Solution Of Master Equation Research Articles

On the earliest jump unravelling of the spatial decoherence master equation

Solution of free particle quantum master equation with spatial decoherence can be unravelled into stochastic quantum trajectories in many ways. The first example (Diósi, 1985) proposed a piecewise deterministic jump process for the wave function. While alternative unravellings, diffusive ones in particular, proved to be tractable analytically, the jump process 1985, also called orthojump, allows for few analytic results, it needs numeric methods as well. Here we prove that, similarly to diffusive unravellings, it is localizing the quantum state.

QmeQ 1.0: An open-source Python package for calculations of transport through quantum dot devices

QmeQ is an open-source Python package for numerical modeling of transport through quantum dot devices with strong electron–electron interactions using various approximate master equation approaches. The package provides a framework for calculating stationary particle or energy currents driven by differences in chemical potentials or temperatures between the leads which are tunnel coupled to the quantum dots. The electronic structures of the quantum dots are described by their single-particle states and the Coulomb matrix elements between the states. When transport is treated perturbatively to lowest order in the tunneling couplings, the possible approaches are Pauli (classical), first-order Redfield, and first-order von Neumann master equations, and a particular form of the Lindblad equation. When all processes involving two-particle excitations in the leads are of interest, the second-order von Neumann approach can be applied. All these approaches are implemented in QmeQ. We here give an overview of the basic structure of the package, give examples of transport calculations, and outline the range of applicability of the different approximate approaches. Program summaryProgram Title:QmeQProgram Files doi:http://dx.doi.org/10.17632/8687mrhgg9.1Licensing provisions: BSD 2-ClauseProgramming language:PythonExternal libraries:NumPy, SciPy, CythonNature of problem: Calculation of stationary state currents through quantum dots tunnel coupled to leads.Solution method: Exact diagonalization of the quantum dot Hamiltonian for a given set of single particle states and Coulomb matrix elements. Numerical solution of the stationary-state master equation for a given approximate approach.Restrictions: Depending on the approximate approach the temperature needs to be sufficiently large compared to the coupling strength for the approach to be valid.

RRKM and master equation kinetic analysis of parallel addition reactions of isomeric radical intermediates in hydrocarbon flames.

We have calculated the temperature-dependent rate coefficients of the addition reactions of butadien-2-yl (C4H5) and acroylyl (C3H3O) radicals with ethene (C2H4), carbon monoxide (CO), formaldehyde (H2CO), hydrogen cyanide (HCN), and ketene (H2CCO), in order to explore the balance between kinetic and thermodynamic control in these combustion-related reactions. For the C4H5 radical, the 1,3-diene form of the addition products is more stable than the 1,2-diene, but the 1,2-diene form of the radical intermediate is stabilized by an allylic delocalization, which may influence the relative activation energies. For the reactions combining C3H3O with C2H4, CO, and HCN, the opposite is true: the 1,2-enone form of the addition products is more stable than the 1,3-enone, whereas the 1,3-enone is the slightly more stable radical species. Optimized geometries and vibrational modes were computed with the QCISD/aug-cc-pVDZ level and basis, followed by single-point CCSD(T)-F12a/cc-pVDZ-F12 energy calculations. Our findings indicate that the kinetics in all cases favor reaction along the 1,3 pathway for both the C4H5 and C3H3O systems. The Rice-Ramsperger-Kassel-Marcus (RRKM) microcanonical rate coefficients and subsequent solution of the chemical master equation were used to predict the time-evolution of our system under conditions from 500 K to 2000 K and from 10-5 bar to 10 bars. Despite the 1,3 reaction pathway being more favorable for the C4H5 system, our results predict branching ratios of the 1,2 to 1,3 product as high as 0.48 at 1 bar. Similar results hold for the acroylyl system under these combustion conditions, suggesting that under kinetic control the branching of these reactions may be much more significant than the thermodynamics would suggest. This effect may be partly attributed to the low energy difference between 1,2 and 1,3 forms of the radical intermediate. No substantial pressure-dependence is found for the overall forward reaction rates until pressures decrease below 0.1 bar.

Efficient simulation of intrinsic, extrinsic and external noise in biochemical systems.

