Reducing a chemical master equation by invariant manifold methods

We study methods for reducing chemical master equations using the Michaelis-Menten mechanism as an example. The master equation consists of a set of linear ordinary differential equations whose variables are probabilities that the realizable states exist. For a master equation with s(0) initial substrate molecules and e(0) initial enzyme molecules, the manifold can be parametrized by s(0) of the probability variables. Fraser's functional iteration method is found to be difficult to use for master equations of high dimension. Building on the insights gained from Fraser's method, techniques are developed to produce s(0)-dimensional manifolds of larger systems directly from the eigenvectors. We also develop a simple, but surprisingly effective way to generate initial conditions for the reduced models.     

Mixed quantum-classical Redfield master equation

Redfield master equation is derived from mixed quantum-classical Liouville equation using product initial conditions. Simple two-level system example is given and comparison with Fermi golden rule is made.     

Time-convolutionless master equation for mesoscopic electron-phonon systems

The time-convolutionless master equation for the electronic populations is derived for a generic electron-phonon Hamiltonian. The equation can be used in the regimes where the golden rule approach is not applicable. The equation is applied to study the electronic relaxation in several models with the finite number of normal modes. For such mesoscopic systems the relaxation behavior differs substantially from the simple exponential relaxation. In particular, the equation shows the appearance of the recurrence phenomena on a time scale determined by the slowest mode of the system. The formal results are quite general and can be used for a wide range of physical systems. Numerical results are presented for a two level system coupled to Ohmic and super-Ohmic baths, as well as for a model of charge-transfer dynamics between semiconducting organic polymers.     

Solution of the master equation for unimolecular reactions

A method is given for computing the rate coefficient of a unimolecular reaction as an eigenvalue solution of an integral master equation, based on Nesbet's algorithm, which overcomes computational difficulties associated with this problem. An illustrative fit to pressure-dependent data on the pyrolysis of azoethane is presented.     

Numerical inversion of the V—V master equation

The information-theoretic approach to vibrational—vibrational (V—V) energy transfer is employed to invert numerically the V—V master equation for vibrational enregy exchange between CO molecules.     

On solving the master equation in spatially periodic systems

We present a new method for solving the master equation for a system evolving on a spatially periodic network of states. The network contains 2(ν) images of a "unit cell" of n states, arranged along one direction with periodic boundary conditions at the ends. We analyze the structure of the symmetrized (2(ν)n) × (2(ν)n) rate constant matrix for this system and derive a recursive scheme for determining its eigenvalues and eigenvectors, and therefore analytically expressing the time-dependent probabilities of all states in the network, based on diagonalizations of n × n matrices formed by consideration of a single unit cell. We apply our new method to the problem of low-temperature, low-occupancy diffusion of xenon in the zeolite silicalite-1 using the states, interstate transitions, and transition state theory-based rate constants previously derived by June et al. [J. Phys. Chem. 95, 8866 (1991)]. The new method yields a diffusion tensor for this system which differs by less than 3% from the values derived previously via kinetic Monte Carlo (KMC) simulations and confirmed by new KMC simulations conducted in the present work. The computational requirements of the new method are compared against those of KMC, numerical solution of the master equation by the Euler method, and direct molecular dynamics. In the problem of diffusion of xenon in silicalite-1, the new method is shown to be faster than these alternative methods by factors of about 3.177 × 10(4), 4.237 × 10(3), and 1.75 × 10(7), respectively. The computational savings and ease of setting up calculations afforded by the new method of master equation solution by recursive reduction of dimensionality in diagonalizing the rate constant matrix make it attractive as a means of predicting long-time dynamical phenomena in spatially periodic systems from atomic-level information.     

A derivation of the master equation from path entropy maximization

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles.     

Exact quantum master equation via the calculus on path integrals

An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.     

Energy transfer in master equation simulations: A new approach

Collisional energy transfer plays a key role in recombination, unimolecular, and chemical activation reactions. For master equation simulations of such reaction systems, it is conventionally assumed that the rate constant for inelastic energy transfer collisions is independent of the excitation energy. However, numerical instabilities and nonphysical results are encountered when normalizing the collision step‐size distribution in the sparse density of states regime at low energies. It is argued here that the conventional assumption is not correct, and it is shown that the numerical problems and nonphysical results are eliminated by making a plausible assumption about the energy dependence of the rate coefficient for inelastic collisions. The new assumption produces a model that is more physically realistic for any reasonable choice of collision step‐size distribution, but more work remains to be done. The resulting numerical algorithm is stable and noniterative. Testing shows that overall accuracy in master equation simulations is better with this new approach than with the conventional one. This new approach is appropriate for all energy‐grained master equation formulations. © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 748–763, 2009     

Solution of the Pauli master equation via the lanczos algorithm

We present a numerical Green's function approach to solving the Pauli master equation which utilizes the recursive residue generation method (RRGM) developed by Wyatt and co-workers. In most practical applications, only a few Green's function matrix elements are required to express all physical quantities of interest. For such systems, computation time reductions of 10²–10³ over direct diagonalization algorithms are achieved even for systems of moderate size. An application to a simple model for trapping in polymers is given.     

Bistability in the chemical master equation for dual phosphorylation cycles

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.     

