Time-dependent Transition Rates Research Articles

Radicalization phenomena: Phase transitions, extinction processes and control of violent activities

In this work, we study a simple mathematical model to analyze the emergence and control of radicalization phenomena. The population consists of core and sensitive subpopulations, and their ways of life may be at least partially incompatible. In such a case, if a conflict exists, core agents act as inflexible individuals about the issue. On the other hand, the sensitive agents choose between two options: live peacefully with core population, or oppose it. This kind of modeling was recently considered by Galam and Javarone (2016) with constant pairwise couplings. Here, we consider the more general case with time-dependent transition rates, with the aim of study the impact of such time dependence on the critical behavior of the model. The analytical and numerical results show that the nonequilibrium active-absorbing phase transition can be suppressed in some cases, with the destruction of the absorbing phase where the radical agents disappear of the population in the stationary states.

Occupancy times for time-dependent stage-structured models

During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by Roth and Caswell for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogenous Markov chains and the discrete-time McKendrick–von Foerster equation, we derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. We apply our approach to study a specific time-dependent model of the Southern Fulmar, and obtain insights into how the number of breeding attempts depends on external conditions that vary through time.

Extended dissipative control for interval type-2 singular systems with nonhomogeneous Markovian switching

This paper investigates the problems of stochastic admissibility and extended dissipativity analysis as well as state feedback controller design for interval type-2 singular systems with nonhomogeneous Markovian switching. By utilizing matrix decomposition technique to deal with the time-dependent transition rates, a sufficient condition is established to guarantee that the systems under consideration are regular, impulse-free, stochastically asymptotically stable and extended dissipative. For developing the state feedback controller in light of the obtained sufficient condition, a novel lemma is proposed inspired by Projection lemma, based on which an approach of controller design is provided. It should be pointed out that no conservatism is introduced in controller design due to the sufficiency and necessity of this lemma. Finally, simulation examples are provided to show the effectiveness of the proposed approach.

Thermodynamic Uncertainty Relation for General Open Quantum Systems

We derive a thermodynamic uncertainty relation for general open quantum dynamics, described by a joint unitary evolution on a composite system comprising a system and an environment. By measuring the environmental state after the system-environment interaction, we bound the counting observables in the environment by the survival activity, which reduces to the dynamical activity in classical Markov processes. Remarkably, the relation derived herein holds for general open quantum systems with any counting observable and any initial state. Therefore, our relation is satisfied for classical Markov processes with arbitrary time-dependent transition rates and initial states. We apply our relation to continuous measurement and the quantum walk to find that the quantum nature of the system can enhance the precision. Moreover, we can make the lower bound arbitrarily small by employing appropriate continuous measurement.

Transient study of Markov models with time-dependent transition rates

We apply the external uniformization (EU) method for the transient study of continuous-time Markov chain models in which the transition rates could be either constant or time-dependent. In the method, we run an exponential clock for watching the Markov chain. Each realization epoch of the random clock is considered as an absorption epoch. The approximate transient probability that the Markov chain is found at a particular state is then found as the probability of absorption from that state. The convergence of the method to the analytical value has been proved. In the case of infinite/huge dimensional Markov chains, implementation of the method needs truncation of the state-space into a finite window of states. A comparison of the results obtained using the EU method with some of the existing analytical results shows that the described method produces highly accurate results on numerical implementation. Several examples considered also reveal that the EU method required much less computational time when compared to the uniformization/randomization (U/R) method.

Mode dependent filtering for semi‐Markovian jump linear systems with sojourn time dependent transition rates

In this study, the H ∞ filtering problem for a class of semi-Markovian jump linear systems with sojourn time dependent transition rates are investigated. The mode dependent filter is obtained to guarantee the stochastic stability of the filtering error system with a prescribed H ∞ performance index. Two examples are presented to illustrate the effectiveness of the authors' results.

Analiza dynamicznej niezawodności wielostanowego systemu produkcyjnego

Dynamic reliability analysis of binary systems has been widely studied in case of homogeneous continuous time Markov process assumption in the literature. In this study, we evaluate dynamic performance of a multi-state rotor line of electric motors manufacturing system under non-homogeneous continuous time Markov process (NHCTMP) degradation by using lifetime distributions of seven workstations within the system. By means of this degradation process assumption we capture the effect of age on the state change of components in the analysis by means of time dependent transition rates between states of the workstations. Essentially this is typical of many systems and more practical to use in real life applications. The working principle is based on a three state structure. If all the machines within each workstation work, the workstation is defined as working with the full performance. Whenever at least one machine fails within each workstation, then the workstation is defined as working with partial performance. If all the machines in the workstation fail then the workstation is defined as failed. The lifetime properties of the workstations under NHCTMP assumption have been studied for this three-state structure of the workstations. The workstations are all working independently and nonidentically from each other and they are connected in series within the system.We especially performed an extensive application study based on the lifetime data regarding the seven workstations within a manufacturing system. Dynamic reliability results are also discussed for the system structure. Some performance characteristics are developed for both workstations and the system as well. Numerical results for the performance characteristics of those workstations and the system are provided and supported with some graphical illustrations.

