Showing papers on "Master equation published in 2001"

Nonequilibrium statistical mechanics

Abstract: 1. Brownian Motion and Langevin equations 2. Fokker-Planck equations 3. Master equations 4. Reaction rates 5. Kinetic models 6. Quantum dynamics 7. Linear response theory 8. Projection operators 9. Nonlinear problems 10. The paradoxes of irreversibility Appendices

Multiple‐Well, multiple‐path unimolecular reaction systems. I. MultiWell computer program suite

Abstract: Unimolecular reaction systems in which multiple isomers undergo simultaneous reactions via multiple decomposition reactions and multiple isomerization reactions are of fundamental interest in chemical kinetics. The computer program suite described here can be used to treat such coupled systems, including the effects of collisional energy transfer (weak collisions). The program suite consists of MultiWell, which solves the internal energy master equation for complex unimolecular reactions systems; DenSum, which calculates sums and densities of states by an exact-count method; MomInert, which calculates external principal moments of inertia and internal rotation reduced moments of inertia; and Thermo, which calculates equilibrium constants and other thermodynamics quantities. MultiWell utilizes a hybrid master equation approach, which performs like an energy-grained master equation at low energies and a continuum master equation in the vibrational quasicontinuum. An adap- tation of Gillespie's exact stochastic method is used for the solution. The codes are designed for ease of use. Details are presented of various methods for treating weak collisions with virtually any desired collision step-size distribution and for utilizing RRKM theory for specific unimolecular rate constants. 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 232- 245, 2001

Ballistic-Diffusive Heat-Conduction Equations

Abstract: We present new heat-conduction equations, named ballistic-diffusive equations, which are derived from the Boltzmann equation. We show that the new equations are a better approximation than the Fourier law and the Cattaneo equation for heat conduction at the scales when the device characteristic length, such as film thickness, is comparable to the heat-carrier mean free path and/or the characteristic time, such as laser-pulse width, is comparable to the heat-carrier relaxation time.

The Renormalization Group Equation for the Color Glass Condensate

Abstract: We present an explicit and simple form of the renormalization group equation which governs the quantum evolution of the effective theory for the Color Glass Condensate (CGC). This is a functional Fokker-Planck equation for the probability density of the color field which describes the CGC in the covariant gauge. It is equivalent to the Euclidean time evolution equation for a second quantized current-current Hamiltonian in two spatial dimensions. The quantum corrections are included in the leading log approximation, but the equation is fully non-linear with respect to the generally strong background field. In the weak field limit, it reduces to the BFKL equation, while in the general non-linear case it generates the evolution equations for Wilson-line operators previously derived by Balitsky and Kovchegov within perturbative QCD.

