Chapman-Enskog development of the multivariate master equation

A systematic development for a Multivariate Master Equation (MME), describing reaction diffusion systems, is presented along the same lines as the Chapman-Enskog development of kinetic gas theory. Diffusion, which occurs at a very fast rate, brings the system near a state of diffusional equilibrium, the analogue of local equilibrium for gases. In diffusional equilibrium, the global Master Equation (ME) is shown to be an exact consequence of the MME. For finite systems, corrections to the global ME, resulting from the finiteness of the diffusion times, are calculated. The development is verified on an exactly solvable model and illustrated on the Schlögl model. The difficulties encountered in the thermodynamic limit are discussed, and possible outcomes suggested.     

Coherent spontaneous emission from large systems

In this paper we discuss the conditions under which coherent spontaneous emission can take place in an arbitrary sample of linear dimensions smaller than a cooperation length. These conditions guarantee that the atoms—field interaction can be treated up to the second order in the Generalized Master Equation for the atoms. An equation describing coherent spontaneous emission is derived and discussed. A comparison is made between this and other known theories.     

Mean-field theory of critical behaviour and first order transition: A comparison between the Master Equation and the nonlinear Fokker-Planck equation

The results of the nonlinear Fokker-Planck equation formalism, which was recently derived as an asymptotic representation of the Master Equation for large system size, are compared with the exact solution of the Master Equation for the Schlögl model. As far as critical behaviour is concerned complete agreement is found. Furthermore at the first order transition points the nonlinear Fokker-Planck equation is shown to constitute an excellent approximation.     

The Argyres-Kelley approach to the master equation and an extended theory of superradiance

A discussion of the Argyres-Kelley approach to the Master Equation is set up, and applied to the Bonifacio-Schwendimann-Haake model of Superradiance. Some properties of the approach and its limits are discussed, and an extension of the BSH theory of Superradiance is obtained.     

Stability of continuous-time quantum filters with measurement imperfections

The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix.     

Die Nicht-Markoffsche Mastergleichung als stochastische Identität

A derivation is given of the Generalized Master Equation which rests entirely on probabilistic reasoning. The probabilistic approach reveals the simplicity and general validity of this Non-Markovian equation which is in fact an identity for the probability vector of any stochastic process with stationary transition mechanism.     

Thermoelectric current and Coulomb-blockade plateaus in a quantum dot

A Generalized Master Equation (GME) is used to study the thermoelectric currents through a quantum dot in both the transient and steady-state regime. The two semi-infinite leads are kept at the same chemical potential but at different temperatures to produce a thermoelectric current which has a varying sign depending on the chemical potential. The Coulomb interaction between the electrons in the sample is included via the exact diagonalization method. We observe a saw-teeth like profile of the current alternating with plateaus of almost zero current. Our calculations go beyond the linear response with respect to the temperature gradient, but are compatible with known results for the thermopower in the linear response regime.     

On the validity of the Fokker-Planck equation for the Heisenberg magnet

We assume an explicit stochastic dynamics for the microscopic variables in the Heisenberg magnet based on experimental data. We obtain additional terms to the Fokker-Planck Equation and also explicit expressions for the numerical coefficients in the Master Equation, near T_c.     

Anomalous isotropic luminescence from anthracene sandwich dimers

A classical statistical probability amplitude is introduced whose square modulus is the distribution function. This enables the analogy between classical statistical mechanics and quantum mechanics to be completed. The analogy is developed until quantum statistical derivations can be used in classical statistical mechanics. Two master equations are found: the classical equivalent of the Pauli Master Equation, and a generally valid master equation. Well-known classical equations are deduced from these in a special representation. Interference terms are found and discussed.     

The factorization approximation for the Master Equation

We examine the validity of the application of the Factorization Approximation to derive the Master Equation for a microscopic system coupled to a reservoir. We developed a formal perturbation expansion for the time evolution of the system reduced density matrix. We employed a diagrammatic schemes to produce each term of the perturbation series. The diagrams in the time domain provide a distinct criteria to distinguish the diagrams which survive the Factorization Approximation. The Feynmann-like diagrams in the energy domain, originated from the Resolvent method, are used for execution of diagram summations to estimate their overall contributions. We demonstrated that for a two level atomic system, interacting with a thermal reservoir, the summation over the diagrams which survived the Factorization Approximation, yields the proper time evolution of the system, in agreement with the solution of the Master Equation. The summation of the diagrams which are excluded by applying the Factorization Approximation are characterized by a dimensionless parameter: Γ/ω₀, where ω₀ is the frequency of the transition line, and Γ is the line width. The Factorization Approximation is thus rigorously justified when this expansion parameter is very small.     

Influence of an external field on the stochastic theory of polymerization

A stochastic model is proposed to take into account the effect of a weak external field associated to diffusion, a discrete Master Equation is written down for the spatial distribution of particles. Approximate solutions are found for the first two cumulants, and applied to an example of polymerization reactions.     

