Master Equation Formulation Research Articles

Master equation analysis of intermolecular energy transfer in multiple-well, multiple-channel unimolecular reactions. I. Basic theory

We present a full theoretical analysis of the master-equation formulation of the problem of intermolecular energy transfer in multiple-well, multiple-channel systems. It is shown that the master equation for chemical or thermal activation possesses a unique steady state, that corresponding to the trivial solution. Rate equations local in time and therefore time-independent rate coefficients for the dissociating processes may be obtained only if a state of secular equilibrium exists. For chemically-activated systems, a general state of secular equilibrium may exist which may contain within it a regime wherein there is a well-separated, nontrivial, least negative eigenvalue of the master equation kernel. The dynamics of thermally activated systems are similarly deduced by treating them as chemically activated systems with appropriate modifications to the inhomogeneous source term of the master equation. A degenerate and nondegenerate perturbation theory analysis of the case of rapid thermalization in the vicinity of the thermodynamic equilibrium state is also enunciated. The special case of negligible thermalization is analyzed. A classification of the ordering of the time scales of thermalization, isomerization, and dissociation is then given.

Application of inverse iteration to 2-dimensional master equations

Recent developments in unimolecular theory have placed great emphasis on the role played by angular momentum in determining the details of the dependence of the rate coefficient on pressure and temperature. The natural way to investigate these dependencies is through the master equation formulation, where the rate coefficient is recovered as the eigenvalue of the smallest magnitude of the spatial operator. Except for very simple cases, the master equation must be solved with numerical methods. For the 2-dimensional master equation this leads to large sparse matrices and correspondingly lengthy computational times in order to determine the eigenvalue of the least magnitude. A reformulation of the problem in terms of a diffusion equation approximates the final matrix with a narrow banded matrix that can easily be factored using a variation of Gaussian elimination. The 2-dimensional master equation can then be solved with inverse iteration, which rapidly converges to the desired eigenpair. This method can be up to 10 times faster than conventional iterative algorithms for finding the desired eigenpair. © 1997 John Wiley & Sons, Inc. J Comput Chem 18:1004–1010, 1997

Application of inverse iteration to 2‐dimensional master equations

Recent developments in unimolecular theory have placed great emphasis on the role played by angular momentum in determining the details of the dependence of the rate coefficient on pressure and temperature. The natural way to investigate these dependencies is through the master equation formulation, where the rate coefficient is recovered as the eigenvalue of the smallest magnitude of the spatial operator. Except for very simple cases, the master equation must be solved with numerical methods. For the 2-dimensional master equation this leads to large sparse matrices and correspondingly lengthy computational times in order to determine the eigenvalue of the least magnitude. A reformulation of the problem in terms of a diffusion equation approximates the final matrix with a narrow banded matrix that can easily be factored using a variation of Gaussian elimination. The 2-dimensional master equation can then be solved with inverse iteration, which rapidly converges to the desired eigenpair. This method can be up to 10 times faster than conventional iterative algorithms for finding the desired eigenpair. © 1997 John Wiley & Sons, Inc. J Comput Chem 18:1004–1010, 1997

Microscopic simulation of chemical bistability in homogeneous systems

Microscopic simulation is used to clarify the status of stochastic theories of homogeneous chemical systems operating in the multiple steady state region. The results demonstrate the failure of the Langevin approach, but show excellent agreement with the master equation formulation.

Collisional Deactivation of CDCl3 Excited with a TEA CO2 Laser

The decay of infrared fluorescence from IR multiphoton excited CDCl3 was studied as a function of incident fluence and pressure of added Ar. The decays were exponential and allowed for the calculation of the average energy transferred per collision, 〈〈ΔE〉〉, with Ar and with CDCl3, as a function of the average internal energy of excited CDCl3 using a direct method. Identical information was obtained following the inversion procedure developed by Barker et al. (Int. Rev. Phys. Chem. 1993, 12, 2, 305). Both sets of values agree well for a single value of the internal energy. The experimental results were also modeled using a master equation formulation to obtain 〈ΔE〉d, confirming the predictive power of the model calculations. The increase of temperature of the gas mixture and its effect on the IR fluorescence signal were also analyzed.

Stochastic dynamics of reinforcement learning

We present a continuous-time master-equation formulation of reinforcement learning. Both non-associative (stochastic learning automaton) and associative (neural network) cases are considered. A Fokker–Planck equation for the stochastic dynamics of the learning process is derived using a small-fluctuation expansion of the master equation. We then show how the Fokker–Planck approximation can be used to determine the global asymptotic behaviour of ergodic learning schemes such as linear reward–penalty (LR−P) and associative reward–penalty (LR−P), in the limit of small learning rates. A simple example of reinforcement learning in a non-stationary environment is studied.

Thermostochastics: Heat conduction and temperature fluctuations

Abstract A stochastic approach to the energy balance equation of nonequilibrium thermodynamics is suggested. The stochastic formulation interprets the temperature field as a multivariate stochastic process which is governed by a master equation. By means of a systematic expansion in powers of the inverse number of degrees of freedom per volume element it is shown that the expectation value of the stochastic process obeys the macroscopic Fourier equation. The expansion reveals that in the linear noise approximation the fluctuations superimposed on the macroscopic dynamics are governed by the equations of fluctuating hydrodynamics. The master equation formulation gives rise to a new stochastic simulation method which is illustrated by applying it to heat conduction and temperature fluctuations in a fluid between two infinite parallel planes which are kept at different temperatures.