Mathematical Methods of Quantum Optics

Abstract: 1. Basic Quantum Mechanics.- 1.1 Postulates of Quantum Mechanics.- 1.1.1 Postulate 1.- 1.1.2 Postulate 2.- 1.1.3 Postulate 3.- 1.1.4 Postulate 4.- 1.1.5 Postulate 5.- 1.2 Geometric Phase.- 1.2.1 Geometric Phase of a Harmonic Oscillator.- 1.2.2 Geometric Phase of a Two-Level System.- 1.2.3 Geometric Phase in Adiabatic Evolution.- 1.3 Time-Dependent Approximation Method.- 1.4 Quantum Mechanics of a Composite System.- 1.5 Quantum Mechanics of a Subsystem and Density Operator.- 1.6 Systems of One and Two Spin-1/2s.- 1.7 Wave-Particle Duality.- 1.8 Measurement Postulate and Paradoxes of Quantum Theory.- 1.8.1 The Measurement Problem.- 1.8.2 Schrodinger's Cat Paradox.- 1.8.3 EPR Paradox.- 1.9 Local Hidden Variables Theory.- 2. Algebra of the Exponential Operator.- 2.1 Parametric Differentiation of the Exponential.- 2.2 Exponential of a Finite-Dimensional Operator.- 2.3 Lie Algebraic Similarity Transformations.- 2.3.1 Harmonic Oscillator Algebra.- 2.3.2 The SU(2) Algebra.- 2.3.3 The SU(1,1) Algebra.- 2.3.4 The SU(m) Algebra.- 2.3.5 The SU(m, n) Algebra.- 2.4 Disentangling an Exponential.- 2.4.1 The Harmonic Oscillator Algebra.- 2.4.2 The SU(2) Algebra.- 2.4.3 SU(1,1) Algebra.- 2.5 Time-Ordered Exponential Integral.- 2.5.1 Harmonic Oscillator Algebra.- 2.5.2 SU (2) Algebra.- 2.5.3 The SU(1,1) Algebra.- 3. Representations of Some Lie Algebras.- 3.1 Representation by Eigenvectors and Group Parameters.- 3.1.1 Bases Constituted by Eigenvectors.- 3.1.2 Bases Labeled by Group Parameters.- 3.2 Representations of Harmonic Oscillator Algebra.- 3.2.1 Orthonormal Bases.- 3.2.2 Minimum Uncertainty Coherent States.- 3.3 Representations of SU(2).- 3.3.1 Orthonormal Representation.- 3.3.2 Minimum Uncertainty Coherent States.- 3.4 Representations of SU(1, 1).- 3.4.1 Orthonormal Bases.- 3.4.2 Minimum Uncertainty Coherent States.- 4. Quasiprobabilities and Non-classical States.- 4.1 Phase Space Distribution Functions.- 4.2 Phase Space Representation of Spins.- 4.3 Quasiprobabilitiy Distributions for Eigenvalues of Spin Components.- 4.4 Classical and Non-classical States.- 4.4.1 Non-classical States of Electromagnetic Field.- 4.4.2 Non-classical States of Spin-1/2s.- 5. Theory of Stochastic Processes.- 5.1 Probability Distributions.- 5.2 Markov Processes.- 5.3 Detailed Balance.- 5.4 Liouville and Fokker-Planck Equations.- 5.4.1 Liouville Equation.- 5.4.2 The Fokker-Planck Equation.- 5.5 Stochastic Differential Equations.- 5.6 Linear Equations with Additive Noise.- 5.7 Linear Equations with Multiplicative Noise.- 5.7.1 Univariate Linear Multiplicative Stochastic Differential Equations.- 5.7.2 Multivariate Linear Multiplicative Stochastic Differential Equations.- 5.8 The Poisson Process.- 5.9 Stochastic Differential Equation Driven by Random Telegraph Noise.- 6. The Electromagnetic Field.- 6.1 Free Classical Field.- 6.2 Field Quantization.- 6.3 Statistical Properties of Classical Field.- 6.3.1 First-Order Correlation Function.- 6.3.2 Second-Order Correlation Function.- 6.3.3 Higher-Order Correlations.- 6.3.4 Stable and Chaotic Fields.- 6.4 Statistical Properties of Quantized Field.- 6.4.1 First-Order Correlation.- 6.4.2 Second-Order Correlation.- 6.4.3 Quantized Coherent and Thermal Fields.- 6.5 Homodvned Detection.- 6.6 Spectrum.- 7. Atom-Field Interaction Hamiltonians.- 7.1 Dipole Interaction.- 7.2 Rotating Wave and Resonance Approximations.- 7.3 Two-Level Atom.- 7.4 Three-Level Atom.- 7.5 Effective Two-Level Atom.- 7.6 Multi-channel Models.- 7.7 Parametric Processes.- 7.8 Cavity QED.- 7.9 Moving Atom.- 8. Quantum Theory of Damping.- 8.1 The Master Equation.- 8.2 Solving a Master Equation.- 8.3 Multi-Time Average of System Operators.- 8.4 Bath of Harmonic Oscillators.- 8.4.1 Thermal Reservoir.- 8.4.2 Squeezed Reservoir.- 8.4.3 Reservoir of the Electromagnetic Field.- 8.5 Master Equation for a Harmonic Oscillator.- 8.6 Master Equation for Two-Level Atoms.- 8.6.1 Two-Level Atom in a Monochromatic Field.- 8.6.2 Collisional Damping.- 8.7 aster Equation for a Three-Level Atom.- 8.8 Master Equation for Field Interacting with a Reservoir of Atoms.- 9. Linear and Nonlinear Response of a System in an External Field.- 9.1 Steady State of a System in an External Field.- 9.2 Optical Susceptibility.- 9.3 Rate of Absorption of Energy.- 9.4 Response in a Fluctuating Field.- 10. Solution of Linear Equations: Method of Eigenvector Expansion.- 10.1 Eigenvalues and Eigenvectors.- 10.2 Generalized Eigenvalues and Eigenvectors.- 10.3 Solution of Two-Term Difference-Differential Equation.- 10.4 Exactly Solvable Two- and Three-Term Recursion Relations.- 10.4.1 Two-Term Recursion Relations.- 10.4.2 Three-Term Recursion Relations.- 11. Two-Level and Three-Level Hamiltonian Systems.- 11.1 Exactly Solvable Two-Level Systems.- 11.1.1 Time-Independent Detuning and Coupling.- 11.1.2 On-Resonant Real Time-Dependent Coupling.- 11.1.3 Fluctuating Coupling.- 11.2 N Two-Level Atoms in a Quantized Field.- 11.3 Exactly Solvable Three-Level Systems.- 11.4 Effective Two-Level Approximation.- 12. Dissipative Atomic Systems.- 12.1 Two-Level Atom in a Quasimonochromatic Field.- 12.1.1 Time-Dependent Evolution Operator Reducible to SU(2).- 12.1.2 Time-Independent Evolution Operator.- 12.1.3 Nonlinear Response in a Bichromatic Field.- 12.2 N Two-Level Atoms in a Monochromatic Field.- 12.3 Two-Level Atoms in a Fluctuating Field.- 12.4 Driven Three-Level Atom.- 13. Dissipative Field Dynamics.- 13.1 Down-Conversion in a Damped Cavity.- 13.1.1 Averages and Variances of the Cavity Field Operators.- 13.1.2 Density Matrix.- 13.2 Field Interacting with a Two-Photon Reservoir.- 13.2.1 Two-Photon Absorption.- 13.2.2 Two-Photon Generation and Absorption.- 13.3 Reservoir in the Lambda Configuration.- 14. Dissipative Cavity QED.- 14.1 Two-Level Atoms in a Single-Mode Cavity.- 14.2 Strong Atom-Field Coupling.- 14.2.1 Single Two-Level Atom.- 14.3 Response to an External Field.- 14.3.1 Linear Response to a Monochromatic Field.- 14.3.2 Nonlinear Response to a Bichromatic Field.- 14.4 The Micromaser.- 14.4.1 Density Operator of the Field.- 14.4.2 Two-Level Atomic Micromaser.- 14.4.3 Atomic Statistics.- Appendices.- A. Some Mathematical Formulae.- B. Hypergeometric Equation.- C. Solution of Twoand Three-Dimensional Linear Equations.- D. Roots of a Polynomial.- References.

