Non-homogeneous Chain Research Articles

Non-homogeneous Chain of Harmonic Oscillators

Non-homogeneous Chain of Harmonic Oscillators

On the Computation of Some Interval Reliability Indicators for Semi-Markov Systems

In this paper, we computed general interval indicators of availability and reliability for systems modelled by time non-homogeneous semi-Markov chains. First, we considered duration-dependent extensions of the Interval Reliability and then, we determined an explicit formula for the availability with a given window and containing a given point. To make the computation of the window availability, an explicit formula was derived involving duration-dependent transition probabilities and the interval reliability function. Both interval reliability and availability functions were evaluated considering the local behavior of the system through the recurrence time processes. The results are illustrated through a numerical example. They show that the considered indicators can describe the duration effects and the age of the multi-state system and be useful in real-life problems.

Optimal novel approach for estimating the pavement transition probabilities used in Markovian prediction models

ABSTRACT An optimal novel approach is proposed to estimate the transition probabilities associated with both homogeneous and non-homogeneous Markov chains. The approach applies an exhaustive optimisation technique to search for the optimal transition probabilities associated with minimal sum of squared errors (SSE), wherein the error defined as the difference between predicted and observed pavement condition ratings. Three state transitions are allowed in constructing the relevant transition probability matrix (TPM). In the homogenous chain, the approach yields one optimal TPM applicable to an analysis period of (n) years. Whereas, one distinct optimal TPM can be derived for each year if non-homogenous chain is deployed. A sequential iterative optimisation approach is proposed wherein the optimal TPM for a given year becomes the input for the subsequent year. Sample results are presented for two projects (A & B) with superior and inferior performances, respectively. The sample results indicate that the non-homogenous chain provided significant reduction in the SSE compared to the homogeneous one. However, the use of 10 condition states instead of 5 resulted in moderate reduction in the SSE considering both homogeneous and non-homogeneous Markov chains. Also, the results indicate that the use of three state transitions made significant impact when deploying 10 condition states instead of 5 especially in the case of inferior performance.

ON DISCRETE STOCHASTIC PROCESSES WITH DISJUNCTIVE OUTCOMES

AbstractWe introduce a class of discrete-time stochastic processes, called disjunctive processes, which are important for reliable simulations in random iteration algorithms. Their definition requires that all possible patterns of states appear with probability 1. Sufficient conditions for nonhomogeneous chains to be disjunctive are provided. Suitable examples show that strongly mixing Markov chains and pairwise independent sequences, often employed in applications, may not be disjunctive. As a particular step towards a general theory we shall examine the problem arising when disjunctiveness is inherited under passing to a subsequence. An application to the verification problem for switched control systems is also included.

Phenomenological mapping and dynamical absorptions in chain systems with multiple degrees of freedom

Using Mihailo Petrović's theory of mathematical phenomenology elements, phenomenological mapping in vibrations, signals, resonance and dynamical absorptions in models of dynamics of chain systems – the abstractions of different real dynamics of a chain system are identified and presented. Using a mathematical description of a chain mechanical system with a finite number of mass particles coupled by linear elastic springs and a finite number of degrees of freedom expressed by corresponding generalized independent coordinates, translator displacements and corresponding analysis of solutions for a free and forced vibrations series of multi-frequency regimes and resonant states as well as dynamical absorption states are identified. Using mathematical analogy and phenomenological mapping, analyses of the dynamics of other chain models are made. Phenomenological mapping is used to explain dynamics in systems with multiple deformable bodies (beams, plates, membranes or belts) through resonance and dynamical absorptions in the system and transfer of mechanical energies between bodies. Amplitude-frequency graphs for homogeneous and non-homogeneous chain systems are presented for a system with 11 degrees of freedom. Expressions for generalized coordinates of a chain non-homogeneous system in resonance regimes for a general case are derived. A theorem is defined and proven.

Spectral origin of non-Markovian open-system dynamics: A finite harmonic model without approximations

Non-homogeneous chain environments (e.g. segmented ion traps) are investigated through an exact diagonalization approach. Different spectral densities, including band-gaps, can be engineered to separately assess memory effects. Environment non-Markovianity is quantified with recently introduced measures of information flow-back and non-divisibility of the system dynamical map. By sweeping the bath spectrum via tuning of the system frequency we show strongest memory effect at band-gap edges and provide an interpretation based on energy flow between system and environment. A system weakly coupled to a stiff chain ensures a Markovian dynamics, while the size of the environment as well as the local density of modes are not substantial factors. We show an opposite effect when increasing the temperature inside or outside the spectral band-gap. Further, non-Markovianity arises for larger (negative and positive) powers of algebraic spectral densities, being the Ohmic case not always the most Markovian one.

