Framework Of Master Equation Research Articles

Reaction-diffusion master equation in the microscopic limit

Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.

Finite-frequency counting statistics of electron transport: Markovian theory

We present a theory of frequency-dependent counting statistics of electron transport through nanostructures within the framework of Markovian quantum master equations. Our method allows the calculation of finite-frequency current cumulants of arbitrary order, as we explicitly show for the second- and third-order cumulants. Our formulae generalize previous zero-frequency expressions in the literature and can be viewed as an extension of MacDonald's formula beyond shot noise. When combined with an appropriate treatment of tunneling using, e.g., the Liouvillian perturbation theory in Laplace space, our method can deal with arbitrary bias voltages and frequencies, as we illustrate with the paradigmatic example of transport through a single resonant level model. We discuss various interesting limits, including the recovery of the fluctuation-dissipation theorem near linear response, as well as some drawbacks inherent to the Markovian description arising from the neglect of quantum fluctuations.

Outbreak properties of epidemic models: The roles of temporal forcing and stochasticity on pathogen invasion dynamics

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R 0, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R 0 is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.

A statistical mechanical theory of proton transport kinetics in hydrogen-bonded networks based on population correlation functions with applications to acids and bases

Extraction of relaxation times, lifetimes, and rates associated with the transport of topological charge defects in hydrogen-bonded networks from molecular dynamics simulations is a challenge because proton transfer reactions continually change the identity of the defect core. In this paper, we present a statistical mechanical theory that allows these quantities to be computed in an unbiased manner. The theory employs a set of suitably defined indicator or population functions for locating a defect structure and their associated correlation functions. These functions are then used to develop a chemical master equation framework from which the rates and lifetimes can be determined. Furthermore, we develop an integral equation formalism for connecting various types of population correlation functions and derive an iterative solution to the equation, which is given a graphical interpretation. The chemical master equation framework is applied to the problems of both hydronium and hydroxide transport in bulk water. For each case it is shown that the theory establishes direct links between the defect's dominant solvation structures, the kinetics of charge transfer, and the mechanism of structural diffusion. A detailed analysis is presented for aqueous hydroxide, examining both reorientational time scales and relaxation of the rotational anisotropy, which is correlated with recent experimental results for these quantities. Finally, for OH(-)(aq) it is demonstrated that the "dynamical hypercoordination mechanism" is consistent with available experimental data while other mechanistic proposals are shown to fail. As a means of going beyond the linear rate theory valid from short up to intermediate time scales, a fractional kinetic model is introduced in the Appendix in order to describe the nonexponential long-time behavior of time-correlation functions. Within the mathematical framework of fractional calculus the power law decay ∼t(-σ), where σ is a parameter of the model and depends on the dimensionality of the system, is obtained from Mittag-Leffler functions due to their long-time asymptotics, whereas (stretched) exponential behavior is found for short times.

Speed of Markovian relaxation toward the ground state

For sufficiently low reservoir temperatures, it is known that open quantum systems subject to decoherent interactions with the reservoir relax toward their ground state in the weak coupling limit. Within the framework of quantum master equations, this is formalized by the Born-Markov-secular (BMS) approximation, where one obtains the system Gibbs state with the reservoir temperature as a stationary state. When the solution to some problem is encoded in the (isolated) ground state of a system Hamiltonian, decoherence can therefore be exploited for computation. The computational complexity is then given by the scaling of the relaxation time with the system size $n$. We study the relaxation behavior for local and nonlocal Hamiltonians that are coupled dissipatively with local and nonlocal operators to a bosonic bath in thermal equilibrium. We find that relaxation is generally more efficient when coherences of the density matrix in the system energy eigenbasis are taken into account. In addition, the relaxation speed strongly depends on the matrix elements of the coupling operators between initial state and ground state. We show that Dicke superradiance is a special case of our relaxation models and can thus be understood as a coherence-assisted relaxation speedup.

Master equations for quantum transport with Franck-Condon blockade

The Master equation framework for single electron transport is extended to include the Franck-Condon effect that can dominate the transport properties of nanoelectromechanical devices. The quantum nature of the resulting vibroelectronic states leaves a signature in the electronic shot noise, which is accounted for by careful treatment of the coherences. Two model systems are shown to demonstrate the relevance of such a formulation.

Chemical Master Equation and Langevin regimes for a gene transcription model

Gene transcription models must take account of intrinsic stochasticity. The Chemical Master Equation framework is based on modelling assumptions that are highly appropriate for this context, and the Stochastic Simulation Algorithm (also known as Gillespie’s algorithm) allows for practical simulations to be performed. However, for large networks and/or fast reactions, such computations can be prohibitatively expensive. The Chemical Langevin regime replaces the massive ordinary differential equation system with a small stochastic differential equation system that is more amenable to computation. Although the transition from Chemical Master Equation to Chemical Langevin Equation can be justified rigorously in the large system size limit, there is very little guidance available about how closely the two models match for a fixed system. Here, we consider a transcription model from the recent literature and show that it is possible to compare first and second moments in the two stochastic settings. To analyse the Chemical Master Equation we use some recent work of Gadgil, Lee and Othmer, and to analyse the Chemical Langevin Equation we use Ito’s Lemma. We find that there is a perfect match—both modelling regimes give the same means, variances and correlations for all components in the system. The model that we analyse involves ‘unimolecular reactions’, and we finish with some numerical simulations involving dimerization to show that the means and variances in the two regimes can also be close when more general ‘bimolecular reactions’ are involved.

