Search PubMed⌕ Search

SEARCH · Search PubMed

Results for “Master equation”

Search indexed PubMed citations on genomics, clinical trials, systematic reviews and public health. Explore titles, authors and supplied subject terms, then open the PubMed record.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 217 records · Page 12Linked to original sources
PubMed2005

Simplified charge separation energetics in a two-dimensional model for polymer-based photovoltaic cells.

An extension of our two-dimensional working model for photovoltaic behavior in binary polymer and/or molecular photoactive blends is presented. The objective is to provide a more-realistic description of the charge generation and charge separation processes in the blend system. This is achieved by assigning an energy to each of the possible occupation states, describing the system according to a simple energy model for exciton and geminate electron-hole pair configurations. The energy model takes as primary input the ionization potential, electron affinity and optical gap of the components of the blend. The underlying photovoltaic model considers a nanoscopic subvolume of a photoactive blend and represents its p- and n-type domain morphology, in terms of a two-dimensional network of donor and acceptor sites. The nearest-neighbor hopping of charge carriers in the illuminated system is described in terms of transitions between different occupation states. The equations governing the dynamics of these states are cast into a linear master equation, which can be solved for arbitrary two-dimensional donor-acceptor networks, assuming stationary conditions. The implications of incorporating the energy model into the photovoltaic model are illustrated by simulations of the short circuit current versus thickness of the photoactive blend layer for different choices of energy parameters and donor-acceptor topology. The results suggest the existence of an optimal thickness of the photoactive film in bulk heterojunctions, based on kinetic considerations alone, and that this optimal thickness is very sensitive to the choice of energy parameters. The results also indicate space-charge limiting effects for interpenetrating donor-acceptor networks with characteristic domain sizes in the nanometer range and high driving force for the photoinduced electron transfer across the donor-acceptor internal interface.

Journal Article↗
PubMed2005

Quantum Brownian motion with large friction.

Quantum Brownian motion in the strong friction limit is studied based on the exact path integral formulation of dissipative systems. In this limit the time-nonlocal reduced dynamics can be cast into an effective equation of motion, the quantum Smoluchowski equation. For strongly condensed phase environments it plays a similar role as master equations in the weak coupling range. Applications for chemical, mesoscopic, and soft matter systems are discussed and reveal the substantial role of quantum fluctuations.

Computer Simulation↗
PubMed1997

Stochastic versions of the Hodgkin-Huxley equations.

A Hodgkin-Huxley model algorithm for the numerical simulation of noise in neurons is contracted from a master equation description (cellular automoton) into a Langevin description. This reduction reduces the time required for a simulation by about two orders of magnitude. Earlier work is summarized, condensed, and made explicit to make the algorithm transparent and facilitate applications. Two approximate treatments are reported. An extension of this approach is presented that includes spatial dependence and the propagation of a noisy action potential along an axon.

Axons↗
PubMed2001

Fractional transport equations for Lévy stable processes.

The influence functional method of Feynman and Vernon is used to obtain a quantum master equation for a system subjected to a Lévy stable random force. The corresponding classical transport equations for the Wigner function are then derived, both in the limits of weak and strong friction. These are fractional extensions of the Klein-Kramers and the Smoluchowski equations. It is shown that the fractional character acquired by the position in the Smoluchowski equation follows from the fractional character of the momentum in the Klein-Kramers equation. Connections among fractional transport equations recently proposed are clarified.

Journal Article↗
PubMed2006

A variational approach to the stochastic aspects of cellular signal transduction.

Cellular signaling networks have evolved to cope with intrinsic fluctuations, coming from the small numbers of constituents, and the environmental noise. Stochastic chemical kinetics equations govern the way biochemical networks process noisy signals. The essential difficulty associated with the master equation approach to solving the stochastic chemical kinetics problem is the enormous number of ordinary differential equations involved. In this work, we show how to achieve tremendous reduction in the dimensionality of specific reaction cascade dynamics by solving variationally an equivalent quantum field theoretic formulation of stochastic chemical kinetics. The present formulation avoids cumbersome commutator computations in the derivation of evolution equations, making the physical significance of the variational method more transparent. We propose novel time-dependent basis functions which work well over a wide range of rate parameters. We apply the new basis functions to describe stochastic signaling in several enzymatic cascades and compare the results so obtained with those from alternative solution techniques. The variational Ansatz gives probability distributions that agree well with the exact ones, even when fluctuations are large and discreteness and nonlinearity are important. A numerical implementation of our technique is many orders of magnitude more efficient computationally compared with the traditional Monte Carlo simulation algorithms or the Langevin simulations.

