: The emergence of spatial correlations around a nonequilibrium steady state is studied by means of a stochastic description based on a multivariate master equation The dependence of the strength and range of the correlations on the distance from equilibrium is determined The formalism is applied to chemically reacting systems and to simple fluids submitted to a temperature gradient

Asymptotic solution of the Kramers-Moyal equation and first-passage times for Markov jump processes

Abstract: We calculate the activation rates of metastable states of general one-dimensional Markov jump processes by calculating mean first-passage times. We employ methods of singular perturbation theory to derive expressions for these rates, utilizing the full Kramers-Moyal expansions for the forward and backward operators in the master equation. We discuss various boundary conditions for the first-passage-time problem, and present some examples. We also discuss the validity of various diffusion approximations to the master equation, and their limitations.

The Quantum Theory of Unimolecular Reactions

Abstract: 1. The observed properties of thermal unimolecular reactions 2. The master equation for internal relaxation in molecules 3. Reaction as a perturbation of the internal relaxtion 4. The specific rate function k(E) as an inverse Laplace transform 5. Unimolecular fall-off in strong collision systems 6. A molecular dynamic approach to specific rate functions 7. Building in the randomisation processes 8. Weak collision processes 9. How well does it all work? Appendix.

Radiative decay of autoionizing states in laser fields. I. General theory

Abstract: A theory describing the radiative decay of an autoionizing state under strong pumping by a coherent field is developed. The theory systematically takes into account the radiative decay of the unperturbed continuum. The problem at hand corresponds to the case of strongly coupled bound states decaying to electron and photon continua with the two continua also weakly coupled to each other. A master equation describing the time evolution of the atomic system is derived, and its exact solution under arbitrary initial conditions is given. The effect of radiative decay on the Fano profiles and photoelectron spectra is analyzed in detail. The time development of the system is also examined. The radiative decay of the autoionizing state and the unperturbed continuum changes the spectra in a significant way. The characteristics of the spectra are correlated with the dressed states (with complex energies) of the system. The changes in the structure of the dressed states as a function of the system parameters such as the spontaneous-emission rate and laser intensity are discussed in detail.

Solution of Kramers–Moyal equations for problems in chemical physics

Abstract: We derive asymptotic solutions of Kramers–Moyal equations (KMEs) that arise from master equations (MEs) for stochastic processes. We consider both one step processes, in which the system jumps from x to x+e or x−e with given probabilities, and general transitions, in which the system moves from x to x+eξ, where ξ is a random variable with a given probability distribution. Our method exploits the smallness of a parameter e, typically the ratio of the jump size to the system size. We employ the full KME to derive asymptotic expansions for the stationary density of fluctuations, as well as for the mean lifetime of stable equilibria. Thus we treat fluctuations of arbitrary size, including large fluctuations. In addition we present a criterion for the validity of diffusion approximations to master equations. We show that diffusion theory can not always be used to study large deviations. When diffusion theory is valid our results reduce to those of diffusion theory. Examples from macroscopic chemical kinetics a...

Laser cooling of trapped ions: dynamics of the final stages

Abstract: We present a master equation that describes the dynamics of laser cooling of a trapped ion. It is valid in the Lamb–Dicke limit and rests on an adiabatic elimination combined with a degenerate perturbation treatment. It describes relaxation of probabilities and coherences in the harmonic-trap degrees of freedom. The eigenvalue spectrum the of the time-evolution operator is derived, and it follows that only one zero eigenvalue exists, giving the unique steady-state probability distribution. The coherences all decay to zero with time. The ultimate steady state is distribution, which can be characterized by a temperature. We also report a numerical calculation that a Planck supports our analytical work. The final energy of the cooling is given and discussed. Finally there is a comparison between the present results and our earlier, approximate, treatments.

Stochastic analysis of symmetry-breaking bifurcations: Master equation approach

Abstract: The multivariate master equation for a general reaction-diffusion system is solved perturbatively, in the vicinity of a bifurcation point leading to symmetry-breaking transitions. The possibility to express the result through a Brazovskii type of potential is examined, and a comparison with the Langevin analysis of Walgraefet al. [Adv. Chem. Phys. 49:311 (1982)] is performed.