Master Equation for Hydrogen Recombination on Grain Surfaces

Abstract: Recent experimental results on the formation of molecular hydrogen on astrophysically relevant surfaces under conditions similar to those encountered in the interstellar medium provided useful quantitative information about these processes Rate equation analysis of experiments on olivine and amorphous carbon surfaces provided the activation energy barriers for the diffusion and desorption processes relevant to hydrogen recombination on these surfaces However, the suitability of rate equations for the simulation of hydrogen recombination on interstellar grains, where there might be very few atoms on a grain at any given time, has been questioned To resolve this problem, we introduce a master equation that takes into account both the discrete nature of the H atoms and the fluctuations in the number of atoms on a grain The hydrogen recombination rate on microscopic grains, as a function of grain size and temperature, is then calculated using the master equation The results are compared to those obtained from the rate equations, and the conditions under which the master equation is required are identified

Feedback-stabilization of an arbitrary pure state of a two-level atom

Abstract: Unit-efficiency homodyne detection of the resonance fluorescence of a two-level atom collapses the quantum state of the atom to a stochastically moving point on the Bloch sphere. Recently, Hofmann, Mahler, and Hess [Phys. Rev. A 57, 4877 (1998)] showed that by making part of the coherent driving proportional to the homodyne photocurrent one can stabilize the state to any point on the bottom-half of the sphere. Here we reanalyze their proposal using the technique of stochastic master equations, allowing their results to be generalized in two ways. First, we show that any point on the upper- or lower-half, but not the equator, of the sphere may be stabilized. Second, we consider nonunit-efficiency detection, and quantify the effectiveness of the feedback by calculating the maximal purity obtainable in any particular direction in Bloch space.