Analytical results for a stochastic model of gene expression with arbitrary partitioning of proteins

In biophysics, the search for analytical solutions of stochastic models of cellular processes is often a challenging task. In recent work on models of gene expression, it was shown that a mapping based on partitioning of Poisson arrivals (PPA-mapping) can lead to exact solutions for previously unsolved problems. While the approach can be used in general when the model involves Poisson processes corresponding to creation or degradation, current applications of the method and new results derived using it have been limited to date. In this paper, we present the exact solution of a variation of the two-stage model of gene expression (with time dependent transition rates) describing the arbitrary partitioning of proteins. The methodology proposed makes full use of the PPA-mapping by transforming the original problem into a new process describing the evolution of three biological switches. Based on a succession of transformations, the method leads to a hierarchy of reduced models. We give an integral expression of the time dependent generating function as well as explicit results for the mean, variance, and correlation function. Finally, we discuss how results for time dependent parameters can be extended to the three-stage model and used to make inferences about models with parameter fluctuations induced by hidden stochastic variables.

Dynamic performance evaluation of multi – state systems under non – homogeneous continuous time Markov process degradation using lifetimes in terms of order statistics

In multi-state modelling a system and its components have a range of performance levels from perfect functioning to complete failure. Such a modelling is more flexible to understand the behaviour of mechanical systems. To evaluate a system’s dynamic performance, lifetime analysis of a multi-state system has been considered in many research articles. The order statistics related analysis for the lifetime properties of multi-state k-out-of-n systems have recently been studied in the literature in case of homogeneous continuous time Markov process assumption. In this paper, we develop the reliability measures for multi-state k-out-of-n systems by assuming a non-homogeneous continuous time Markov process for the components which provides time dependent transition rates between states of the components. Therefore, we capture the effect of age on the state change of the components in the analysis which is typical of many systems and more practical to use in real life applications.

Entropy production and large deviation function for systems with microscopically irreversible transitions

We obtain the large deviation function for entropy production of the medium and its distribution function for the two-site totally asymmetric simple exclusion process (TASEP) and the three-state unicyclic network. Since such systems are described through microscopic irreversible transitions, we obtain time-dependent transition rates by sampling the states of these systems at a regular short time interval τ. These transition rates are used to derive the large deviation function for the entropy production in the nonequilibrium steady state and its asymptotic distribution function. The shapes of the large deviation function and the distribution function depend on the value of the mean entropy production rate which has a non-trivial dependence on the particle injection and withdrawal rates in the case of TASEP. Further, it is argued that in case of a TASEP, the distribution function tends to be like a Poisson distribution for smaller values of particle injection and withdrawal rates.

A Multi-State Physics Modeling approach for the reliability assessment of Nuclear Power Plants piping systems

A Multi-State Physics Modeling (MSPM) approach is here proposed for degradation modeling and failure probability quantification of Nuclear Power Plants (NPPs) piping systems. This approach integrates multi-state modeling to describe the degradation process by transitions among discrete states (e.g., no damage, micro-crack, flaw, rupture, etc.), with physics modeling by (physic) equations to describe the continuous degradation process within the states. We propose a Monte Carlo (MC) simulation method for the evaluation of the time-dependent transition rates between the states of the MSPM. Accountancy is given for the uncertainty in the parameters and external factors influencing the degradation process. The proposed modeling approach is applied to a benchmark problem of a piping system of a Pressurized Water Reactor (PWR) undergoing thermal fatigue. The results are compared with those obtained by a continuous-time homogeneous Markov Chain Model.

Quantization and fractional quantization of currents in periodically driven stochastic systems. I. Average currents

This article studies Markovian stochastic motion of a particle on a graph with finite number of nodes and periodically time-dependent transition rates that satisfy the detailed balance condition at any time. We show that under general conditions, the currents in the system on average become quantized or fractionally quantized for adiabatic driving at sufficiently low temperature. We develop the quantitative theory of this quantization and interpret it in terms of topological invariants. By implementing the celebrated Kirchhoff theorem we derive a general and explicit formula for the average generated current that plays a role of an efficient tool for treating the current quantization effects.

On parameter estimation in population models III: Time-inhomogeneous processes and observation error

Essential to applying a mathematical model to a real-world application is calibrating the model to data. Methods for calibrating population models often become computationally infeasible when the population size (more generally the size of the state space) becomes large, or other complexities such as time-dependent transition rates, or sampling error, are present. Continuing previous work in this series on the use of diffusion approximations for efficient calibration of continuous-time Markov chains, I present efficient techniques for time-inhomogeneous chains and accounting for observation error. Observation error (partial observability) is accounted for by joint estimation using a scaled unscented Kalman filter for state-space models. The methodology will be illustrated with respect to models of disease dynamics incorporating seasonal transmission rate and in the presence of observation error, including application to two influenza outbreaks and measles in London in the pre-vaccination era.

