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PubMed2004

A semiclassical generalized quantum master equation for an arbitrary system-bath coupling.

The Nakajima-Zwanzig generalized quantum master equation (GQME) provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a, possibly anharmonic, quantum bath. In this equation, a memory kernel superoperator accounts for the influence of the bath on the dynamics of the system. In a previous paper [Q. Shi and E. Geva, J. Chem. Phys. 119, 12045 (2003)] we proposed a new approach to calculating the memory kernel, in the case of arbitrary system-bath coupling. Within this approach, the memory kernel is obtained by solving a set of two integral equations, which requires a new type of two-time system-dependent bath correlation functions as input. In the present paper, we consider the application of the linearized semiclassical (LSC) approximation for calculating those correlation functions, and subsequently the memory kernel. The new approach is tested on a benchmark spin-boson model. Application of the LSC approximation for calculating the relatively short-lived memory kernel, followed by a numerically exact solution of the GQME, is found to provide an accurate description of the relaxation dynamics. The success of the proposed LSC-GQME methodology is contrasted with the failure of both the direct application of the LSC approximation and the weak coupling treatment to provide an accurate description of the dynamics, for the same model, except at very short times. The feasibility of the new methodology to anharmonic systems is also demonstrated in the case of a two level system coupled to a chain of Lennard-Jones atoms.

Journal Article↗
PubMed2006

Master equation methods in gas phase chemical kinetics.

In this article, we discuss the application of master equation methods to problems in gas phase chemical kinetics. The focus is on reactions that take place over multiple, interconnected potential wells and on the dissociation of weakly bound free radicals. These problems are of paramount importance in combustion chemistry. To illustrate specific points, we draw on our experience with reactions we have studied previously.

Journal Article↗
PubMed2002

Stochastic unraveling of time-local quantum master equations beyond the Lindblad class.

A method for stochastic unraveling of general time-local quantum master equations (QME) which involve the reduced density operator at time t only is proposed. The present kind of jump algorithm enables a numerically efficient treatment of QMEs that are not of Lindblad form. So it opens large fields of application for stochastic methods. The unraveling can be achieved by allowing for trajectories with negative weight. We present results for the quantum Brownian motion and the Redfield QMEs as test examples. The algorithm can also unravel non-Markovian QMEs when they are in a time-local form like in the time-convolutionless formalism.

Journal Article↗
PubMed2004

Pair contact process with diffusion: failure of master equation field theory.

We demonstrate that the "microscopic" field theory representation, directly derived from the corresponding master equation, fails to adequately capture the continuous nonequilibrium phase transition of the pair contact process with diffusion (PCPD). The ensuing renormalization group (RG) flow equations do not allow for a stable fixed point in the parameter region that is accessible by the physical initial conditions. There exists a stable RG fixed point outside this regime, but the resulting scaling exponents, in conjunction with the predicted particle anticorrelations at the critical point, would be in contradiction with the positivity of the equal-time mean-square particle number fluctuations. We conclude that a more coarse-grained effective field theory approach is required to elucidate the critical properties of the PCPD.

Journal Article↗
PubMed1983

A master equation theory of fluorescence induction, photochemical yield, and singlet-triplet exciton quenching in photosynthetic systems.

A master equation theory is formulated to describe the dependence of the fluorescence yield (phi) in photosynthetic systems on the number of photons (Y) absorbed per photosynthetic unit (or domain). This theory is applied to the calculation of the dependence of the fluorescence yield on Y in (a) fluorescence induction, and (b) singlet exciton-triplet excited-state quenching experiments. In both cases, the fluorescence yield depends on the number of previously absorbed photons per domain, and thus evolves in a nonlinear manner with increasing Y. In case a, excitons transform the photosynthetic reaction centers from a quenching state to a nonquenching state, or a lower efficiency of quenching state; subsequently, absorbed photons have a higher probability of decaying by radiative pathways and phi increases as Y increases. In case b, ground-state carotenoid molecules are converted to long-lived triplet excited-state quenchers, and phi decreases as Y increases. It is shown that both types of processes are formally described by the same theoretical equations that relate phi to Y. The calculated phi (Y) curves depend on two parameters m and R, where m is the number of reaction centers (or ground-state carotenoid molecules that can be converted to triplets), and R is the ratio phi (Y leads to infinity)/(Y leads to 0). The finiteness of the photosynthetic units is thus taken into account. The m = 1 case corresponds to the "puddle" model, and m leads to infinity to the "lake," or matrix, model. It is shown that the experimental phi (Y) curves for both fluorescence induction and singlet-triplet exciton quenching experiments are better described by the m leads to infinity cases than the m = 1 case.