Continuous quantum measurement of two coupled quantum dots using a point contact: A quantum trajectory approach

Abstract: We obtain the finite-temperature unconditional master equation of the density matrix for two coupled quantum dots (CQD's) when one dot is subjected to a measurement of its electron occupation number using a point contact (PC). To determine how the CQD system state depends on the actual current through the PC device, we use the so-called quantum trajectory method to derive the zero-temperature conditional master equation. We first treat the electron tunneling through the PC barrier as a classical stochastic point process (a quantum-jump model). Then we show explicitly that our results can be extended to the quantum-diffusive limit when the average electron tunneling rate is very large compared to the extra change of the tunneling rate due to the presence of the electron in the dot closer to the PC. We find that in both quantum-jump and quantum-diffusive cases, the conditional dynamics of the CQD system can be described by the stochastic Schrodinger equations for its conditioned state vector if and only if the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements.

Generalized continuity equation and modified normalization in PT-symmetric quantum mechanics

Abstract: The continuity, equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

Theory of pseudomodes in quantum optical processes

Abstract: This paper deals with non-Markovian behavior in atomic systems coupled to a structured reservoir of quantum electromagnetic field modes, with particular relevance to atoms interacting with the field in high-Q cavities or photonic band-gap materials. In cases such as the former, we show that the pseudomode theory for single-quantum reservoir excitations can be obtained by applying the Fano diagonalization method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two, and many discrete quasimodes are made. For a simple photonic band-gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.

Environment-Induced Decoherence and the Transition from Quantum to Classical

Abstract: We study dynamics of quantum open systems, paying special attention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection einselection in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equations we study einselection employing the general strategy of the predictability sieve. Assumptions that are usually made in the discussion of decoherence are critically reexamined along with the ``standard lore'' to which they lead. Restoration of quantum-classical correspondence in systems that are classically chaotic is discussed. The dynamical second law -it is shown- can be traced to the same phenomena that allow for the restoration of the correspondence principle in decohering chaotic systems (where it is otherwise lost on a very short time-scale). Quantum error correction is discussed as an example of an anti-decoherence strategy. Implications of decoherence and einselection for the interpretation of quantum theory are briefly pointed out.

Exact solution of the Hu-Paz-Zhang master equation

Abstract: The Hu-Paz-Zhang equation is a master equation for an oscillator coupled to a linear passive bath. It is exact within the assumption that the oscillator and bath are initially uncoupled. Here an exact general solution is obtained in the form of an expression for the Wigner function at time t in terms of the initial Wigner function. The result is applied to the motion of a Gaussian wave packet and to that of a pair of such wave packets. A serious divergence arising from the assumption of an initially uncoupled state is found to be due to the zero-point oscillations of the bath and not removed in a cutoff model. As a consequence, worthwhile results for the equation can only be obtained in the high temperature limit, where zero-point oscillations are neglected. In that limit closed form expressions for wave packet spreading and attenuation of coherence are obtained. These results agree within a numerical factor with those appearing in the literature, which apply for the case of a particle at zero temperature that is suddenly coupled to a bath at high temperature. On the other hand very different results are obtained for the physically consistent case in which the initial particle temperature is arranged to coincide with that of the bath.

