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.     

Metabasin approach for computing the master equation dynamics of systems with broken ergodicity

We propose a technique for computing the master equation dynamics of systems with broken ergodicity. The technique involves a partitioning of the system into components, or metabasins, where the relaxation times within a metabasin are short compared to an observation time scale. In this manner, equilibrium statistical mechanics is assumed within each metabasin, and the intermetabasin dynamics are computed using a reduced set of master equations. The number of metabasins depends upon both the temperature of the system and its derivative with respect to time. With this technique, the integration time step of the master equations is governed by the observation time scale rather than the fastest transition time between basins. We illustrate the technique using a simple model landscape with seven basins and show validation against direct Euler integration. Finally, we demonstrate the use of the technique for a realistic glass-forming system (viz., selenium) where direct Euler integration is not computationally feasible.     

Steady-state master equation methods

This paper investigates the application of the steady-state master equation (SSME) to complex unimolecular reactions. It is shown how the steady-state approximation and the Boltzmann reservoir approximation can be applied, both together or separately, to improve the efficiency and the numerical robustness of simple master equation calculations. The case of a two-well isomerization is considered in detail, and two versions of the SSME are derived for it, one of which shows very clearly and analytically the relationship between the rate coefficients and the flux coefficients. It is shown for a reversible second order recombination reaction that the second order phenomenological equation can be regained in either version of the SSME. It is also shown how, under steady state conditions, the SSME may be used to determine rate coefficients for each reaction in a multiple well scheme.     

Quantum master equation approach to the second hyperpolarizability of nanostar dendritic systems

We investigate the dynamic second hyperpolarizability (gamma) of nanostar dendritic systems using the quantum master equation approach. In the nanostar dendritic systems composed of three-state monomers, the multistep exciton states are obtained by the dipole-dipole interactions, and the directional energy transport, i.e., exciton migration, from the periphery to the core is predicted to occur by the relaxation between exciton states originating in the exciton-phonon coupling. The effects of the intermolcecular interaction and the exciton migration, i.e., exciton relaxation, on the gamma in the third-harmonic generation (THG) are examined in the three-photon off- and on- resonance regions using the two-exciton model. Furthermore, the method for analysis of spatial contributions of excitons to gamma is presented by partitioning the total gamma into the one- and two-exciton contributions. It turns out that the exciton relaxation between exciton states causes significant broadening of the spectra of gamma and their mutual overlap as well as the relative increase of two-exciton contributions in the nanostar dendritic system.     

Polaronic quantum master equation theory of inelastic and coherent resonance energy transfer for soft systems

This work extends the theory of coherent resonance energy transfer [S. Jang, J. Chem. Phys. 131, 164101 (2009)] by including quantum mechanical inelastic effects due to modulation of donor-acceptor electronic coupling. Within the approach of the second order time local quantum master equation (QME) in the polaron picture and under the assumption that the bath degrees of freedom modulating the electronic coupling are independent of other modes, a general time evolution equation for the reduced system density operator is derived. Detailed expressions for the relaxation operators and inhomogeneous terms of the QME are then derived for three specific models of modulation in distance, axial angle, and dihedral angle, which are all approximated by harmonic oscillators. Numerical tests are conducted for a set of model parameters. Model calculation shows that the torsional modulation can make significant contribution to the relaxation and dephasing mechanisms.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

A constitutive equation for composite systems

By the differential operator representation of constitutive equations for linearly viscoelastic materials, general theoretical expressions are developed for the physical parameters of a composite system in relation to those of the constituents. The interfacial tension between the dispersed phase and continuous matrix is included in the derivation. The results show that it has effect only on the shear parameter but not on the bulk parameter of the composite system. The general expressions developed in this article represent a unified theory for composite systems. The results are shown to reduce to many special cases obtained by other investigators.     

Formation of mesoscopic water networks in aqueous systems

Formation of the macroscopically-infinite hydrogen-bonded water network in various aqueous systems occurs via 3D percolation transition when the probability of finding a spanning water cluster exceeds 95%. As a result, in a wide interval of water content below the percolation threshold, rarefied quasi-2D water networks span over the mesoscopic length scale. Formation and topology of spanning water networks, which affect various properties of aqueous systems, can be described within the framework of the percolation theory.     

The finite state projection algorithm for the solution of the chemical master equation

This article introduces the finite state projection (FSP) method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation (CME). Unlike Monte Carlo methods such as the stochastic simulation algorithm (SSA) or tau leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and tau leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than tau leaping methods.     

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.     

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.