Showing papers on "Master equation published in 1985"

Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation.

Abstract: We develop a formulation of quantum damping theory in which the explicit nature of inputs from a heat bath, and of outputs into it, is taken into account. Quantum Langevin equations are developed, in which the Langevin forces are the field operators corresponding to the input modes. Time-reversed equations exist in which the Langevin forces are the output modes, and the sign of damping is reversed. Causality and boundary conditions relating inputs to system variables are developed. The concept of ``quantum white noise'' is formulated, and the formal relationship between quantum Langevin equations and quantum stochastic differential equations (SDE's) is established. In analogy to the classical formulation, there are two kinds of SDE's: the Ito and the Stratonovich forms. Rules are developed for converting from one to the other. These rules depend on the nature of the quantum white noise, which may be squeezed. The SDE's developed are shown to be exactly equivalent to quantum master equations, and rules are developed for computing multitime-ordered correlation functions with use of the appropriate master equation. With use of the causality and boundary conditions, the relationship between correlation functions of the output and those of the system and the input is developed. It is possible to calculate what kind of output statistics result, provided that one knows the input statistics and provided that one can compute the system correlation functions.

Damping of quantum coherence: The master-equation approach.

Abstract: We solve the master equation for the coordinate-coordinate damped harmonic oscillator for initial superpositions of coherent states. In the zero-temperature case the solution remains a simple superposition of coherent states. While the underdamped oscillator evolves all initial superpositions into mixtures of coherent states the overdamped oscillator does so selectively. For finite temperatures coherent states are no longer preserved, and we find a decrease in the variance of the off-diagonal coordinate-basis density-matrix elements below the coherent-state value. This variance decreases with increasing bath temperature. In the overdamped case there is negligible associated spreading of the diagonal coordinate-basis density-matrix elements. Thus the coordinate basis is an example of Zurek's pointer basis and the coordinate damped oscillator models the coordinate-basis density-matrix diagonalization which occurs in a coordinate measurement.

On the coupling of electron and vibrational energy distributions in H2, N2, and CO post discharges

Abstract: A coupled solution of the Boltzmann equation, of the vibrational master equation, and of the plasma chemistry describing the dissociation process has been performed in H2 post discharges in the μs and ms regimes. The results in H2 show that the superelastic vibrational gain tends to compensate both the inelastic and elastic (including rotational) energy losses, thereby yielding a quasistationary situation characterized by an electron ‘‘temperature’’ smaller than the vibrational temperature θ1(Te θ1 in N2 and CO as a result of the deviation of the actual vibrational distributions of these species from the Boltzmann one.

Exact solutions for random coagulation processes

Abstract: Smoluchowski's coagulation equation can only derive physical validity as the limit of a random coagulation process. For coagulation rateKij=a+b(i+j) and no fragmentation, random coagulation is exactly solvable and has the coagulation equation as its limit as the system sizeM→∞, with deviations for finite systems of relative orderO(M−1/2). The same is true for constant coagulation and non-zero fragmentation rate, independent ofi andj (the sizes of the coagulation clusters) in which case the stochastic equations (master equation) are also exactly solvable.

Diffusive and jump description of hindered motions

Abstract: Site jump models are often used to interpret spectroscopic effects of molecular motions occurring in the presence of potential wells. Continuous diffusion equations, albeit complex to handle, are expected to give a more detailed picture of the dynamics and to provide molecular interpretation of the kinetic parameters. In the paper we show how the results of random walk models can be recovered from the correct solutions of the diffusion equations. To this purpose, two routes are followed. First, a procedure is developed for the exact calculation of the time integral of pertinent correlation functions, to be compared with the time constant for the kinetic process of interest. Secondly, the asymptotic solutions of the diffusion equations, valid in the limit of high potential gradients, are used to derive ‘localized functions’, which lead quite naturally to master equations for jumps among discrete sites. Rotational diffusion in uniaxial liquid crystals, translational motions across smectic layers, hindered i...