Cells↗
PubMed2007

Mass fluctuation kinetics: capturing stochastic effects in systems of chemical reactions through coupled mean-variance computations.

The intrinsic stochastic effects in chemical reactions, and particularly in biochemical networks, may result in behaviors significantly different from those predicted by deterministic mass action kinetics (MAK). Analyzing stochastic effects, however, is often computationally taxing and complex. The authors describe here the derivation and application of what they term the mass fluctuation kinetics (MFK), a set of deterministic equations to track the means, variances, and covariances of the concentrations of the chemical species in the system. These equations are obtained by approximating the dynamics of the first and second moments of the chemical master equation. Apart from needing knowledge of the system volume, the MFK description requires only the same information used to specify the MAK model, and is not significantly harder to write down or apply. When the effects of fluctuations are negligible, the MFK description typically reduces to MAK. The MFK equations are capable of describing the average behavior of the network substantially better than MAK, because they incorporate the effects of fluctuations on the evolution of the means. They also account for the effects of the means on the evolution of the variances and covariances, to produce quite accurate uncertainty bands around the average behavior. The MFK computations, although approximate, are significantly faster than Monte Carlo methods for computing first and second moments in systems of chemical reactions. They may therefore be used, perhaps along with a few Monte Carlo simulations of sample state trajectories, to efficiently provide a detailed picture of the behavior of a chemical system.

Algorithms↗
PubMed1977

A general treatment of relaxation phenomena.

The singular perturbation method has been applied to solve the quantum mechanical Liouville equation for the relaxation phenomenon of the system in thermal contact with a heat bath. The master equation derived gives the proper expressions for both diagonal and off-diagonal elements of the density matrix and is capable of describing the time-dependent behavior of the system in the time range comparable with the reciprocal of the damping constants and the time range t --> infinity compared with the reciprocal of the damping constants.

Journal Article↗
PubMed2002

Spatial distribution of ion channel activity in biological membranes: the role of noise.

A new approach is proposed to model a collective ion channel dynamics. We have assumed that ion channels create a two-component spatio-temporal interaction field. Every channel at its current spatial location in membrane contributes permanently to this field with its state (open or closed) and coupling strength to other channels. This field is described by a reaction-diffusion equation, the transition of ion channel from closed to open state (and vice versa) is described by a master equation, and migration of channels in membrane is described by a set of Langevin equations coupled by the interaction field. Within this model, we have investigated critical conditions for spatial distribution of ion channel activity.

Cell Membrane↗
PubMed2001

Fokker-Planck equation for lattice deposition models.

An asymptotically exact Fokker-Planck equation for the height fluctuations of lattice deposition models is derived from a Van Kampen expansion of the master equation. Using an Edwards-Wilkinson-type model as an example, the solution of the equivalent Langevin equation reproduces the surface roughness and lateral height correlations obtained with kinetic Monte Carlo (KMC) simulations. Our discrete equations of motion thereby provide an exact analytic and computational alternative to KMC simulations of these models.

Journal Article↗
PubMed2002

Non-Gaussian equilibrium distributions arising from the Langevin equation.

We study the Langevin equation of a point particle driven by random noise, modeled as a two-state Markov process. The corresponding master equation differs from the Fokker-Planck equation. In equilibrium, the velocity of the particle is distributed according to a binomial power law. We discuss transient (i.e., nonequilibrium) behavior, and the consequences of non-Markovian noise statistics.

Journal Article↗
PubMed2002

Urn model of separation of sand.

We introduce an urn model that describes spatial separation of sand. In this dynamical model, in a certain range of parameters spontaneous symmetry breaking takes place and equipartitioning of sand into two compartments is broken. The steady-state equation for an order parameter, a critical line, and the tricritical point on the phase diagram are found exactly. The master equation and the first-passage problem for the model are solved numerically and the results are used to locate first-order transitions. Exponential divergence of a certain characteristic time shows that the model can also exhibit very strong metastability. In certain cases characteristic time diverges as N(z), where N is the number of balls and z=1 / 2 (critical line), 2 / 3 (tricritical point), or 1 / 3 (limits of stability).

Journal Article↗
PubMed2005

Modeling of negative autoregulated genetic networks in single cells.

We discuss recent developments in the modeling of negative autoregulated genetic networks. In particular, we consider the temporal evolution of the population of mRNA and proteins in simple networks using rate equations. In the limit of low copy numbers, fluctuation effects become significant and more adequate modeling is then achieved using the master equation formalism. The analogy between regulatory gene networks and chemical reaction networks on dust grains in the interstellar medium is discussed. The analysis and simulation of complex reaction networks are also considered.

