Showing papers on "Master equation published in 1990"

Theory of Unimolecular and Recombination Reactions

Abstract: Introduction Elementary Transition State Theory Microscopic Rate Coefficients Practical Implementation of RRKM Theory Collisional Energy Transfer The Master Equation Conclusions.

Environmental and spontaneous localization

Abstract: The dynamical elimination or reduction of macroscopic superpositions has long been of interest, particularly with regard to the quantum theory of measurement. A number of models have demonstrated this for the reduced density matrix of a system interacting with an environment. Alternatively, Ghirardi, Rhimini, and Weber [Phys. Rev. D 34, 470 (1986)] have proposed a fundamental modification of the Schr\"odinger equation, quantum mechanics with spontaneous localization (QMSL), which provides a master equation similar in mathematical form. In this paper we consider an isotropic environment that is elastically scattered by the system of interest with negligible momentum transfer, extending a previous result of Joos and Zeh [Z. Phys. B. 59, 223 (1985)] from small length scales to all length scales. We discuss the physical nature and relevance of the differences between our result and similar open systems calculations. We describe the mathematical similarity between our extended environmental model and the QMSL dynamics determining the QMSL parameters that allow calculations using QMSL to be used as a model for the effect of the environment. That gives us access to a number of interesting results obtained for the QMSL master equation. Finally, we discuss some experimental considerations for the purposes of detecting effective nonunitary evolution of this form.

The weak coupling limit as a quantum functional central limit

Abstract: We show that, in the weak coupling limit, the laser model process converges weakly in the sense of the matrix elements to a quantum diffusion whose equation is explicitly obtained. We prove convergence, in the same sense, of the Heisenberg evolution of an observable of the system to the solution of a quantum Langevin equation. As a corollary of this result, via the quantum Feynman-Kac technique, one can recover previous results on the quantum master equation for reduced evolutions of open systems. When applied to some particular model (e.g. the free Boson gas) our results allow to interpret the Lamb shift as an Ito correction term and to express the pumping rates in terms of quantities related to the original Hamiltonian model.

Cooperative behavior of atoms irradiated by broadband squeezed light.

Abstract: We consider the dynamics of a collection of atoms interacting with a coherent field and a broadband squeezed vacuum. We obtain an exact solution of the master equation and study in detail the types of nonequilibrium steady states that can be generated. We show that in the absence of coherent drive the atoms are in a state whose properties are similar to those of the squeezed vacuum for photons. We demonstrate that the steady state for certain discrete values of the external field strength and detuning is a pure state which is the eigenstate of the non-Hermitian operator cosh(‖ξ‖)S-+sinh(‖ξ‖)S+, where ξ is the squeezing parameter associated with the input radiation field. These eigenstates play a very fundamental role in the theory and satisfy the equality sign in the Heisenberg uncertainty relation Δ SxΔSy≥ ½‖Sz‖. We also present detailed numerical results for the characteristics of the field generated by the collective system.

Transitions induced in a double minimum system by interaction with a quantum mechanical heat bath

Abstract: A quantum mechanical treatment of a double minimum system interacting with a heat bath is presented for the purpose of interpreting experimental data on transfer kinetics in condensed hydrogen‐bonded systems. The model describes the transfer motion in one or two dimensions. The heat bath is represented by a set of harmonic oscillators and the interaction by a term linear in the system coordinates and in the bath coordinates. Extending an earlier random field approach, the present treatment consistently accounts for the quantum nature of the total system. With crystalline benzoic acid dimer used as an example, the master equation for the populations of the energy levels of the hydrogen transfer motion is derived. Transition probabilities consistent with the principle of detailed balance are obtained, based on a representation with explicit off‐diagonal tunnel interactions for pairs of states localized on different sides of the barrier and with diagonal terms describing the rearrangement of the heat bath as a consequence of the tunneling motion. The activation of the double minimum transfer process with increasing temperature is related to the excitation of the local vibrations in the two potential wells.

Ito, Stratonovich and kinetic forms of stochastic equations

Abstract: The well known Ito and Stratonovich forms of presentation of stochastic equations are not, in general, physically equivalent. From the point of view of the statistical theory of nonequilibrium processes the third is most natural-the “kinetic form” of presentation of the Langevin and corresponding Fokker-Planck equations. Only in this case exist fluctuation- dissipation relations (the Einstein formula) for nonlinear systems. For the confirmation of this point of view the following different concrete systems are considered: Brownian motion of particles in a medium with nonlinear friction, of the Van der Pol oscillators and others. The connection between the master equation and the Fokker-Planck one is also considered.

Quantum Kramers model: Solution of the turnover problem.

Abstract: The quantum-mechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within first-order perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.

Spin in contact with thermostat: Exact reduced dynamics

Abstract: New aspects of exact reduced dynamics of a simple model of an open system (spin- 1 2 in a magnetic field and coupled to a Bose reservoir) are presented. A relaxation problem, final states of the system, Schrodinger-like and Heisenberg-like representations of dynamics are considered. A master equation for a statistical operator of the system and its solution are obtained, and the related question of the construction of a semigroup for the time evolution is discussed.