Electron-impact vibrational excitation rates in the flow field of aeroassisted orbital transfer vehicles

Abstract: This paper examines the vibrational excitation rate processes expected in the flow field of aeroassisted orbital transfer vehicles (AOTVs). An analysis of the multiple-quantum vibrational excitation processes by electron impact is made to predict the vibrational excitation cross sections, rate coefficients, and relaxation times which control vibrational temperature. The expression for the rate of electron-vibration energy transfer is derived by solving the system of master equations which account for the multiple-level transitions. The vibrational excitation coefficients, which are the prerequisite physical quantities in solving the obtained vibrational equation, are calculated based on the theoretically predicted cross sections. These cross sections are obtained from quantum mechanical calculations, based on the concept that vibrational excitation of molecules by electron impact occurs through formation of an intermediate negative ion state. Finally, the modified Landau-Teller-type rate equation, which is suitable for the numerical calculations for the AOTV flow fields, is suggested.

Precompound decay calculations for reactions induced by 10--100 MeV/nucleon heavy ions

Abstract: The Boltzmann master equation model has been applied to the question of precompound nucleon deexcitation of reactions induced by 10--100 MeV/nucleon (c.m.) heavy ions. Test systems of /sup 16/O+ /sup 60/Ni and /sup 27/Al+ /sup 86/Kr were selected. Experimental neutron spectra in coincidence with evaporation residue and fission fragments from the /sup 20/Ne+ /sup 165/Ho system (due to Holub et al.) were reproduced quite well by the master equation with exciton numbers between 20 and 23. Exciton values of projectile mass and projectile mass plus 3 were therefore used in the extrapolations of the master equation. Results show major fractions of the excitation and up to 35 nucleons removed during the coalescence-equilibration period. The linear momentum transfer predicted by the master equation is shown to be in good agreement with a broad range of data. Calculations are provided as to the range of angular momenta which may be carried off by the precompound cascade.

Quantum mechanical theory of isotope effect on thermally activated hydrogen migration on W(110)

Abstract: A quantum mechanical model is introduced to explain the exponential increase with isotopic mass of the preexponential factor for the thermally activated diffusion rate for atomic hydrogen and its isotopes on the W(110) surface as recently observed by Gomer and co‐workers. The adsorbed hydrogen atom is taken to have self‐trapped quantum levels at low energies, with mobile ones above the barrier to hopping. The kinetics of transitions among these levels is analyzed using a master equation, and the individual transition rates are evaluated from a model of the coupling between the adsorbed hydrogen atom and its environment. The preexponential factor is found to vary exponentially with terms having a −1/2 and −1 power of the hydrogen mass (the latter is multiplied by a more slowly varying function of mass), in excellent agreement with the experiments on hydrogen, deuterium, and tritium.

Random dimer filling of lattices: Three-dimensional application to free radical recombination kinetics

Abstract: The recombination of nearest neighbors in a condensed matrix of free radicals was modeled by Jackson and Montroll as irreversible, sequential, random dimer filling of nearest-neighbor sites on an infinite, three-dimensional lattice. Here we analyze the master equations for random dimer filling recast as an infinite hierarchy of rate equations for subconfiguration probabilities using techniques involving truncation, formal density expansions (coupled with resummation), and spectral theory. A detailed analysis for the cubic lattice case produces, e.g., estimates for the fraction of isolated empty sites (i.e., free radicals) at saturation. We also consider the effect of a stochastically specified distribution of nonadsorptive sites (i.e., inert dilutents).

Global and local dissipation in a quantum map

Abstract: For a given 2-dimensional dissipative discrete map generating chaotic dynamics we present the phenomenological construction of a quantum mechanical master equation which reduces to the given map in the classical limit. Global dissipation, caused by the non-invertibility of the map, and local dissipation, caused by the local contraction of the map, are both incorporated in the description. The behavior in the two opposite limits of vanishing local dissipation and of strong local dissipation is analyzed exactly. Using the representation of the statistical operator by the Wigner distribution, the classical and semi-classical limit is studied. An estimate of the critical time is obtained, which determines the crossover between classical and quantum mechanical behavior in the chaotic state. This critical time diverges logarithmically for ħ→0.