Conductivity of a generalized hopping system with internal dynamics

We derive explicit exact expressions for the frequency dependent conductivity of a system whose dynamics is described by a Master Equation in which the transition rates _ab can be functions of other dynamical variables. We apply this method to study the hopping of particles in a lattice with the hopping rate modulated by an additional harmonic variable. The results of this model are related to the observed microwave conductivity of superionic conductors. In particular we show that the modulation of the hopping rate can produce a structure similar to the one observed in AgI type systems.     

Analysis of noise-induced bimodality in a Michaelis–Menten single-step enzymatic cycle

In this paper we study noise-induced bimodality in a specific circuit with many biological implications, namely a single-step enzymatic cycle described by Michaelis–Menten equations. We study the biological feasibility of this phenomenon, which allows for switch-like behavior in response to graded stimuli, considering a small and discrete number of molecules involved in the circuit, and we characterize the conditions necessary for it. We show that intrinsic noise (due to the stochastic character of the Master Equation approach) of a one-dimensional substrate reaction is not sufficient to achieve bimodality, then we characterize analytically the necessary conditions on enzyme number fluctuations. We implement numerically two model circuits that show bimodality over different parameter windows, that depend critically on system size as predicted by our results, providing hints about how such a phenomenon could be exploited in real biological systems.     

Static and time dependent fields at nonmagnetic impurities in ferromagnetic samples

Several techniques are used for the measurement of hf-interactions. One of them, the PAC-method, gives the value of the effective field acting on a decaying nucleus, assuming a static force. Using the Master Equation, this paper shows if that there exists in addition a strong fluctuating field, the angular correlation is wiped out.     

MASTER EQUATION FOR QUANTUM SYSTEMS

The Master equation for quantum systems with strong interaction is derived, and the condition for its validity is discussed.     

A single pulse shock tube study of pentene isomer pyrolysis

A single-pulse shock tube study of the four pentene isomers is carried out at 2 ± 0.16 bar and 900–1600 K. C₁ to C₆ species profiles were recorded using gas chromatography mass spectrometry analyses. The species are identified using mass spectrometry and quantified by flame ionization detection. High-pressure limiting and pressure-dependent rate constants for 2M1B, 2M2B and 3M1B + ? were calculated using RRKM theory with a Master Equation (ME) analysis using the Master Equation System Solver, MESS. A mechanism was formulated based on rate rules and theoretical calculations. Comparisons between experimental results and model simulations are provided for all of the five pentene isomers investigated with satisfactory agreement. Furthermore, an insight is provided into the influence of molecular structure on the reactivity of pyrolysis chemistry. Interestingly, it is found that the HACA mechanism is much less prominent for benzene formation compared to the role of cyclopentadienyl radical recombination with methyl radicals and also the recombination of propargyl radicals.     

Energy basis via decoherence

The question of the emergence of a preferred basis which is generally understood as that basis in which the reduced density matrix is driven to a diagonal (classically interpretable) form via environment induced decoherence is addressed. The exact solutions of the Caldeira-Leggett Master Equation are analyzed for a free particle and a harmonic oscillator system. In both cases, we see that the reduced density matrix is driven diagonal in the energy basis, which is momentum for the free particle and the number states for the harmonic oscillator. This seems to single out the energy basis as the preferred basis which is contrary to the general notion that it is the position basis which is selected since the coupling to the environment is via the position coordinates     

From the Von-Neumann Equation to the Quantum Boltzmann Equation in a Deterministic Framework

In this paper, we investigate the rigorous convergence of the Density Matrix Equation (or Quantum Liouville Equation) towards the Quantum Boltzmann Equation (or Pauli Master Equation). We start from the Density Matrix Equation posed on a cubic box of size L with periodic boundary conditions, describing the quantum motion of a particle in the box subject to an external potential V. The physics motivates the introduction of a damping term acting on the off-diagonal part of the density matrix, with a characteristic damping time ^–1. Then, the convergence can be proved by letting successively L tend to infinity and to zero. The proof relies heavily on a lemma which allows to control some oscillatory integrals posed in large dimensional spaces. The present paper improves a previous announcement [CD].     

The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory

In the framework of perturbative quantum field theory a new, universal renormalization condition (called Master Ward Identity) was recently proposed by one of us (M.D.) in a joint paper with F.-M. Boas. The main aim of the present paper is to get a better understanding of the Master Ward Identity by analyzing its meaning in classical field theory. It turns out that it is the most general identity for classical local fields which follows from the field equations. It is equivalent to a generalization of the Schwinger-Dyson Equation and is closely related to the Quantum Action Principle of Lowenstein and Lam. As a byproduct we give a self-contained treatment of Peierls manifestly covariant definition of the Poisson bracket. Work supported by the Deutsche Forschungsgemeinschaft.