Molecular geometry fluctuations and field-dependent mobility in conjugated polymers

Abstract: Many conjugated polymers exhibit an electric field-dependent mobility of approximately the Poole-Frenkel form. We propose a model to describe transport in dense films of these materials in which thermal fluctuations in the molecular geometry modify the energy levels of localized electronic charged states in the material. Based on quantum chemistry calculations we argue that the primary restoring force for these fluctuations in molecular geometry is steric in origin, which leads to spatially correlated fluctuations in the on-site energy of the charged electronic states. The phenylene ring torsion, in PPV-like conjugated polymers, is an example of this kind of spatially correlated thermal fluctuation. Using a Master equation approach to calculate the mobility, we show that the model can quantitatively explain the experimentally observed field-dependent mobility in conjugated polymers. We examine typical paths taken by carriers and find that at low fields, the paths are three-dimensional, whereas at high fields the paths become essentially one-dimensional along the applied field. Thus, one-dimensional transport models can be valid at high fields but not at low fields. Effects of deep traps, the site energy correlation length, temperature, and asymmetric and small polaron rates are studied.

MassKinetics: a theoretical model of mass spectra incorporating physical processes, reaction kinetics and mathematical descriptions

Abstract: A theoretical framework and an accompanying computer program (MassKinetics, www.chemres.hu/ms/ masskinetics) is developed for describing reaction kinetics under statistical, but non-equilibrium, conditions, i.e. those applying to mass spectrometry. In this model all the important physical processes influencing product distributions are considered: reactions, including the effects of acceleration, collisions and photon exchange. These processes occur simultaneously and are taken into account by the master equation approach. The system is described by (independent) product, kinetic energy and internal energy distributions, and the time development of these distributions is studied using transition probability functions. The product distribution at the end of the experiment corresponds to the mass spectrum. Individual elements in this scheme are mostly well known: internal energy-dependent reaction rates are calculated by transition state theory (RRK or RRKM formalisms). In the course of collisions, energy transfer and other processes may occur (the latter usually resulting in the 'loss' of ion signal). Collisions are characterized by their probability and by energy transfer in a single collision. To describe single collisions, three collision models are used: long-lived collision complexes, partially inelastic collisions and partially inelastic collisions with cooling. The latter type has been developed here, and is capable of accounting for cooling effects occurring in collision cascades. Descriptions of photon absorption and emission are well known in principle, and these are also taken into account, in addition to changes in kinetic energy due to external (electric) fields. These changes in the system occur simultaneously, and are described by master equations (a set of differential equations). The usual form of the master equation (taking into account reactions and collisional excitation) was extended to consider also radiative energy transfer, kinetic energy changes, energy partitioning and ion loss collisions. Initial results show that close to experimental accuracy can be obtained with MassKinetics, using few or no adjustable parameters. The model/program can be used to model almost all types of mass spectrometric experiments (e.g. MIKE, CID, SORI and resonant excitation). Note that it was designed for mass spectrometric applications, but can also be used to study reaction kinetics in other non-equilibrium systems.

From completely positive maps to the quantum Markovian semigroup master equation

Abstract: A central problem in the theory of the dynamics of open quantum systems is the derivation of a rigorous and computationally tractable master equation for the reduced system density matrix. Most generally, the evolution of an open quantum system is described by a completely positive linear map. We show how to derive a completely positive Markovian master equation (the Lindblad equation) from such a map by a coarse-graining procedure. We provide a novel and explicit recipe for calculating the coefficients of the master equation, using perturbation theory in the weak-coupling limit. The only parameter external to our theory is the coarse-graining time-scale. We illustrate the method by explicitly deriving the master equation for the spin-boson model. The results are evaluated for the exactly solvable case of pure dephasing, and an excellent agreement is found within the time-scale where the Markovian approximation is expected to be valid. The method can be extended in principle to include non-Markovian effects.

Decoherence, pointer engineering, and quantum state protection.

Abstract: We show that motional states of trapped ions can be protected against decoherence by generating, through transitions involving unstable electronic states, artificial reservoirs for the vibrational motion, which have the states to be protected as their pointer states. We exemplify our procedure with qubits, squeezed states, approximate phase eigenstates and superpositions of coherent states. We show that these states may be efficiently protected against decoherence from thermal reservoirs and random fields.