Optical dephasing in solution: A line shape and resonance light scattering study of azulene in isopentane and cyclohexane

Abstract: Results of a line shape and resonance light scattering study of the S1←S0 and S2←S0 electronic transitions of azulene in isopentane and cyclohexane are reported. The results are analyzed using two different non‐Markovian master equations that make different assumptions about the statistical properties of the bath. For both these origin transitions we find that the solution dynamics fall in the so‐called intermediate modulation regime. If exponential decay is assumed for the bath correlation function we obtain a correlation time of the bath of 25 fs for the S1←S0 transition and of 13 fs for the S2←S0 transition at room temperature. From the frequency dependence of the ratio of fluorescence to Raman yields of the S1←S0 transition we calculate an excited state lifetime of 1.4±0.2 ps using the parameters of the bath derived from the line shape analysis, and irrespective of which master equation is used.

A theoretical prediction of hydrogen molecule dissociation-recombination rates including an accurate treatment of internal state nonequilibrium effects

Abstract: The dissociation and recombination of H2 over the temperature range 1000-5000 K are calculated in a nonempirical manner. The computation procedure involves the calculation of the state-to-state energy transfer rate coefficients, the solution of the 349 coupled equations which form the master equation, and the determination of the phenomenological rate coefficients. The nonempirical results presented here are in good agreement with experimental data at 1000 and 3000 K.

Microscopic Simulation of Chemical Oscillations in Homogeneous Systems

Abstract: A microscopic computer experiment is set up to investigate the statistical properties of far from equilibrium homogeneous chemical systems undergoing instabilities. Sustained periodic behavior of limit cycle type is observed. Both the frequency and the amplitude of the oscillations are found to be in good agreement with the macroscopic description. A comparison with the stochastic theory of chemical systems based on master equation formalism is also carried out. The dynamical and static correlation functions obtained by these two procedures are in very good agreement.

Exact quantum statistics of a nonlinear dissipative oscillator evolving from an arbitrary state.

Abstract: The statistical properties of the third-order nonlinear dissipative oscillator, which evolves from any state, are derived on the basis of the exact solution to the master equation. Some important features of the nonlinear oscillator model, such as the recurrences of the initial state, are related to the properties of quasidistributions connected with the phase of the complex field amplitude.

Electron transport in the presence of a Coulomb field

Abstract: We analyze the modifications of the transport behavior of electrons in dense media due to the presence of a strong Coulomb field generated by an ion moving initially in close phase-space correlation with the electrons. These modifications play a profound role in convoy electron emission in ion-solid collisions. The transport behavior is studied within the framework of a classical phase-space master equation. The nonseparable master equation is solved numerically using test-particle discretization and Monte Carlo sampling. In the limit of vanishing Coulomb forces the master equation becomes separable and can be reduced to standard one-dimensional kinetic equations for free-electron transport that can be solved exactly. The comparison to free-electron transport is used to gauge both the reliability of test-particle discretization and the significance of Coulomb distortion of the distribution functions. Applications to convoy-electron emission are discussed.

Canonical transformation and decay into phase-sensitive reservoirs.

Abstract: We show that the master equation describing the dissipative evolution of a bosonic quantum system coupled to a phase-sensitive reservoir may be reduced to a standard form of dissipation for a thermal reservoir by canonical transformations. The solution to the master equation for complex phase-sensitive interaction induced, for example, by broadband squeezed light, may then be obtained by a transformation from the known solutions for the simpler thermal case. We show that canonical transformations have a particularly useful form when describing the evolution of quantum dissipative systems in phase space using the Wigner function. We illustrate these ideas with two phase-sensitive dissipative problems: that of the correlated-emission laser and of decay induced by a broadband squeezed vacuum field.

Noise and dissipation in quantum theory

Abstract: A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrodinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.

Kinetic lattice gas model: Langmuir, Ising and interaction kinetics

Abstract: The kinetic lattice gas model is formulated properly to account for adsorption, desorption, and diffusion at surfaces. We examine three choices for the transition probabilities in the master equation, which we term Langmuir, Ising and interaction kinetics, and show how they lead to different sticking coefficients and desorption rates.

Electron energy distribution functions and vibrational population densities of excited electronic states in DC discharges through nitrogen

Abstract: An augmented collisional-radiative model which includes the vibrational kinetics of excited electronic states of nitrogen is presented. The Boltzmann transport equation which includes vibrational and electronic superelastic collisions is solved, coupled to a system of vibrational master equations with eight excited electronic states. The model explains satisfactorily the trends of vibrational distributions of several excited electronic states observed under non-equilibrium conditions. The results also show that energy pooling reactions can play an important role in determining the vibrational distributions of A3 Sigma u+, B3 Pi g, and C3 Pi u.