Random walk on a fractal: Eigenvalue analysis

Abstract: The eigenvalues of the master equation describing the motion on a nested hierarchy ofd-dimensional intervals with selfsimilar scaling of spatial extension as well as of the level dependent transition rates are derived. Based on this spectrum the diffusion behaviour is obtained, which is anomalous, either exponential or obeying a power law with various exponents. Emphasis is put on the insight into the mechanism of the anomalous diffusion, in particular the geometrical structure of the decay rate spectrum.

Energy transfer in condensed media. II. Comparison of stochastic Liouville equations

Abstract: The linearized form of the transport theory recently developed by West and Lindenberg is used to obtain an approximate equation of motion for the density matrix of an exciton confined to a dimer. The resulting equation has a form which may be readily compared with others derived from both dynamical and phenomenological approaches. Comparisons are made between the equations of motion, optical spectra, and generalized master equation memory functions which result from the theories considered.

Master Equation Description of External Poisson White Noise in Finite Systems

Abstract: We consider finite systems with random control parameters. A theory for a unified description of internal fluctuations and external noise is presented. Internal fluctuations are modeled by a one-step Markovian master equation. External noise is introduced by random parameters in the master equation. It is modeled by a Poisson white noise. The unified description of fluctuations features a Markovian master equation with nonvanishing transition probabilities for all steps in the state space. Alternative formulations are given in terms of the generating function, Poisson representation and the equations for the factorial moments. An expansion around the thermodynamic limit is considered. The theory permits the calculation of finite-size effects. It predicts the existence of a coupling of the two types of fluctuations leading to “crossed-fluctuation” contributions. Two examples are considered: (i) a Poisson counting process with fluctuating parameter, (ii) a creation and annihilation process with source terms and fluctuations in each of the creation, annihilation, and source parameters. In the second example a complete analysis is given for the stationary distribution and associated moments for a finite system and also in the thermodynamic limit. The different role of the fluctuations of the three parameters is discussed. Explicit “crossed-fluctuations” contributions are found. The effect of the system size on the type of transitions induced by external noise in the thermodynamic limit is discussed.

Quantization of Hénon's map with dissipation

Abstract: Henon's map with dissipation is suspended to the nonlinearly kicked damped harmonic oscillator and then quantized. The ensuing master equation between two subsequent kicks is solved exactly in the representation by the Wigner distribution, resulting in a quantized version of Henon's dissipative map. The semi-classical limit of the map is studied. The leading quantum corrections are shown to be associated with dissipation and can be formulated as a classical map with classical stochastic perturbations. The next-to-leading quantum corrections, arising from the nonlinearity of the kicks, are similar as in the area conserving map and cannot be described within the framwork of classical statistics. The Wigner distribution in the steady state is investigated in the limit of strong dissipation, where Henon's map is reduced to the logistic map. The insensitivity of the main results against details of the quantization procedure is demonstrated by comparing with the results of a different phenomenological quantization procedure.

The systematic adiabatic elimination of fast variables from a many-dimensional Fokker-Planck equation

Abstract: The Chapman-Enskog method for the adiabatic elimination of fast variables is applied to a general Fokker-Planck equation linear in the fast variables. This equation is the counterpart of the generalized Haken-Zwanzig model, a system of coupled Langevin equations often encountered in quantum optics and in the theory of non-equilibrium phase transitions. After a few equilibration times for the fast variables the time dependence of smooth solutions of this Fokker-Planck equation is completely governed by the reduced distribution function of the slow variables, which obeys a closed evolution equation. We obtain an explicit perturbation series for the generator of this reduced evolution. The system considered here is the most general system for which the Chapman-Enskog hierarchy can be solved explicitly by exploiting an analogy between the unperturbed operator and the Hamiltonian for coupled harmonic oscillators.

Non-equilibrium scaling in the Schlogl model

Abstract: From the reaction-diffusion master equation formulation of the Schlogl model the authors develop a comprehensive description of the non-equilibrium dynamics. Using the Poisson transformation and the dynamical renormalisation group they firstly show how the equilibrium limit is recovered, and secondly describe the low-frequency, low-momenta scaling behaviour of the concentration fluctuations. The approach extends, simplifies or corrects previous studies and is easily generalised to treat more complicated systems.