A stochastic approach to grain surface chemical kinetics

Abstract: A stochastic model of grain surface chemistry, based on a master equation description of the probability distributions of reactive species on grains, is developed. For an important range of conditions, rates of molecule formation are limited by low accretion rates, so that the probability that a grain contains more than one reactive atom or molecule is small. We derive simple approximate expressions for these circumstances, and explore their validity through comparison with numerical solutions of the master equation for H, O and H, N, O reaction systems. A more detailed analysis of the range of validity of several analytic approximations and numerical solutions, based on exact analytical results for a model in which H and H2 are the only species, is also made. Though the use of our simple approximate expressions is computationally ecient, the solution of the master equation under the assumption that no grain contains more than two particles of each species usually gives more accurate results in the parameter regimes where the deterministic rate equation approach is inappropriate. The implementation of sparse matrix inversion techniques makes the use of such a truncated master equation solution method feasible for considerably more complicated surface chemistries than the ones we have examined here.

Hazards of reservoir memory

Abstract: We add memory effects to the master equation describing a harmonic oscillator embedded in a reservoir. Solving the time evolution exactly, we show that the model is sensible only in the Markovian limit. Thus we issue a warning against indiscriminate introduction of memory effects in master equations and call for a systematic method to obtain corrections to Markovian time evolution.

Complete parameterization, and invariance, of diffusive quantum trajectories for Markovian open systems

Abstract: The state matrix ρ for an open quantum system with Markovian evolution obeys a master equation. The master equation evolution can be unraveled into stochastic nonlinear trajectories for a pure state P , such that on average P reproduces ρ . Here we give for the first time a complete parameterization of all diffusive unravelings (in which P evolves continuously but non-differentiably in time). We give an explicit measurement theory interpretation for these quantum trajectories, in terms of monitoring the system's environment. We also introduce new classes of diffusive unravelings that are invariant under the linear operator transformations under which the master equation is invariant. We illustrate these invariant unravelings by numerical simulations. Finally, we discuss generalized gauge transformations as a method of connecting apparently disparate descriptions of the same trajectories by stochastic Schrodinger equations, and their invariance properties.

Microscopic Quantum Hydrodynamics of Systems of Fermions: Part I

Abstract: The fundamental equations of the microscopic quantum hydrodynamics of fermions in an external electromagnetic field (i.e., the particle balance equation, the momentum balance equation, the energy balance equation, and the magnetic moment balance equation) are derived using the Schrodinger equation. The form of the spin–spin interaction Hamiltonian is specified. To close the system of the balance equations for a multiparticle fermion system, the effective one-particle Schrodinger equation must be introduced.

Self-Consistent Model of Chemical, Vibrational, Electron Kinetics in Nozzle Expansion

Abstract: A self-consistent model to describe vibrational, electronically excited states (master equations) and free electron kinetics (Boltzmann equation) has been applied to study N 2 expansion through a converging-diverging conic nozzle. Strong departures from equilibrium can be observed for both vibrational, electronically excited states and electron energy distributions. In particular, the role of electronically excited states of nitrogen molecules and free electrons has been investigated. The strong interaction between these two systems, by means of inelastic and superelastic collisions, influences not only the internal state kinetics, but also the macroscopic quantities such as Mach number and gas temperature profile

Vibration-Dissociation Coupling Using Master Equations in Nonequilibrium Hypersonic Blunt-Body Flow

Abstract: Numerical simulations are presented of a steady-state hypersonic flow past a hemisphere cylinder. Two types of models, one a lumped Landau-Teller vibrational relaxation model (Landau, L., and Teller, E.) and the other a discrete state kinetic relaxation model (DSKR), were used to study effects of vibration-dissociation coupling on the flow physics. The widely used Park's dissociation model was used as baseline for coupling vibration and dissociation processes (Park, C.). For a Mach 8.6 flow, both relaxation models matched experimental data. At Mach 11.18, however, the underprediction of shock-standoff distance by both relaxation models using Park's model for dissociation coupling provided the motivation to implement a new master equation-based (DSKR) depletion model. The new model was used to study the effect of dissociation on population depletion in the vibrational states of the nitrogen molecule. The new model helps explain the restricted success of Park's dissociation model in certain temperature ranges of hypersonic flow past a blunt body. In the range of 5000-15,000 K, the new model yielded a substantial rate reduction relative to Park's equilibrium rate at lower temperatures and a consistent value at the high end

Fractional master equation: non-standard analysis and Liouville–Riemann derivative

Abstract: Fractional master equations may be defined either by means of Liouville–Riemann (L–R) fractional derivative or via non-standard analysis. The first approach describes processes with long-range dependence whilst the second approach deals with processes involving independent increments. The present papers put in evidence some of the differences between these two modellings, and to this end it especially considers more fractional Poisson processes.

