Expected Return Time Research Articles

Equilibrium return times of small fluctuating clusters and vacancies.

The expected return time of a fluctuating two-dimensional cluster or vacancy to a given configuration is studied in thermodynamic equilibrium. We define a family of bond-breaking models that preserve the number of particles. This family includes edge diffusion and surface diffusion inside vacancies in the limit of fast particle diffusion and slow attachment-detachment kinetics. Within the frame of these bond-breaking models, the expected return time is found to depend on the energies of the configurations and on the energies of the excited states formed by removing a single particle from the cluster. High- and low-temperature regimes are studied. We clarify the conditions under which the return time is a nonmonotonous function of temperature: a minimum is found when the energy obtained by the average over the excited states of the configuration weighted by their attachment probabilities is lower than the energy averaged over all states. In addition, we show that the optimal temperature at which the return time is minimum is shifted to a higher temperature as compared to the temperature at which the equilibrium probability is maximum. This shift is influenced by the average curvature of the cluster edge, and is therefore larger for vacancies.

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

IoT and BDA in the Brazilian future logistics 4.0 scenario

Abstract Paper aims This paper aims to identify the degree of interest and expected return time on investment in Internet of Things (IoT) and Big Data Analytics (BDA) technologies by Brazilian logistics companies. Originality A logistics system that meets the requirements imposed by Industry 4.0 is known as Logistics 4.0. According to the current scientific literature, IoT and BDA technologies are the most promising for the Logistics 4.0 applications. Studies on the adoption of these technologies in Brazil are still embryonic and this paper collaborates to narrow this knowledge gap. Research method Exploratory research was conducted using the quantitative approach. Main findings The evidence of the interest in investment in IoT and BDA by Brazilian logistics companies, confirming the current literature. Implications for theory and practice The identification of the intended practical applications for Iot and BDA, as well as the expected difficulties in the implementation of these Technologies.

Markov chain modeling for air pollution index based on maximum a posteriori method

Air pollution is a major environmental problem, which brings about a threat to human health and the natural environment. Thus, determination and assessment of the level of air pollution is an important component in monitoring of the air quality. This study involves estimating the transition probability matrix of the Markov chain model based on maximum a posteriori (MAP) method using hourly data of air pollution index (API). The API data has been collected from seven air-monitoring stations in peninsular Malaysia. The estimated transition probability matrix is used to determine the characteristics of air pollution such as the steady-state probability and the expected return time. The transition probabilities of the Markov chain are fitted based on maximum a posteriori method under three different priors, which are Dirichlet, Jeffreys, and uniform. The results found show that the maximum a posteriori method under the Dirichlet prior produced the most precise estimates as compared with the other priors. In addition, for the areas considered in the study, the moderate state is more persistent as opposed to unhealthy states indicating that the problem of air quality is not very serious. In general, this study could provide important implications for developing proper strategies for managing air quality and for improving public health.

THE REASONING OF CORRECTIVE COEFFICIENTS FOR DETERMINING THE TIME OF PROTECTIVE ACTION OF SELF-CONTAINED BREATHING APPARATUS FOR FIREFIGHTERS

According to regulatory requirements and rules of safety work, every firefighter who works in the SСBA while working in a gasified and smoke environment should monitor the readings of the pressure gauge and be able to calculate the air consumption (oxygen) and the duration of work in the SCBA. The main indicators that are taken into account during calculations of working time in gaseous and smoke environments are control pressure of air (oxygen) into the SCBA, in which it is necessary to go out to fresh air; working time in gassed and smoke environments; expected return time to fresh air. The supply of air will directly depend on, firstly, the volume of the cylinder (cylinders) and the working pressure, secondly from the working conditions and physiological features of the firefighter. Since, in determining the average work time of firefighters in SCBA in a gasified and smoked environment, according to the formula specified in the regulatory requirement, the capacity of the SCBA cylinder has not taken into account, the purpose of the work was to determine the correction coefficients, which would consider this. Considering the providing of fire rescue units of the SES of Ukraine with SCBA, we can conclude that the largest number of SCBA in the operative calculation, this is the SCBA of leading European firms, namely Drager and MSA AUER. The analysis of the load of the work of the firefighters in the gasified and smoked environment has shown that 50% is the work of the average degree of gravity corresponding to pulmonary ventilation of 40-liter per minute. Analysis of the technical characteristics of the SCBA of Drager and MSA AUER firms has shown that in this SCBA, the same performance of the pressure demand regulators 40-liter per minute and working pressure P = 300 bar. However, different capacities of cylinders 6, 6.8, 7 litres, respectively, which definitely affect the supply of air that is necessary for the work of the firefighter. Fire Safety, №34, 2019 65 Therefore, having performed the calculations of the average time of operation of the firefighters in SCBA link in the gassed and smoked environment in two formulas. The first one - according to the regulatory requirement defining such calculation, the second – according to Boyle–Mariotte law, which takes into account the cylinder capacity, working pressure and air consumption at medium load, we got the following data:  for cylinders of 6 l capacity, this difference is not significant (only 2 minutes), which is only 5% of the error, thus the correction coefficient to the base formula is not required;  for cylinders with a capacity of 6.8 litres this difference is significant (7 minutes) which is 20% of the error, respectively, the introduction of a correction coefficient + 7 minutes is proposed;  for cylinders of 7 litres, this difference is significant (8 minutes) which is 23% of the error, respectively, the introduction of a correction coefficient + 8 minutes is proposed for the base formula. Additionally is proposing, for SCBA of double-cylinder design, such as Drager PSS-5000 in the proposed formulas additionally average working time of the firefighters in the SCBA multiply by two. Thus, based on the above, in the practical calculations for the determination of the average time of operation of firefighters in a gasified and smoked environment, during using SCBA with cylinders of 6.8 and 7 litres, or a doublecylinder construction, it is proposed to apply correction coefficient to the base formula. However, for an SCBA with a cylinder capacity of 6 litres leave the formula unchanged. The introduction of the correction factors in the basic formula will enable firefighters units safely carry out intelligence during fire extinguishing, elimination of emergencies and their consequences, saving people and evacuation of material values from gassed and smoky premises.