MotivationBiological cells operate in a noisy regime influenced by intrinsic, extrinsic and external noise, which leads to large differences of individual cell states. Stochastic effects must be taken into account to characterize biochemical kinetics accurately. Since the exact solution of the chemical master equation, which governs the underlying stochastic process, cannot be derived for most biochemical systems, approximate methods are used to obtain a solution.ResultsIn this study, a method to efficiently simulate the various sources of noise simultaneously is proposed and benchmarked on several examples. The method relies on the combination of the sigma point approach to describe extrinsic and external variability and the τ-leaping algorithm to account for the stochasticity due to probabilistic reactions. The comparison of our method to extensive Monte Carlo calculations demonstrates an immense computational advantage while losing an acceptable amount of accuracy. Additionally, the application to parameter optimization problems in stochastic biochemical reaction networks is shown, which is rarely applied due to its huge computational burden. To give further insight, a MATLAB script is provided including the proposed method applied to a simple toy example of gene expression.Availability and implementationMATLAB code is available at Bioinformatics online.Supplementary information Supplementary data are available at Bioinformatics online.

Deterministic submanifolds and analytic solution of the quantum stochastic differential master equation describing a monitored qubit

This paper studies the stochastic differential equation (SDE) associated with a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non-demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves.

Pressure Dependence of Rate Coefficients for Formation and Dissociation of Pentachlorodisilane and Related Chemical Activation Reactions

Four entrance channels to the pentachlorodisilane (Si2HCl5) potential energy well, namely, SiCl2 + SiHCl3, SiCl4 + SiHCl, Cl3SiSiCl + HCl, and SiCl3 + SiHCl2, were analyzed in detail through transition state theory, Rice–Ramsperger–Kassel–Marcus (RRKM) theory, and solution of the multichannel master equation. The stationary points in the potential energy surface were optimized, and their vibrational frequencies and rotational constants calculated at the (U)B3LYP/6–31+G(d,p) level of theory; the (U)CCSD(T)/aug-cc-pVTZ level was then used for accurate estimation of activation energies. The pressure and temperature dependence of the rate coefficients of the channels related to Si2HCl5 stabilization/dissociation was determined along a wide range of conditions, for the first time. All channels showed strong pressure dependence in the four cases, at least at low-to-moderate pressure conditions. Each entrance channel leads to the formation of different products under different conditions, and the mechanism was analyzed in detail. The results indicated that at atmospheric pressure the reactions are in the falloff region, and therefore do not support the adoption of high-pressure limit rate coefficients in reaction models designed for simulation of systems at atmospheric or subatmospheric pressure conditions.

The MV formalism for $${\mathrm{IBL}}_\infty $$ IBL ∞ - and $${\mathrm{BV}}_\infty $$ BV ∞ -algebras

We develop a new formalism for the quantum master equation \(\Delta e^{S/\hbar } = 0\) and the category of \(\mathrm{IBL}_\infty \)-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of \(\mathrm{IBL}_\infty \)-algebras and \(\mathtt{L}_\infty \)-algebras and, on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.

Solutions of generic bilinear master equations for a quantum oscillator—Positive and factorized conditions on stationary states

We obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the Gaussian form. The well-behavedness and positive semidefiniteness of the stationary states could be characterized by a three-dimensional Minkowski vector. By requiring the stationary states to satisfy a factorized condition, we obtain a generic class of master equations that includes the well-known ones and their generalizations, some of which are completely positive. A further subset of the master equations with the Gibbs states as stationary states is also obtained. For master equations with not completely positive generators, an analysis on the stationary states for a given initial state suggests conditions on the coefficients of the master equations that generate positive evolution.

Numerical Simulations Reveal Randomness of Cu(II) Induced Aβ Peptide Dimerization under Conditions Present in Glutamatergic Synapses.

The interactions between the Aβ1–40 molecules species and the copper ions (Cu(II)) were intensively investigated due to their potential role in the development of the Alzheimer Disease (AD). The rate and the mechanism of the Cu(II)—Aβ complexes formation determines the aggregation pathway of the Aβ species, starting from smaller but more cytotoxic oligomers and ending up in large Aβ plaques, being the main hallmark of the AD. In our study we exploit the existing knowledge on the Cu(II)—Aβ interactions and create the theoretical model of the initial phase of the copper- driven Aβ aggregation mechanism. The model is based on the direct solution of the Chemical Master Equations, which capture the inherent stochastics of the considered system. In our work we argue that due to a strong Cu(II) affinity to Aβ and temporal accessibility of the Cu(II) ions during normal synaptic activity the aggregation driven by Cu(II) dominates the pure Aβ aggregation. We also demonstrate the dependence of the formation of different Cu(II)—Aβ complexes on the concentrations of reagents and the synaptic activity. Our findings correspond to recent experimental results and give a sound hypothesis on the AD development mechanisms.