Computational Modeling and Numerical Methods for Spatiotemporal Calcium Cycling in Ventricular Myocytes

Intracellular calcium (Ca) cycling dynamics in cardiac myocytes is regulated by a complex network of spatially distributed organelles, such as sarcoplasmic reticulum (SR), mitochondria, and myofibrils. In this study, we present a mathematical model of intracellular Ca cycling and numerical and computational methods for computer simulations. The model consists of a coupled Ca release unit (CRU) network, which includes a SR domain and a myoplasm domain. Each CRU contains 10 L-type Ca channels and 100 ryanodine receptor channels, with individual channels simulated stochastically using a variant of Gillespie’s method, modified here to handle time-dependent transition rates. Both the SR domain and the myoplasm domain in each CRU are modeled by 5 × 5 × 5 voxels to maintain proper Ca diffusion. Advanced numerical algorithms implemented on graphical processing units were used for fast computational simulations. For a myocyte containing 100 × 20 × 10 CRUs, a 1-s heart time simulation takes about 10 min of machine time on a single NVIDIA Tesla C2050. Examples of simulated Ca cycling dynamics, such as Ca sparks, Ca waves, and Ca alternans, are shown.

Synchronization of weakly perturbed Markov chain oscillators

Rate processes are simple and analytically tractable models for many dynamical systems that switch stochastically between a discrete set of quasistationary states; however, they may also approximate continuous processes by coarse-grained, symbolic dynamics. In contrast to limit-cycle oscillators that are weakly perturbed by noise, in such systems, stochasticity may be strong, and topologies more complicated than a circle can be considered. Here we apply a second-order time-dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete-state oscillators coupled or driven through weakly time-dependent transition rates. We also describe a method of global control to optimize the response of the mean frequency in complex transition networks.

General no-go condition for stochastic pumping

The control of chemical dynamics requires understanding the effect of time-dependent transition rates between states of chemomechanical molecular configurations. Pumping refers to generating a net current, e.g., per period in the time dependence, through a cycle of consecutive states. The work of artificial machines or synthesized molecular motors depends on it. In this paper we give short and simple proofs of no-go theorems, some of which appeared before but here with essential extensions to non-Markovian dynamics, including the study of the diffusion limit. It allows to exclude certain protocols in the working of chemical motors where only the depth of the energy well is changed in time and not the barrier height between pairs of states. We also show how pre-existing steady state currents are, in general, modified with a multiplicative factor when this time dependence is turned on.

Generating function of spin correlation functions for kinetic Glauber–Ising model with time-dependent transition rates

The exact analytical solution in quadratures have been found for the generating function of spin correlation functions for the Reiss variant of the kinetic Ising model that is the extension of the Glauber–Ising model to account for the dependence of heat bath temperature on time.

Fluctuation theorem for birth–death or chemical master equations with time-dependent rates

For systems described by univariate birth–death or chemical master equations driven out of equilibrium by externally controlled time-dependent transition rates, a nonlinear fluctuation theorem is derived. For paradigmatic chemical reactions, this theorem acquires a particularly transparent form.

Memory effects and oscillations in single-molecule kinetics.

An exactly solvable model for single-molecule kinetics is suggested, based on the following assumptions: (i) A single molecule can exist in different chemical states and the random transitions from one chemical state to another can be described by a local master equation with time-dependent transition rates. (ii) Because of conformational and other intramolecular fluctuations the rate coefficients in the master equation are random functions of time; their stochastic properties are represented in terms of a set of control parameters. We assume that the fluctuating rate coefficients fulfill a separability condition, that is, they are made up of the multiplicative contributions of two factors: (a) a universal factor, which depends on the vector of control parameters and is the same for all chemical transformation processes and (b) process-dependent factors, which depend on the initial and final chemical states of the molecule but are independent of the control parameters. For systems with two chemical states the condition of separability is automatically fulfilled. We introduce an intrinsic time scale, which makes it possible to compute theoretically various experimental observables, such as the correlation functions of the fluorescent signal. We analyze the connections between the condition of separability and detailed balance, and discuss the possible cause of chemical oscillations in single molecule kinetics. We show that the intrinsic dynamics of the molecule, expressed by the fluctuations of the control parameters, may lead to damped oscillations of the correlation functions of the fluorescent signal. The influence of the random fluctuations on the control parameters may be described by a renormalized master equation with nonfluctuating apparent rate coefficients. The apparent rate coefficients do not have to obey a condition of detailed balance, even though the real rate coefficients do obey such a condition. It follows that the renormalized master equation may have damped oscillatory solutions.

Hybrid analysis of SGSPNs with time-dependent transition rates

For the class superposed generalized stochastic Petri nets (SGSPNs), we present a new analysis algorithm which combines the ideas of transient numerical analysis via uniformization and discrete event simulation. Simulation is used to determine the firing times of synchronized transitions describing interactions between the components of an SGSPN. Numerical analysis is used to compute the distribution of states between firing times of synchronized transitions. The approach avoids handling vectors of the size of the state space and can therefore be used for models with a huge state space which cannot be analyzed with numerical analysis techniques. On the other hand, the approach allows the computation of reliable estimates for results based on small probabilities or rare events which cannot be analyzed with simulation. As a byproduct of the handling of synchronized transitions using simulation, it is possible to extend the class of SGSPNs by allowing synchronized transitions with time-dependent transition rates.