Performance analysis of radio spectrum for cognitive radio wireless networks using discrete time Markov chain

Cognitive radio (CR) arises as an important technique for wireless network with the aim to provide better utilization of radio spectrum, which is being utilized sporadically in some areas. IEEE 802.22 wireless regional area network (WRAN) is the first standard for CR technology designed to opportunistically utilize the unused or under-utilized TV bands. A WRAN cell normally consists of a number of customer premises equipments (CPEs)/CR users and a base station (BS) having master/slave architecture. When a CPE is powered on, it first attempts to associate with the BS. In this paper, a discrete time Markov chain model (DTMC) is presented to show the association process of CPEs with the BS. To the best of our knowledge, this paper is the first in which DTMC Model is analyzed and investigated for WRAN. Using this model, various parameters such as association time, expected return time to the respective backoff stage, first passage time, etc., are derived. Finally, the evaluation results are provided to analyze the system's performance.

Expected return time to the initial state for biochemical systems with linear cyclic chains: unidirectional and bidirectional reactions

Biochemical systems are robust in nature. We define robustness of a biochemical system as the property where during time evolution, a closed system returns to its initial state. In this study, we propose some mathematical formulations to analyse the robustness of a closed biochemical system. We have provided a tentative guideline towards applying the theory to a non-closed system. We know that a biochemical system evolves with time as a continuous-time Markov process. When this Markov chain is irreducible, it can be proved theoretically that the system will always return to its initial state, and also the expected time of return can be determined. This return time depends upon the stationary probability distribution, which is determined as the solution of an eigenvalue equation \(xQ=0\) where Q is the transition rate matrix. We calculate this expected return time for five different closed systems: unidirectional cyclic linear chains, bidirectional cyclic linear chains and three real biological systems, and verify the theoretical results against the average return time obtained by stochastic simulation.

Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster’s Theorem

In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler's ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster's Theorem for the expected return time together with applications.

Site recurrence of open and unitary quantum walks on the line

We study the problem of site recurrence of discrete-time nearest-neighbor open quantum random walks (OQWs) on the integer line, proving basic properties and some of its relations with the corresponding problem for unitary (coined) quantum walks (UQWs). For both kinds of walks, our discussion concerns two notions of recurrence, one given by a monitoring procedure (Grünbaum et al. in Commun Math Phys 320:543–569, 2013; Lardizabal and Souza in J Stat Phys 159:772–796, 2015), and we study their similarities and differences. In particular, by considering UQWs and OQWs induced by the same pair of matrices, we discuss the fact that recurrence of these walks is related by an additive interference term in a simple way. Based on a previous result of positive recurrence, we describe an open quantum version of Kac’s lemma for the expected return time to a site.

Generalized Kac lemma for recurrence time in iterated open quantum systems

We consider recurrence to the initial state after repeated actions of a quantum channel. After each iteration a projective measurement is applied to check recurrence. The corresponding return time is known to be an integer for the special case of unital channels, including unitary channels. We prove that for a more general class of quantum channels the expected return time can be given as the inverse of the weight of the initial state in the steady state. This statement is a generalization of the Kac lemma for classical Markov chains.

Quantized recurrence time in unital iterated open quantum dynamics

The expected return time to the original state is a key concept characterizing systems obeying both classical or quantum dynamics. We consider iterated open quantum dynamical systems in finite dimensional Hilbert spaces, a broad class of systems that includes classical Markov chains and unitary discrete time quantum walks on networks. Starting from a pure state, the time evolution is induced by repeated applications of a general quantum channel, in each timestep followed by a measurement to detect whether the system has returned to the original state. We prove that if the superoperator is unital in the relevant Hilbert space (the part of the Hilbert space explored by the system), then the expectation value of the return time is an integer, equal to the dimension of this relevant Hilbert space. We illustrate our results on partially coherent quantum walks on finite graphs. Our work connects the previously known quantization of the expected return time for bistochastic Markov chains and for unitary quantum walks, and shows that these are special cases of a more general statement. The expected return time is thus a quantitative measure of the size of the part of the Hilbert space available to the system when the dynamics is started from a certain state.