Obtaining effective rate coefficients to describe the decomposition kinetics of the corannulene oxyradical at high temperatures

Unimolecular reactions play an important role in combustion kinetics. An important task of reaction kinetic analysis is to obtain the phenomenological rate coefficients for unimolecular reactions based on the master equation approach. In most cases, the eigenvalues of the transition matrix describing collisional internal energy relaxation are of much larger magnitude than and well separated from the chemically significant eigenvalues, so that phenomenological rate coefficients may be unequivocally derived for incorporation in combustion mechanisms. However, when dealing with unimolecular reactions for a large molecule, especially at high temperatures, the large densities of states of the reactant cause the majority of the population distribution to lie at very high energy levels where the microcanonical reaction rate constants are large and the relaxation and chemical eigenvalues overlap, so that well-defined phenomenological rate coefficients cannot be determined. This work attempts to analyze the effect of overlapping eigenvalues on the high-temperature kinetics of a large oxyradical, based on microcanonical reaction rates and population distributions as well as the eigenvalue spectrum of the transition matrix from the master equation. The aim is to provide a pragmatic method for obtaining the most effective rate coefficients for competing elimination, dissociation, and bimolecular reactions for incorporation in combustion mechanisms. Our approach is demonstrated with a representative example, thermal decomposition and H addition reactions of the corannulene oxyradical.

Exploration of CPT violation via time-dependent geometric quantities embedded in neutrino oscillation through fluctuating matter

We propose a new approach to explore CPT violation of neutrino oscillations through a fluctuating matter based on time-dependent geometric quantities. By mapping the neutrino oscillations onto a Poincaré sphere structure, we obtain an analytic solution of master equation and further define the geometric quantities, i.e., radius of Poincaré sphere and geometric phase. We find that the mixing process between electron and muon neutrinos can be described by the radius of Poincaré sphere that depends on the intrinsic CP-violating angle. Such a radius reveals a dynamic mechanism of CPT-violation, i.e., both spontaneous symmetry breaking and Majorana–Dirac neutrino confusion. We show that the time-dependent geometric phase can be used to find the neutrino nature and observe the CPT-violation because it is strongly enhanced under the neutrino propagation. We further show that the time-dependent geometric phase can be easily detected by simulating the neutrino oscillation based on fluctuating magnetic fields in nuclear magnetic resonance, which makes the experimental observation of CPT-violation possible in the neutrino mixing and oscillations.

Kinetic Models of Oxygen Thermochemistry Based on Quasi-Classical Trajectory Analysis

Recent progress in state-resolved kinetic models of thermal relaxation and dissociation of oxygen based on high-fidelity transition rate coefficients is presented. Specifically, three types of collisions encountered in high-temperature flows are discussed: , , and . For these molecular systems, the thermal relaxation times and dissociation rate coefficients, obtained from extensive trajectory simulations, are compared with existing experimental data. A new set of calculations for on an ab-initio potential energy surface is presented. Accuracy of multitemperature models is assessed based on comparison with the solution of master equations. Recommendations for adjustable parameters employed in multitemperature models are provided.

Kinetics of nanosize ferroelectrics

We analyze theoretically the finite-temperature polarization dynamic in displacive-type ferroelectrics. In particular we consider the thermally-activated switching time of a single-domain ferroelectric polarization studied by means of the Landau-Khalatnikov equation, extended as to capture thermal fluctuations. The results are compared with the switching time formula that follows from the analytical solution of Pauli master equations. In a second step we focus on the phase diagram of a prototypical ferroelectric as described by the temperature-dependent Landau-Devonshire model including thermal fluctuations. Our simulations show the emergence of phase instability at reduced sizes which we attribute to thermal fluctuations in the order parameter in the respective phase. We conclude that, along with the temperature-dependent potential coefficients, thermal fluctuations should be taken into account to achieve a comprehensive description of the thermal behavior of reduced-size ferroelectrics. To endorse our conclusions, we simulated the electric-field activated switching time for a multi-domain system and compared the results to the predictions of well-established models such as the Kolmogorov-Avrami-Ishibashi.