The master equation for laser cooling of trapped particles

Abstract: The authors consider the laser cooling of a trapped two-level system during its final approach to equilibrium. Then it moves only within one optical wavelength and an expansion in the Lamb-Dicke parameter is possible. This allows an adiabatic elimination of the internal degrees of freedom. There remains a slow time evolution on a new time scale related to the Lamb-Dicke parameter. This scale is determined by an effective time evolution operator, which the authors derive using the method of degenerate perturbation theory. The ensuing master equation can be completely solved both for its time evolution and the ultimate steady state. In addition to providing a complete description of the final stages of the laser cooling, the calculation can be seen to add one more soluble case to the discussion of non-equilibrium statistical mechanics.

Erratum: Relaxation of quantum systems weakly coupled to a bath. I. Total-time-ordering-cumulant and partial-time-ordering-cumulant non-Markovian theories

Abstract: The present paper studies the formal aspects of the total-time-ordering-cumulant (TTOC) and the partial-time-ordering-cumulant (PTOC) relaxations of a quantum system, of few degrees of freedom, in weak interaction with a bath, both within and outside the Markovian limit. To this end, the general expressions connecting the matrix elements of the TTOC and PTOC relaxation superoperators with the bath correlation functions are determined. Special attention is paid to two particular cases: a system with a nonequidistant energy spectrum and a system with an equidistant energy spectrum. Discussions revolve mainly around the possibility of applying the secular approximation to the TTOC and PTOC master equations for the off-diagonal matrix elements of the reduced-density operator of the system.

Bethe States for the Quantum Three Wave Interaction Equation

Abstract: The quantum three wave interaction equation is introduced. The exact eigen-states of the Hamiltonian are constructed. Furthermore, the existence of the bound states is shown. It is concluded that the quantum three wave interaction equation is completely integrable.

Quantal Brownian motion in stationary and nonstationary fermionic reservoirs

Abstract: A model for collective mode damping in nuclei is devised in the frame of a theory of irreversible evolution. The decay width of a fast nuclear vibration, originated in its coupling to the remaining nuclear degrees of freedom, is calculated in a dynamical fashion. To this aim, a set of equations is proposed that describes the simultaneous dynamics of the oscillation or its associated array of bosons and of the interacting fermions that play the role of a heat reservoir. These are, respectively, a quantal master equation and modified kinetic one. The two of them exhibit their mutual coupling in the non-Hermitian terms of their generators of motion. The equations are worked out in detail in (a) the weak-coupling approximation plus (b) the very-close-to-equilibration regime plus (c) the energy-conserving description of intermediate processes. With hypothesis (c) the heat bath can be regarded as lying in a steady state at all times and the master equation is solved for different temperatures and phonon energies. The damping width of the oscillations is thus quantitatively predicted.

Hydrogen bond statistics and dynamics in water: Self‐diffusion and dielectric relaxation

Abstract: Dynamical properties of water up to the supercooled region are analyzed by explicitly taking into account the dynamics of the hydrogen bonds. The H‐bond dynamics is described by means of a stochastic variable η exploring different states according to the number of H bonds established by the tagged molecule with its neighbors. Application of the ‘‘reduced’’ model theory leads to a master equation for the long‐time diffusional properties of water, depending only on meaningful physical quantities that are obtained either from experiments and computer simulations or on the ground of physical considerations. Dielectric relaxation and self‐diffusion behaviors are successfully reproduced and also recent results from incoherent neutron scattering can be rationalized.

Theory of external two-state Markov noise in the presence of internal fluctuations

Abstract: A theory is presented to take into account internal fluctuations in the study of stochastically driven systems. Internal fluctuations are modeled by a master equation in which external noise is introduced. External noise is modeled by a two-state Markov process. A unified theory of internal and external fluctuations is described in terms of an effective integrodifferential master equation or its equivalent generating function representation. Two examples for which exact analytical results can be obtained are presented. A discussion of the white noise limit of the theory is also given.

Breakdown of linear response theory in one-dimensional classical diffusion problems (charge transport)

Abstract: The authors examine charge transport in one-dimensional disordered systems which can be described by a master equation. The diffusivity is not a meaningful quantity to study because in general the linear response assumption does not hold. The authors formulate a scaling theory for the current and examine three classes of distribution functions which lead to localised, quasilocalised and delocalised behaviour in the long-time limit. Whereas linear response is valid as t to infinity for delocalised motion (finite DC conductivity), interesting anomalies are found and predicted for the electric field dependence of the current in the first two situations.