Quantum Recurrence of a Subspace and Operator-Valued Schur Functions

A notion of monitored recurrence for discrete-time quantum processes was recently introduced in Grünbaum et al. (Commun Math Phys (2), 320:543–569, 2013) taking the initial state as an absorbing one. We extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension. The generating function approach leads to a connection with the well-known theory of operator-valued Schur functions. This is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in Grünbaum et al. (Commun Math Phys (2), 320:543–569, 2013). The spectral decomposition of the unitary step operator driving the evolution yields a spectral measure, which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent, and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time. This notion of subspace recurrence also links the concept of expected return time to an Aharonov–Anandan phase that, in contrast to the case of state recurrence, can be non-integer. Even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning, so that the averaged expected return time is always a rational number. Moreover, state recurrence can occasionally give higher return probabilities than subspace recurrence, a fact that reveals once more the counter-intuitive behavior of quantum systems. All these phenomena are illustrated with explicit examples, including as a natural application the analysis of site recurrence for coined walks.

Excursions of excited random walks on integers

Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter $\delta\in\mathbb R$. For recurrence/transience the critical threshold is $|\delta|=1$, for ballisticity it is $|\delta|=2$ and for diffusivity $|\delta|=4$. In this paper we establish a phase transition at $|\delta|=3$. We show that the expected return time of the walker to the starting point, conditioned on return, is finite iff $|\delta|>3$. This result follows from an explicit description of the tail behaviour of the return time as a function of $\delta$, which is achieved by diffusion approximation of related branching processes by squared Bessel processes.

Modeling the effects of eruptive and seismic activities on flank instability at Mount Etna, Italy

AbstractThe identification and evaluation of trigger mechanisms for volcano flank instabilities and/or collapse represent a key issue for risk assessment in densely populated volcanic areas, as well as in long‐distance settings, particularly in case of island or coastal volcanoes. Here we address quantitatively the effects of external (seismic) and inner (magmatic) forcing on the stress‐strain state associated to flank instabilities at Mount Etna (Sicily, southern Italy) by means of a 2‐D finite difference method numerical modeling. Modeled seismic actions include strong near‐field, strong far‐field, and low‐magnitude near‐field earthquakes. Magmatic actions consider the inner pressure changes induced by energetic lava fountains in the summit crater area and subvertical and oblique dike ascent below the summit area. Model results are validated in light of available monitoring data and recent eruptive activity. Numerical results show that the main strain effects are produced by high‐magnitude near‐field earthquakes (expected return time of ~103 yrs) and by vertical uprise of a magma dike below the volcano summit area. Maximum displacements in the order of tens of centimeters may involve the summit area, up to some 106 m3/m over some kilometers laterally. Stress releases up to 107 Pa may affect a limited portion of the magmatic conduit, thus favoring major effusive flank eruptions. Major catastrophic events, such as volcano flank collapse, should not be expected by applying, either individually or combined, the aforementioned actions.

An Abramov formula for stationary spaces of discrete groups

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

Fixation probability of beneficial mutations in a fluctuating population

We compute an accurate approximation to the probability of fixation for a beneficial mutation in a population fluctuating with a stationary distribution of population size. The population dynamics are described by the theta-logistic model with environmental variance, assuming that the population size is large enough to ignore demographic variance. We show that stochastic fluctuations of population size reduce the probability of fixation. However, it is not the magnitude of the population fluctuations per se that creates this reduction. Only the environmental variance has a substantial effect on the probability of fixation. The strength of density dependence (or expected return time to equilibrium) and the functional form of density-regulation, given by the parameter theta in the theta-logistic model, have little effect on the fixation probability. Effective population size based on harmonic mean population size will therefore underestimate the expected fixation rate of beneficial mutations in fluctuating populations.

Sharp error terms and neccessary conditions for exponential hitting times in mixing processes

We prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in α-mixing processes with exponential decay, ϕ-mixing processes with a summable function ϕ and for general ψ-mixing processes with a finite alphabet. In the first case the bound is uniform as a function of the measure of the event. In the last two cases the bound depends also on the time scale t. This allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time. A uniform bound is a consequence. We present an example that shows that this bound is sharp. We also prove that second moments are not necessary for having the exponential law. Moreover, we prove a necessary condition for having the exponential limit law.

Classes of Mixing Stable Processes

Every measurable stationary α-stable process with 0<α<2 can be related to a non-singular flow on a σ-finite measure space. We establish the relationship between properties of the flow and mixing of the stationary stable process. We provide the first example of a mixing stationary stable process corresponding to a conservative flow. We show further the connection between the expected return time of the flow to sets of finite positive measure and the mixing properties of the process.

On the limit behavior of certain quantities in a subcritical storage model

The limit behavior of the content of a subcriticai storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point (i 0,0) of the process. By showing that the expected return time to (i 0, 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (0,t) is then established by showing the asymptotic independence of these two.

On the limit behavior of certain quantities in a subcritical storage model

The limit behavior of the content of a subcriticai storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point (i0,0) of the process. By showing that the expected return time to (i0, 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (0,t) is then established by showing the asymptotic independence of these two.