Epidemiological models in semiclassical approximation: an analytically solvable model as a test case

Stochastic models that can describe real epidemiological processes can become very quickly quite complex. Approximation schemes are a useful tool to understand the qualitative behaviour of such processes. In this paper, we investigate the semiclassical approximation, performed in the context of the Hamilton–Jacobi formalism, for solutions of master equations of stochastic epidemiological systems. In a test case of a previously investigated process, the linear infection model, we can analytically solve Hamilton's equations of motion. This helps to understand generalizations to more complex epidemiological systems as needed to describe realistic cases like multi‐strain systems applicable to dengue fever, for example. The connection between the semiclassical approach for epidemiological systems and the Wentzel–Kramers–Brillouin approximation in quantum mechanics is also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

A novel method to compute the time dependence of state distributions in the stochastic kinetic description of an autocatalytic system

A novel method is introduced to estimate the solution of the stochastic kinetic master equation of an autocatalytic reaction scheme. The method first computes an accurate probability master distribution for relatively low particle numbers, then uses a time shift property of the distributions to extend this solution to the entire state space and time scale. A limited comparison with accurately calculated probability distributions suggests that the approximation works exceptionally well and is able to provide a very reliable way to obtain the solution of the stochastic master equation of a system without any limitation on its size. The method will most probably be applicable with minor modifications for the stochastic kinetic description of other irreversible reactions as well.

Quantum localization of classical mechanics

Quantum localization of classical mechanics within the BRST-BFV and BV (or field–antifield) quantization methods are studied. It is shown that a special choice of gauge fixing functions (or BRST-BFV charge) together with the unitary limit leads to Hamiltonian localization in the path integral of the BRST-BFV formalism. In turn, we find that a special choice of gauge fixing functions being proportional to extremals of an initial non-degenerate classical action together with a very special solution of the classical master equation result in Lagrangian localization in the partition function of the BV formalism.

An Analytical Framework for Studying Small-Number Effects in Catalytic Reaction Networks: A Probability Generating Function Approach to Chemical Master Equations.

Cell activities primarily depend on chemical reactions, especially those mediated by enzymes, and this has led to these activities being modeled as catalytic reaction networks. Although deterministic ordinary differential equations of concentrations (rate equations) have been widely used for modeling purposes in the field of systems biology, it has been pointed out that these catalytic reaction networks may behave in a way that is qualitatively different from such deterministic representation when the number of molecules for certain chemical species in the system is small. Apart from this, representing these phenomena by simple binary (on/off) systems that omit the quantities would also not be feasible. As recent experiments have revealed the existence of rare chemical species in cells, the importance of being able to model potential small-number phenomena is being recognized. However, most preceding studies were based on numerical simulations, and theoretical frameworks to analyze these phenomena have not been sufficiently developed. Motivated by the small-number issue, this work aimed to develop an analytical framework for the chemical master equation describing the distributional behavior of catalytic reaction networks. For simplicity, we considered networks consisting of two-body catalytic reactions. We used the probability generating function method to obtain the steady-state solutions of the chemical master equation without specifying the parameters. We obtained the time evolution equations of the first- and second-order moments of concentrations, and the steady-state analytical solution of the chemical master equation under certain conditions. These results led to the rank conservation law, the connecting state to the winner-takes-all state, and analysis of 2-molecules M-species systems. A possible interpretation of the theoretical conclusion for actual biochemical pathways is also discussed.

Edge-state blockade of transport in quantum dot arrays

We propose a transport blockade mechanism in quantum dot arrays and conducting molecules based on an interplay of Coulomb repulsion and the formation of edge states. As a model we employ a dimer chain that exhibits a topological phase transition. The connection to a strongly biased electron source and drain enables transport. We show that the related emergence of edge states is manifest in the shot noise properties as it is accompanied by a crossover from bunched electron transport to a Poissonian process. For both regions we develop a scenario that can be captured by a rate equation. The resulting analytical expressions for the Fano factor agree well with the numerical solution of a full quantum master equation.

Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model

In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.

An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems

Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating the solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.