Cell Physiological Phenomena↗
PubMed2000

Heuristic derivation of continuum kinetic equations from microscopic dynamics.

We present an approximate and heuristic scheme for the derivation of continuum kinetic equations from microscopic dynamics for stochastic, interacting systems. The method consists of a mean-field-type, decoupled approximation of the master equation followed by the "naive" continuum limit. The Ising model and driven diffusive systems are used as illustrations. The equations derived are in agreement with other approaches, and consequences of the microscopic dependences of coarse-grained parameters compare favorably with exact or high-temperature expansions. The method is valuable when more systematic and rigorous approaches fail, and when microscopic inputs in the continuum theory are desirable.

Journal Article↗
PubMed2006

Bidding process in online auctions and winning strategy: Rate equation approach.

Online auctions have expanded rapidly over the last decade and have become a fascinating new type of business or commercial transaction in this digital era. Here we introduce a master equation for the bidding process that takes place in online auctions. We find that the number of distinct bidders who bid k times up to the tth bidding progresses, called the k-frequent bidder, seems to scale as n(k)(t) approximately tk(-2.4). The successfully transmitted bidding rate by the k-frequent bidder is likely to scale as q(k)(t) approximately k(-1.4), independent of t for large t. This theoretical prediction is close to empirical data. These results imply that bidding at the last moment is a rational and effective strategy to win in an eBay auction.

Journal Article↗
PubMed2002

Statistical theory for the Kardar-Parisi-Zhang equation in (1+1) dimensions.

The Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions dynamically develops sharply connected valley structures within which the height derivative is not continuous. We develop a statistical theory for the KPZ equation in (1+1) dimensions driven with a random forcing that is white in time and Gaussian-correlated in space. A master equation is derived for the joint probability density function of height difference and height gradient P(h-h*, partial differential(x)h,t) when the forcing correlation length is much smaller than the system size and much larger than the typical sharp valley width. In the time scales before the creation of the sharp valleys, we find the exact generating function of h-h* and partial differential(x)h. The time scale of the sharp valley formation is expressed in terms of the force characteristics. In the stationary state, when the sharp valleys are fully developed, finite-size corrections to the scaling laws of the structure functions left angle bracket(h-h*)(n)(partial differential(x)h)(m)right angle bracket are also obtained.

Journal Article↗
PubMed2001

Nonlinear reactive systems on a lattice viewed as Boolean dynamical systems.

We present a stochastic, time-discrete Boolean model that mimics the mesoscopic dynamics of the desorption reactions A+A-->A+S and A+A-->S+S in a one-dimensional lattice. In the continuous-time limit, we derive a hierarchy of dynamical equations for the subset of moments involving contiguous lattice sites. The solution of the hierarchy allows to compute the exact dynamics of the mean coverage for both microscopic and coarse-grained initial conditions, which turn out to be different from the mean field predictions. The evolution equations for the mean coverage and the second-order moments are shown to be equivalent to those provided by a time-continuous master equation. The important role of higher-order fluctuations is brought out by the failure of a truncation scheme retaining only two-particle fluctuation correlations.

Journal Article↗
PubMed2004

Vibrational relaxation in methyl hydrocarbons at high temperatures: propane, isobutene, isobutane, neopentane, and toluene.

Vibrational relaxation has been seen in shock waves in propane, isobutene, isobutane, neopentane, and toluene dilute in krypton with the laser-schlieren technique. These experiments cover about 600-2200 K and post-shock pressures from 5 to 29 Torr. The process cannot be resolved in any for T<600 K, or in any for large molecule fraction. All the ultrasonic measurements of relaxation in these at room temperature show characteristic times in the 1-5 ns atm range, corresponding to fewer than five collisions, whereas the relaxation times in the shock waves range from 20 to 200 ns atm, with a clearly defined negative or "inverted" temperature dependence. It would seem the observed slowdown of relaxation with increasing T is simply a consequence of the much increased energy transfer required at high temperature in such large polyatomics when this is combined with a collision efficiency, here interpreted as down, already so large it cannot much increase. The simple method for the extraction of a down from relaxation data offered here by consideration of the energy relaxation equation for Evib=0 appears to be original and should prove quite useful in connecting thermal relaxation data to values obtained from spectroscopy and master-equation analyses. Here it is found that the derived down extrapolate well to room temperature ultrasonic measurements, showing a slight increase with temperature.

Journal Article↗
PubMed2005

Non-Markovian stochastic Schrödinger equations in different temperature regimes: a study of the spin-boson model.

Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.

Journal Article↗