On the Numerical Accuracy of the Fokker-Planck Approximation to the Hierarchy of Master Equations

Many physical processes are described by birth and death phenomena and can generally be treated with either a master equation or Fokker-Planck formulation. We present a numerical study for one such process which describes the agglomeration of atomic clusters on clean surfaces during the early stages of thin film formation. We develop moment equations which describe the evolving size distribution of these clusters during an atom deposition process. These moments are derived from both the hierarchical master equations and the equivalent Fokker-Planck approximation. We limit our treatment to growth and decay mechanisms which occur via single-atom transitions. It is shown that the Fokker-Planck approximation is surprisingly good for sizes down to a few atoms. Analysis of the numerical accuracy of the Fokker-Planck approximation is given by comparing results for the total density of clusters, their average size, and the first four moments of the size distribution function. Results are presented for growth laws associated with two- and three-dimensional cluster shapes.

Kinetic paths in two order parameters: Theory

It is shown that when a crystalline alloy is characterized by two order parameters, the alloy may develop various combinations of the two order parameters en route to a final equilibrium state. The kinetic evolution of the two order parameters in a ternary alloy with B2 order is analyzed with an activated state rate theory. Elementary vacancy jump processes are considered in a Master Equation formulation. A two dimensional transport equation describing the evolution of the alloy through the two order parameters is developed, and the chain of states in this two dimensional order parameter space is found by numerical solution of the transport equation. It is shown that either different interatomic interactions or different activation barrier heights for the diffusive jumps of the different atoms will cause the alloy to evolve through different nonequilibrium states, or along different “kinetic paths”.

Fluence and wavelength dependence of the IRMPA and IRMPD of CDCl3 with a CO2 laser

A TEA CO2 laser was used to study the infrared multiple-photon absorption (IRMPA) and dissociation (IRMPD) spectra of CDCl3 in the fluence ranges 0.01–1.4 and 7–45 J/cm2, respectively, for different sample pressures. Experimental results were modeled with a master equation formulation which includes rotational and anharmonic bottlenecks and collisional effects. Experimental and calculated results show that CDCl3 has great rotational and anharmonic restrictions at the first stages of excitation. The IRMPD spectrum falls more slowly than the linear absorption spectrum at the blue wing due to intramolecular vibrational relaxation at the quasi-continuum level of excitation.

Thermal fluctuations in a Knudsen flow system

Abstract Spatial correlations of thermal fluctuations in a model system are examined. The system is a one-dimensional chain of cells containing a dilute gas and connected by Knudsen apertures. A Monte Carlo simulation is described in which the long range correlations observed agree quantitatively with a general master equation formulation.

Non-equilibrium scaling in the Schlogl model

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.

A theory of kinematics for the Potts model

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.

Comment: Strict Lindemann behavior as a limiting form

The master equation formulation for a thermal unimolecular reaction is extended to a global form. Deviations from strict Lindemann behavior are discussed.(AIP).

Stochastic Analysis of Nonlinear Point Reactor Systems with Colored Multiplicative Noise

This paper presents a master equation formulation of the fluctuations of neutron density in a nonlinear power reactor model system perturbed by colored multiplicative noise using the Van Kampen's master equation. The reactivity and power reactivity feedback coefficient are assumed to be stationary stochastic processes with short but finite correlation times. Some statistical characteristics of the neutron density fluctuations, such as moment equations, stationary distribution and stationary moments, are derived and compared with the results calculated for the system with Gaussian white parametric noise. It is shown that in the limit of Gaussian white noise our results converge to those obtained by using the Stratonovich calculus. Some remarks on the divergence between the Stratonovich calculus and the Ito calculus are also given.

Stochastic analysis of nonlinear point reactor systems with colored multiplicative noise.

This paper presents a master equation formulation of the fluctuations of neutron density in a nonlinear power reactor model system perturbed by colored multiplicative noise using the Van Kampen's master equation. The reactivity and power reactivity feedback coefficient are assumed to be stationary stochastic processes with short but finite correlation times. Some statistical characteristics of the neutron density fluctuations, such as moment equations, stationary distribution and stationary moments, are derived and compared with the results calculated for the system with Gaussian white parametric noise. It is shown that in the limit of Gaussian white noise our results converge to those obtained by using the Stratonovich calculus. Some remarks on the divergence between the Stratonovich calculus and the Ito calculus are also given.

Random walk model with correlated jumps: Self-correlation function and frequency-dependent diffusion coefficient

In this paper we report the results of a generalization of the continuous-time random walk theory of Montroll and Weiss to include correlations over two jumps. We expand our results of a previous letter by discussing the multi-state master equation formulation for Bravais and non-Bravais lattices. We present results for these equations and discuss the ease of tetrahedral interstitial sites in body centered cubic lattices which have a realization in hydrogen in metal systems. We prove that the Bravais lattices always reduce to a second-order differential equation and we give a general solution for the frequency-dependent diffusion coefficient in this case.

Stochastic theory of nonlinear rate processes with multiple stationary states

A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of O ( N 0) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is O (e N ). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.

The Master-Equation Formulation of Chromatography Theory

Abstract De Clerk et al. have recently proposed that if chromatography theory were based on a master equation rather than a Fokker-Planck equation, asymmetries could readily arise [Separation Sci., 1, 443(1966)]. It is shown here that the master-equation description of an infinitely long homogeneous column also leads to a Gaussian limiting distribution.