Probability currents as principal characteristics in the statistical mechanics ofnon-equilibrium steady states

One of the key features of non-equilibrium steady states (NESS) is the presence ofnon-trivial probability currents. We propose a general classification of NESS in which thesecurrents play a central distinguishing role. As a corollary, we specify the transformations ofthe dynamic transition rates which leave a given NESS invariant. The formalism ismost transparent within a continuous-time master equation framework since itallows for a general graph-theoretical representation of the NESS. We discuss theconsequences of these transformations for entropy production, present several simpleexamples, and explore some generalizations, to discrete time and continuous variables.

Damping of a harmonic oscillator in a squeezed vacuum without rotating-wave approximation

A single harmonic oscillator interacting with a broadband squeezed reservoir is analyzed within the framework of master equation without invoking the rotating-wave approximation. The dynamical evolution and photon statistics of the system are investigated by studying mean photon number and second order intensity–intensity correlation function, respectively, under resonance condition which show transient oscillations at twice the harmonic oscillator frequency. The transient fluorescent spectrum reveals asymmetric features. Inclusion of vacuum and field-dependent frequency shifts affects the thermal equilibrium value of the average photon number of the harmonic oscillator.

Evolution towards criticality in an epidemiological model for meningococcal disease

In a model for bacterial infections with various mutants we find the epidemiological system evolving towards criticality without outer tuning of a control parameter. This is an indication for self-organized criticality. The epidemic model is a susceptible–infected–recovered hosts system (SIR) for the harmless agent infecting hosts I, acting as a background to a mutant strain Y which occasionally creates severely affected hosts X. The full system of SIRYX is described in the master equation framework, confirming limiting assumptions about a reduced YX-system with the SIR-system in stationarity. In this limiting case we can analytically show convergence to power law scaling typical of critical states. Furthermore, in this approximation we can show analytically that the control parameter, the pathogenicity in this model, evolves to be predominantly in its critical value zero. These findings are then confirmed by simulations of the full SIRYX-system.

Meningitis, pathogenicity near criticality: the epidemiology of meningococcal disease as a model for accidental pathogens

We formulate and analyse a model for infectious diseases transmitted by asymptomatic carriers finding, that if harmless and pathogenic strains of the infected agent compete, frequent outbreaks of the pathogenic strains can occur. A counterintuitively high number of clustered outbreaks at low pathogenicity in our model compares well with observations in diseases with severe and often fatal results for the host, as for example in meningitis. These clustered outbreaks can be described by the typical scaling behaviour around criticality. The epidemic model is a susceptible-infected-recovered system (SIR) for the harmless infective agent, acting as a background to a mutant strain Y which occasionally creates severely affected hosts X. The full system of SIRYX is described in the master equation framework, confirming limiting assumptions about a reduced YX-system with the SIR-system in stationarity. In this limiting case we can analytically show convergence to power law scaling typical for critical states, as well as the divergence of the variance of outbreaks near criticality. These large fluctuations of outbreaks of accidental pathogens as mutants of otherwise harmless commensal organisms is the challenging new feature of our model for future epidemiology of diseases like meningococcal disease.

Dephasing Times in Closed Quantum Dots

Dephasing of one-particle states in closed quantum dots is analyzed within the framework of random matrix theory and the master equation. The combination of this analysis with recent experiments on the magnetoconductance allows, for the first time, the evaluation of the dephasing times of closed quantum dots. These dephasing times turn out to be dependent on the mean level spacing and significantly enhanced as compared with the case of open dots. Moreover, the experimental data available are consistent with the prediction that the dephasing of one-particle states in finite closed systems disappears at low enough energies and temperatures.

Robustness of decoherence-free subspaces for quantum computation

It was shown recently [D.A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes.

Collective coordinates as open systems: (I). Existence conditions for frictional hamiltonians

We examine the time evolution of an object coupled to a heat reservoir at zero temperature in the framework of general master and kinetic equations for open systems. As we specialize to the particular set of solutions that represents pure states, we are led to a frictional Schrödinger equation that is free from either heuristic hypotheses or perturbation-like approximations. The generator of the evolution of damped pure states, namely the frictional hamiltonian. can be seen to be non-hermitian and to satisfy an integral, non-linear equation. The Schrödinger equation for damped wave packets is valid at all times, provided that a strongly restrictive condition holds. This condition offers an interesting interpretation in the general case. while in the weak-coupling limit, it could be said to provide a quantal version of a fluctuation-dissipation theorem.

Master Equation Approach to the Hopping Transport Theory

AbstractThe fundamentals of the hopping transport theory are analyzed within the Master equation framework, that allows a clarification of the underlying assumptions and approximations. The common nature of the approximations, inherent both in hopping and weak scattering problems, is revealed. Some of the delicate limit problems are illustrated in exactly solvable models. The dangers in applying ordinary perturbation theory to the d.c. conductivity are shown. Through the introduction of the Master equation with external sources a proof of the Miller‐Abrahams resistance‐network analogy is given. There is also a discussion of the “local equilibrium” distribution, with some reference to thermoelectric phenomena.