Kinetics↗
PubMed2002

Master equation approach to the conjugate pairing rule of Lyapunov spectra for many-particle thermostated systems.

The master equation approach to Lyapunov spectra for many-particle systems is applied to nonequilibrium thermostated systems to discuss the conjugate pairing rule. We consider iso-kinetic thermostated systems with a shear flow sustained by an external restriction, in which particle interactions are expressed as a Gaussian white randomness. Positive Lyapunov exponents are calculated by using the Fokker-Planck equation to describe the tangent vector dynamics. We introduce another Fokker-Planck equation to describe the time-reversed tangent vector dynamics, which leads to the calculation of the negative Lyapunov exponents. Using the Lyapunov exponents provided by these two Fokker-Planck equations we show the conjugate pairing rule is satisfied for thermostated systems with a shear flow in the thermodynamic limit which allow us to replace the friction coefficient with a constant number. We also give an explicit form to connect the Lyapunov exponents with the time correlation of the interaction matrix in a thermostated system with a color field.

Journal Article↗
PubMed2007

Methyl vinyl ketone+OH and methacrolein+OH oxidation reactions: a master equation analysis of the pressure- and temperature-dependent rate constants.

High-level electronic structure calculations and master equation analyses were carried out to obtain the pressure- and temperature-dependent rate constants of the methyl vinyl ketone+OH and methacrolein+OH reactions. The balance between the OH addition reactions at the high-pressure limit, the OH addition reactions in the fall-off region, and the pressure-independent hydrogen abstractions involved in these multiwell and multichannel systems, has been shown to be crucial to understand the pressure and temperature dependence of each global reaction. In particular, the fall-off region of the OH addition reactions contributes to the inverse temperature dependence of the rate constants in the Arrhenius plots, leading to pressure-dependent negative activation energies. The pressure dependence of the methyl vinyl ketone+OH reaction is clearly more important than in the case of the methacrolein+OH reaction owing to the weight of the hydrogen abstraction process in this second system. Comparison of the theoretical rate constants and the experimental measurements shows quite good agreement.

Journal Article↗
PubMed2003

Single stranded DNA translocation through a nanopore: a master equation approach.

We study voltage driven translocation of a single stranded DNA through a membrane channel. Our model, based on a master equation approach, investigates the probability density function of the translocation times, and shows that it can be either double peaked or mono peaked, depending on the system parameters. We show that the most probable translocation time is proportional to the polymer length, and inversely proportional to the first or second power of the voltage, depending on the initial conditions. The model recovers experimental observations on hetropolymers when using their properties inside the pore, such as stiffness and polymer-pore interaction.

Base Composition↗
PubMed2005

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.

Models, Chemical↗
PubMed2005

Force spectroscopy of single multidomain biopolymers: a master equation approach.

Experiments using atomic force microscopy for unfolding single multidomain biopolymers cover a broad range of time scales from equilibrium to non-equilibrium. A master equation approach allows to identify and treat coherently three dynamical regimes for increasing linear ramp velocity: i) an equilibrium regime, ii) a transient regime where refolding events still occur, and iii) a saw-tooth regime without any refolding events. For each regime, analytical approximations are derived and compared to numerically investigated examples. We analyze in the framework of this model also a periodic experimental protocol instead of a linear ramp. In this case, a major simplification arises if the dynamics can be restricted to an effectively two-dimensional subspace. For transitions with an intermediate meta-stable state, like Immunoglobulin27, a refined model allows to extract previously unknown molecular parameters related to this meta-stable state.

Algorithms↗
PubMed1999

Escape from a fluctuating system: a master equation and trapping approach.

We present a general solution for the mean exit time in a system with on-site fluctuations between two configurations described by a master equation. The coupled configurations represent a spatially discretized version of an escape over a fluctuating barrier [C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992)], and passage through modulating channels. Based on the general properties of the mean exit time, we obtain a simple solution for a coupled "birth" and "death" case that exhibits resonant activation. Within this exactly solvable model we derive analytically the optimal fluctuating rate, which is sensitive to the initial condition and scales as 1/n, where n is the system size. Our approach unifies a number of escape problems and points towards the generality of resonant activation.