Application of the MLD “toy” model to extensional flows of broadly polydisperse linear polymers: Part I – Model development

When no relative tube stretch is present, all topological entanglements are equivalent in their impact on the dynamics of nearby, primary chain segments independent of molecular weight, molecular architecture, position along the chain or flow conditions. In this no relative tube stretch limit, the average lifetime of an entanglement is its only distinguishing feature [1]. However, when tube stretch commences and non-homogeneous chain tension is present, two rational bases for entanglement discrimination manifest themselves. One is based upon the entanglement orientational relaxation time relative to the test chain stretch relaxation time and the other on the relative tension (incremental moduli) of the test chain and the entangling chain. In this paper we focus on the first criteria for entanglement discrimination and demonstrate its fundamental importance in predicting the transient extensional stresses of broadly polydisperse polymer systems typical of commercial resins. Entanglement microstructure modification, i.e. reduction of the number of viable “stretch” entanglements for a given MWD component, determined by the underlying MWD is critical to properly predicting transient and steady material properties in extensional (stretching) flows (Mishler et al. (2000) [2]; Mishler (2001) [3]). A new molecular model is created incorporating these new ideas along with the new extensional rheometry experimental results of Auhl et al. into the polydisperse MLD model structure at the “toy” level [1–4]. Multiple fundamental length scales in polydisperse melts are predicted by the new model, a fundamental paradigm shift from the original Doi–Edwards model. An alternative model for diluted tube stretch and orientation proposed by Auhl et al. has also been cast into the generalized MLD format for polydisperse systems [4]. Predictions of the Auhl et al. model and the new diluted stretch tube MLD polydispersity model proposed in this paper are in close accord for model bidisperse systems but disagree for systems with general polydispersity such as the data of Minegishi et al. [5–7]. The origins of the differences between the two models can be traced to the distinction between mean field tube descriptions of the entanglement effect used by Auhl et al. and the pair wise discrete description of entanglement constraints used in the MLD model. These seemingly subtle differences in interpretation of the entanglement effect become important when stress is calculated in polydisperse systems.

Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators

We study different types of asymptotic behaviour in the set of (infinite dimensional) nonhomogeneous chains of stochastic operators acting on L1(μ) spaces. In order to examine its structure we consider different norm and strong operator topologies. To describe the nature of the set of nonhomogeneous chains of Markov operators with a particular limit behaviour we use the category theorem of Baire. We show that the geometric structure of the set of those stochastic operators which have asymptotically stationary density differs depending on the considered topologies.

Occupancy distributions in Markov chains via Doeblin's ergodicity coefficient

We state and prove new properties about Doeblin's ergodicity coefficient for finite Markov chains. We show that this coefficient satisfies a sub-multiplicative type inequality (analogous to the Markov-Dobrushin's ergodicity coefficient), and provide a novel but elementary proof of Doeblin's characterization of weak-ergodicity for non-homogeneous chains. Using Doeblin's coefficient, we illustrate how to approximate a homogeneous but possibly non-stationary Markov chain of duration $n$ by independent and short-lived realizations of an auxiliary chain of duration of order $\ln (n)$. This leads to approximations of occupancy distributions in homogeneous chains, which may be particularly useful when exact calculations via one-step methods or transfer matrices are impractical, and when asymptotic approximations may not be yet reliable. Our findings may find applications to pattern problems in Markovian and non-Markovian sequences that are treatable via embedding techniques.

The crossing barrier of a non-homogeneous semi-Markov chain

In this paper, we derive the probability distribution function of the first entry time in a time varying subset of the phase space of a non-homogeneous semi-Markov chain. Deterministic and stochastic time varying subsets are considered and results with a ready algorithmic implementation are provided. A possible application is debated by defining and analysing deterministic and stochastic time variable level crossing order for semi-Markov chains.

A note on a variation of Doeblin's condition for uniform ergodicity of Markov chains

We introduce a simple variation of Doeblin's condition, Condition D*, that assures the uniform ergodicity of a Markov chain. It is also shown that for non-homogeneous chains our conditions are equivalent to Dobrushin's weak ergodic coefficient.