A master equation investigation of coagulation reactions: Sol-gel transition

The kinetics of irreversible coagulation phenomena in spatially homogeneous systems is formulated in terms of a multivariate stochastic process. The latter is governed by a master equation for the joint probability distribution of the numbers of reacting species. An efficient numerical algorithm is used to simulate the complete time evolution of the stochastic process. The method is illustrated by simulating the coagulation reaction with configuration-dependent reaction kernels, K_ij = (ij)^ω, for clusters of mass i and j with 1/2 < ω ⩽ 1, which are designed to model gelation phenomena. It is demonstrated that the stochastic simulation allows the determination of critical exponents and the gel point directly from the master equation. The results are compared to predictions of the rate equation approach to the sol-gel transition.     

Computational aspects of master equation transformation in terms of moments

The master equation describing the temporal evolution of a gaseous system in contact with a heat bath can be transformed into a system of linear, constant-coefficient, first-order differential equations of moments of the population distribution. While it has the advantage that populations are obtained directly from observables (moments), this system of equations is not too well-conditioned and unless precautions are taken, unsurmountable numerical problems appear. These are principally associated with manipulations (inversion and taking the exponential of a matrix) involving slightly modified Vandermonde matrices whose elements span a very wide range of orders of magnitude. This article discusses ways to avoid these pitfalls which consist principally of a suitable matrix normalization.     

Development of calculation and analysis methods for the dynamic first hyperpolarizability based on the ab initio molecular orbital-quantum master equation method

We develop novel calculation and analysis methods for the dynamic first hyperpolarizabilities β [the second-order nonlinear optical (NLO) properties at the molecular level] in the second-harmonic generation based on the quantum master equation method combined with the ab initio molecular orbital (MO) configuration interaction method. As examples, we have evaluated off-resonant dynamic β values of donor (NH(2))- and/or acceptor (NO(2))-substituted benzenes using these methods, which are shown to reproduce those by the conventional summation-over-states method well. The spatial contributions of electrons to the dynamic β of these systems are also analyzed using the dynamic β density and its partition into the MO contributions. The present results demonstrate the advantage of these methods in unraveling the mechanism of dynamic NLO properties and in building the structure-dynamic NLO property relationships of real molecules.     

Coarse master equation from Bayesian analysis of replica molecular dynamics simulations

We use Bayesian inference to derive the rate coefficients of a coarse master equation from molecular dynamics simulations. Results from multiple short simulation trajectories are used to estimate propagators. A likelihood function constructed as a product of the propagators provides a posterior distribution of the free coefficients in the rate matrix determining the Markovian master equation. Extensions to non-Markovian dynamics are discussed, using the trajectory "paths" as observations. The Markovian approach is illustrated for the filling and emptying transitions of short carbon nanotubes dissolved in water. We show that accurate thermodynamic and kinetic properties, such as free energy surfaces and kinetic rate coefficients, can be computed from coarse master equations obtained through Bayesian inference.     

Solving the unimolecular master equation with a weighted subspace projection method

A weighted subspace projection method for solving the unimolecular master equation over a wide range of temperatures and pressures is developed. Sample calculations modeling the dissociation of ethane at 300 K and pressures as low as 0.65 Torr demonstrates the utility of the method in regimes where standard projection methods fail. For the sample calculations the weighted Arnoldi method was able to reliably calculate the smallest eigenvalue of the rate matrix in excellent agreement with calculations using the Nesbet algorithm. Extremely small eigenvalues of the order of −10⁻⁴⁸ could be calculated without difficulty. The formal equivalence between various weighting schemes and common matrix transformations is shown. The point that merely taking the transpose of the rate matrix can be extremely beneficial is made, commenting on the relationship between the left and right eigenvectors of the rate matrix. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 592–606, 2000     

Electronic excitation dynamics in multichromophoric systems described via a polaron-representation master equation

We derive a many-site version of the non-Markovian time-convolutionless polaron master equation [Jang et al., J. Chem Phys. 129, 101104 (2008)] to describe electronic excitation dynamics in multichromophoric systems. By treating electronic and vibrational degrees of freedom in a combined frame (polaron frame), this theory is capable of interpolating between weak and strong exciton-phonon coupling and is able to account for initial non-equilibrium bath states and spatially correlated environments. Besides outlining a general expression for the expected value of any electronic system observable in the original frame, we also discuss implications of the Markovian and Secular approximations highlighting that they need not hold in the untransformed frame despite being strictly satisfied in the polaron frame. The key features of the theory are illustrated using as an example a four-site subsystem of the Fenna-Mathews-Olson light-harvesting complex. For a spectral density including a localised mode, we show that oscillations of site populations may only be observed when non-equilibrium bath effects are taken into account. Furthermore, we illustrate how this formalism allows us to identify the electronic and vibrational components of the oscillatory dynamics.     

Laguerre polynomial solutions to master equation for vibrational energy relaxation in gases

Collisional energy transfer between highly vibrationally excited molecules and a bath gas is considered as a stochastic process occurring in energy space. An exact solution to master equation for the conditional probability is given in terms of simple analytical formulas for weak and strong collisions. The strong collisions are shown to manifest themselves in the distribution pattern composed of maxima and minima in the energy dependence of conditional probability. This effect is explained in detail on physical grounds.