Urban Dynamics : Designing an Integrated Model

Abstract: 1. Introduction 2. General economic principles for building comprehensive urban models 3. Approaches to stock dynamics based on spatial interaction models 4. Master equations 5. Stochastic processes 6. The elements for an integrated approach 7. Housing 1: A dynamic economic model of the regulated housing market 8. Housing 2: A master equation approach 9. Housing 3: A stochastic assignment approach 10. Services 1: A spatial-interaction-dynamic approach 11. Services 2: A master equation approach 12. The land market in economic urban models 13. Labour market 1: an economic model 14. Labour market 2: A master equation approach 15. Labour market 3: A stochastic assignment approach 16. Transport 17.The first example of an integrated operational model

Nonequilibrium transitions in driven AB3 compounds on the fcc lattice: A multivariate master-equation approach.

Abstract: We studied the order-disorder transition in an Ising-type alloy on a fcc lattice with ${\mathit{AB}}_{3}$ stoichiometry with atomic exchanges due to two competing processes: thermally activated jumps and ballistic jumps, as, for example, is the case under irradiation with high-energy particles. The latter favor disordered configurations, while the former tend to restore a certain degree of order. The state of order is described by a four-dimensional parameter, the occupation of the four simple cubic sublattices into which the fcc lattice may be decomposed. In a mean-field approximation the kinetic equations for the evolution of this order parameter can be found. For a stochastic description, the master equation for the probability of a given state of order is approximated using Kubo's ansatz. The resulting partial differential equation is solved taking advantage of symmetry properties of the order-parameter space. A dynamical-equilibrium phase diagram is constructed, and it is shown that new phases, not found under thermal conditions, can be stabilized for a certain model for the saddle-point energy of the thermal jumps.

Renormalized equations for linear transport in stochastic media

Abstract: The master equation is used to obtain a model describing the ensemble‐averaged intensity corresponding to linear particle transport in randomly mixed immiscible fluids. An asymptotic limit corresponding to small amounts of opaque fluids admixed with large amounts of transparent fluids is employed to reduce the complexity of the description. In the limit of a single transparent fluid, a renormalized transport equation is obtained, involving an effective source and effective interaction coefficients that account, in a simple way, for the statistical nature of the problem in this asymptotic limit.

Relaxation and aging in spin glasses and other complex systems

Abstract: We describe how a hierachical model of spin glass relaxation can display aging behaviour, similarly to what is found in spin glasses and other complex systems out of thermodynamic equilibrium. Since we deal with a nonequilibrium situation, the usualF(luctuation)D(issipation)T(heory) does not apply. We therefore derive a general relation between the linear response function and the non equilibrium propagator of the unperturbed system. The relation is shown to be very similar to the equilibrium FDT under certain conditions, which one can reasonably assume for spin glass systems. Having thus related the linear response of the system to a small external field to the autocorrelation function of the magnetization, we calculate the latter quantity by a master equation on a set of states which have the topology of a tree. The model can reproduce the main qualitative features of theZ(ero)F(ield)C(ooled) spin glass experiments, i.e. the maximum in the logarithmic time derivative of the magnetization, with only two free parameters.

Generation of squeezed microwave states by a dc-pumped degenerate parametric Josephson junction oscillator.

Abstract: The master equation for a dc-pumped degenerate Josephson parametric amplifier is derived. It is shown that the Wigner distribution representation of this master equation can be approximated by a Fokker-Planck equation. By using this equation, the dynamical behavior of this degenerate Josephson amplifier with respect to squeezing of the radiation field is investigated. It is shown that below threshold of parametric oscillation, a squeezed vacuum state can be generated, and above threshold a second bifurcation point exists, where the device generates amplitude squeezed radiation. Basic relations between the achievable amplitude squeezing, the output power, and the operation frequency are derived.

Chaos, molecular fluctuations, and the correspondence limit

Abstract: Chaos is characterized by sensitive dependence on initial conditions. Trajectories determined by coupled, ordinary differential equations show sensitive dependence when their associated Liapunov exponent is positive. The Liapunov exponent is positive if the Jacobi matrix associated with the coupled differential equations has an eigenvalue with a positive real part, on the average, as the Jacobi matrix evolves along the trajectory. For macrovariable equations, there are also fluctuation equations which follow the macrovariable trajectories. The covariance matrix for these fluctuations evolves according to an equation in which the Jacobi matrix for the deterministic motion plays the dominant role. For a chaotic trajectory, the covariance matrix grows exponentially. This means that for macrovariable equations that imply chaos, the construction of the macrovariable equations out of an underlying master equation is no longer valid. The macrovariable equations cease to be physical, and the physical description must be done entirely at the master equation level of description where the fluctuations, which are very large scale, can be properly treated. In parallel with this analysis, the correspondence limit connecting the time evolution of the Wigner distribution with Liouville's equation breaks down when the classical motion is strongly chaotic. This implies that strongly chaotic classical dynamics must be treated quantum mechanically in order to properly treat the quantum fluctuations which have grown macroscopically large. Experimental confirmation of these ideas is discussed.