Lattice theory of solvation in macromolecular fluids. III. Monte Carlo simulations

Monte Carlo simulations are used to calculate the energy, free energy, and entropy of solvation in a lattice model of polymer host. The solute particle interacts with specific beads in the host chain at nearest neighbor sites. The results are used to check the accuracy of the quasichemical approximation (QCA) recently used [Olender and Nitzan, J. Chem. Phys. 101, 2338 (1994)] to study ion solvation and ion pair dissociation in polymer hosts. For noninteracting chains we find that the QCA is very accurate when the solvent consists of homogeneous chains (all beads interact equally with the impurity), and give errors of up to 20% when nonhomogeneous chains (with some of the beads interacting with the impurity) are used. For interacting chains this trend is reversed and the QCA works better for nonhomogeneous chains. Deviations of the QCA prediction from the ‘‘exact’’ numerical results are traced to three-body and higher order correlations. The success of the QCA for interacting solvents of nonhomogeneous chains is associated with cancellation of opposing effects of such correlations.

Occupation times for nonhomogeneous Markov chains with discrete time

The results, published by the author in Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., Vol. 119, on finite-dimensional distributions of the field of occupation time for Markov chains with discrete time are carried over to the nonhomogeneous case. There is given a scheme for the description of conditional finite-dimensional distributions of such a random field, generalizing to nonhomogeneous chains the previously obtained analogue of the “Ray description” of the local time for a one-dimensional Brownian motion.

Markovian Dynamics of Simple and Complex Desert Plant Communities

Many simple plant communities in extremely arid regions of the Sonoran Desert consist essentially of two shrub species, Ambrosia dumcsa and Larrea tridentata. These species exhibit completely different recruitment patterns and rates of population turnover. In these communities, the majority of space is devoid of perennial plant cover, and Ambrosia readily colonize this open space. Recruitment of Larrea occurs less frequently and is largely limited to areas beneath the canopies of large, older Ambrosia individuals. The dynamics of these systems can be modeled as Markov chains. Probabilities of transitions among states (open space, space covered by Ambrosia canopies, and space covered by Larrea canopies) are computed from data on spatial and temporal patterns of recruitment and mortality. Although the stationary distributions of states predicted by a homogeneous Markov chain are similar to those observed in undisturbed vegetation at two sites, a nonhomogeneous chain more accurately describes vegetational change following disturbance. The dynamics of species-rich plant communities in less xeric parts of the Sonoran Desert appear to be qualitatively similar. One shrub species in particular, A. deltoidea, is capable of colonizing open space, whereas for most other shrub, tree, and succulent species, recruitment is largely limited to areas beneath canopies of other plants, especially canopies of A. deltoidea. Although seemingly more complex, a digraph model showing the transitions among states possesses the same fundamental characteristics as does a digraph of the simple two-species system. A diverse set of mechanisms underlie these dynamics and include, in part, factors that influence seed distributions, germination success, and patterns of post-germination mortality. Even in the simple two-species systems, it is likely that no individual mechanism such as competition can be singled out as the predominant determinant of dynamics and, hence, of the patterns of community structure.

The exchangeable sigma-field of markov chains

Blackwell and Freedman [2] proved that the exchangeable σ-field ℰ of a homogeneous recurrent Markov chain is atomic. If the chain is finite, the atoms can be found explicitly by means of an algorithm given below. The approach in [2] cannot be extended to a non-homogeneous chain, but a description of ℰ can be obtained in this case by using coupling methods, provided the chain is finite and satisfies certain conditions.

Convergence of non-homogenous stochastic chains. II

AbstractEarlier study by one of the authors on the convergence of non-homogeneous stochastic chains is continued here. Several results more general than those considered earlier relating convergence of such chains to ergodicity of certain smaller chains are presented.

Geometrical aspects of the theory of non-homogeneous Markov chains

AbstractA geometrical representation of the transition matrices of a non-homogeneous chain with N states, in terms of certain convex subsets of , is used to describe aspects of the chain. For example, an important theorem of Cohn on the structure of the tail σ-field is a simple corollary. The embedding problem is shown to be entirely geometrical in character. The representation extends to Markov processes on quite general state spaces, and the tail is then represented by the projective limit of these convex sets.

Chain orientation and crystallization

Abstract

Some theorems concerning the law of large numbers for non-homogeneous Markoff chains

Some theorems concerning the law of large numbers for non-homogeneous Markoff chains

Weak ergodicity in non-homogeneous Markov chains

ABSTRACTThe paper deals with weak ergodicity, i.e. the tendency for a chain to ‘forget’ the distant past. This may occur in non-homogeneous chains even if the probabilities of being in a given state do not tend to a limit as the number of trials increases.Investigation of the general conditions of weak ergodicity leads to the definition of a special subclass of regular matrices. Conditions for chains involving regular matrices are also given (§§2,3).Section 4 discusses similarities in asymptotic behaviour between chains whose transition matrices differ only by small amounts.