Journal Article↗
PubMed2000

Master equation approach to synchronization in diffusion-coupled nonlinear oscillators

We study the influence of internal fluctuations on phase synchronization in oscillatory reaction-diffusion systems through a master equation approach. In the limit of large system size, the probability density is analyzed by means of the eikonal approximation. This approximation yields a Hamilton-Jacobi equation for the stochastic potential, which may be reduced to coupled nonlinear diffusion equations for the phase of oscillation and (conjugate) "momentum." We give explicit expressions for the coefficients of these equations in terms of averages over the deterministic periodic orbit. For one-dimensional systems, we obtain an explicit solution for the stationary stochastic potential: the width in phase, which is defined as the root mean square fluctuation in phase, characterizes the roughness of phase locking, and diverges with the system size L according to a power law w approximately Lalpha, with alpha=1/2. To study higher-dimensional systems, we show that the eikonal approximations of the diffusion-coupled oscillator problem and the Kardar-Parisi-Zhang (KPZ) equation (in the limit of small noise intensity) are equivalent. The KPZ equation governs some forms of surface growth, and the height of a growing front corresponds to the phase (the 2pi periodicity in phase is ignored) in the diffusion-coupled oscillator problem. From the equivalence, we obtain the result that spatially synchronized states may exist only in systems with a spatial dimension greater than or equal to 3; for dimensions 1 and 2, a "rough" state exists in which the width (in phase) diverges algebraically with the system size, alpha>0.

Journal Article↗
PubMed2005

Master-equation approach to the study of phase-change processes in data storage media.

We study the dynamics of crystallization in phase-change materials using a master-equation approach in which the state of the crystallizing material is described by a cluster size distribution function. A model is developed using the thermodynamics of the processes involved and representing the clusters of size two and greater as a continuum but clusters of size one (monomers) as a separate equation. We present some partial analytical results for the isothermal case and for large cluster sizes, but principally we use numerical simulations to investigate the model. We obtain results that are in good agreement with experimental data and the model appears to be useful for the fast simulation of reading and writing processes in phase-change optical and electrical memories.

Journal Article↗
PubMed1977

Random walks and generalized master equations with internal degrees of freedom.

We present an extension of the continuous-time random walk formalism to include internal states and to establish the connection to generalized master equations with internal states. The theory allows us to calculate physical observables from which we can extract the characteristic parameters of the internal states of the system under study.

Journal Article↗
PubMed2006

Solving the chemical master equation for monomolecular reaction systems analytically.

The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce easily many properties of the solution. The model class includes many interesting examples. For more complex reaction systems, our results can be seen as a first step towards the construction of new numerical integrators, because solutions to the monomolecular case provide promising ansatz functions for Galerkin-type methods.

Kinetics↗
PubMed2006

Master equation approach to the assembly of viral capsids.

The distribution of inequivalent geometries occurring during self-assembly of the major capsid protein in thermodynamic equilibrium is determined based on a master equation approach. These results are implemented to characterize the assembly of SV40 virus and to obtain information on the putative pathways controlling the progressive build-up of the SV40 capsid. The experimental testability of the predictions is assessed and an analysis of the geometries of the assembly intermediates on the dominant pathways is used to identify targets for anti-viral drug design.

Capsid↗
PubMed2005

The effect of memory in the stochastic master equation analyzed using the stochastic Liouville equation of motion. Electronic energy migration transfer between reorienting donor-donor, donor-acceptor chromophores.

This paper discusses the process of energy migration transfer within reorientating chromophores using the stochastic master equation (SME) and the stochastic Liouville equation (SLE) of motion. We have found that the SME over-estimates the rate of the energy migration compared to the SLE solution for a case of weakly interacting chromophores. This discrepancy between SME and SLE is caused by a memory effect occurring when fluctuations in the dipole-dipole Hamiltonian (H(t)) are on the same timescale as the intrinsic fast transverse relaxation rate characterized by (1/T(2)). Thus the timescale critical for energy-transfer experiments is T(2) approximately 10(-13) s. An extended SME is constructed, accounting for the memory effect of the dipole-dipole Hamiltonian dynamics. The influence of memory on the interpretation of experiments is discussed.

Electrons↗
PubMed2006

Master equation for a quantum particle in a gas.

The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.

Journal Article↗