Boson stochastic calculus

Abstract: This is a brief expositary account of Boson stochastic calculus which includes a description of quantum Ito’s formula and its application to the integration of a master equation in statistical mechanics.

Dynamics of trapped particle cooling in the Lamb–Dicke limit

Abstract: This paper reviews our current understanding of the dynamics of a laser-cooled trapped particle in the Lamb–Dicke regime, where the quantum structure of the energy levels cannot be ignored. The derivation and validity of a master equation are surveyed, and its physical interpretation is discussed in some detail. The structure and physical nature of the ultimate steady state are discussed. Using a generating function method, we can solve for both the complete eigenvalue spectrum and the general time-dependent solution of the master equation. These results are derived and interpreted physically. They have earlier been scattered in our various publications and are presented here in a coherent way for the first time. Also included are some new results and a physical discussion of the situation. The paper concludes with a discussion of the validity and limitations of the model as treated so far.

Separation of variables for the Rarita--Schwinger equation on all type D vacuum backgrounds

Abstract: We present a separable master equation governing Rarita–Schwinger spin‐ (3)/(2) fields, valid in the whole class of type D vacuum backgrounds.

Stochastic Analysis of a Hopf Bifurcation: Master Equation Approach

Abstract: The effect of inhomogeneous fluctuations in a reaction-diffusion system exhibiting a Hopf bifurcation is analyzed using the master equation approach. A Taylor expansion of the logarithm of the stationary probability, known as the stochastic potential, is calculated. This procedure displays marked analogies with the theory of normal forms. The critical potential, reduced to its local expansion around an arbitrary point of the limit cycle, brings out the essential role played by the phase of the oscillating variables. A comparison with the Langevin analysis of Walgraefet al. [J. Chem. Phys.78(6):3043 (1983)] is performed.

Photon statistics in the three-photon hyper-Raman process.

Abstract: An exact solution of the master equation for the three-photon hyper-Raman process is obtained by a matrix method. Several cases of the photon statistics are considered and the appearance of photon-antibunching effect and the violation of the classical Cauchy inequality are discussed. I. INTRODUCTION The photon statistics of various nonlinear optical processes has been a subject of increasing interest in recent years. In particular, photon statistics for the usual Raman process has been analyzed theoretically by Simaan in detail. ' The similar hyper-Raman (HR) process has also been studied already by Simaan, who formulated the master equation of the process giving the solution by the Laplace transform method. In this paper we consider again the HR process and give the solution of the master equation by a new matrix method similar to that used by Zubairy for multiphoton absorption processes. Our results complete the previous work by giving rather complete explicit numerical results in a number of cases of interest. In the interaction picture, the equation of motion for the density operator p of the coupled atoms-field system is given by the Liouville equation

A theory of kinematics for the Potts model

Abstract: A master equation formulation of the kinetic, q-state Potts model is presented. It is shown that this formulation reduces to Glauber's dynamics in the Ising limit. The single-spin problem is considered and the properties of its q-1, temperature dependent relaxation times are studied. The linear chain problem is much harder than in the Glauber case-a variational method is used to obtain a lower bound for the dynamical critical exponent (z>or=3 for q>2). A discussion of these and related problems is presented.

Exact macroscopic dynamics in non-equilibrium chemical systems

Abstract: Non-equilibrium reaction-diffusion models show an exciting array of critical phenomena ranging from the continuous transitions reminiscent of equilibrium systems to the spectacular chemical oscillators. Transforming the underlying stochastic master equations using 'Poisson' techniques the author obtains for the first time a comprehensive description of the dynamics, including in particular the problematical multitime correlation functions. The formalism is ideally adapted for the use of scaling and renormalisation group ideas.

Bloch equations and kinetic equations in multiphoton ionization

Abstract: In this paper we investigate the effect of laser phase fluctuations on the ionization dynamics of a two-level system. The conditions for validity of the kinetic master equation for the description of a resonant two-photon ionization process of an isolated molecule were established on the basis of analytical results and numerical simulations.