Output spectrum of a detector measuring quantum oscillations

Abstract: We consider a two-level quantum system (qubit) which is continuously measured by a detector, and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the frequency of quantum (Rabi) oscillations and a Lorentzian-shape increase of the detector noise at low frequency. As the coupling increases, the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed.

Determination of the Spectral Gap for Kac's Master Equation and Related Stochastic Evolutions

Abstract: We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model due to Mark Kac of energy conserving collisions with one dimensional velocities. It is also sufficiently robust to provide qualitatively sharp bounds also in the case of more physically realistic momentum and energy conserving collisions in three dimensions, as well as a range of related models.

Dynamics of thermal Bose fields in the classical limit

Abstract: We develop an approximate formalism suitable for performing simulations of the thermal dynamics of interacting Bose gases. The method is based on the observation that when the lowest-energy modes of the Bose field operator are highly occupied, they may be treated classically to a good approximation. We derive a finite-temperature Gross-Pitaevskii equation (FTGPE) for these modes which is coupled to an effective reservoir described by quantum kinetic theory. We discuss each of the terms that arise in this GPE, and their relevance to experimental systems. We then describe a simpler projected GPE that may be useful in simulating thermal Bose condensates. This classical method could be applied to other Bose fields.

Hydrodynamic equations for mixed quantum states. I. General formulation

Abstract: Quantum-mechanical hydrodynamic equations are considered for mixed quantum states, and the corresponding equations for pure quantum states are derived as a particular case. A generalization of the “quantum potential” of Bohmian mechanics is formulated. In the mixed-state case, an infinite hierarchy of kinetic equations arises that may be truncated by introducing suitable approximations. The influence of dissipation on the kinetic equations is discussed.

The influence of atomic and molecular metastable states in high-enthalpy nozzle expansion nitrogen flows

Abstract: The role of atomic and molecular electronically excited states on the whole kinetics of an high-enthalpy nozzle flow has been examined by using a self-consistent model which couples Euler equations with appropriate master equations and with the Boltzmann equation for the electron energy distribution function (eedf). The results show that in high-enthalpy flows metastable atomic nitrogen can form structures in the eedf through superelastic collisions, partially smoothed by electron-electron Coulomb collisions.

Hydrodynamic equations for mixed quantum states. II. Coupled electronic states

Abstract: A hydrodynamic approach is developed to describe nonadiabatic nuclear dynamics. We derive a hierarchy of hydrodynamic equations which are equivalent to the exact quantum Liouville equation for coupled electronic states. It is shown how the interplay between electronic populations and coherences translates into the coupled dynamics of the corresponding hydrodynamic fields. For the particular case of pure quantum states, the hydrodynamic hierarchy terminates such that the dynamics may be described in terms of the local densities and momentum fields associated with each of the electronic states.

Branching Ratio and Pressure Dependent Rate Constants of Multichannel Unimolecular Decomposition of Gas-Phase α-HMX: An Ab Initio Dynamics Study†

Abstract: The dynamics of the initial thermal decomposition step of gas-phase α-HMX is investigated using the master equation method. Both the NO2 fission and HONO elimination channels were considered. The structures, energies, and Hessian information along the minimum energy paths (MEP) of these two channels were calculated at the B3LYP/cc-pVDZ level of theory. Thermal rate constants at the high-pressure limit were calculated using the canonical variational transition state theory (CVT), microcanonical variational transition state theory (μVT). The pressure-dependent multichannel rate constants and the branching ratio were calculated using the master equation method. Quantum tunneling effects in the HONO elimination are included in the dynamical calculations and found to be important at low temperatures. At the high-pressure limit, the NO2 fission channel is found to be dominant in the temperature range (500−1500 K). Both channels exhibit strong pressure dependence at high temperatures. Both reach the high-pressur...