Markoff Process Research Articles

When is a Markovian Semigroup Induced by a Semiflow?

, for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating the points of X and satisfying Zf2 = 2fZf. If either (1) there exists δ > 0 such that (Tt f)2∈ D if 0 ≤ t ≤δ for each f in this family; or (1′) for some core D′ of Z, g ∈ D′ implies g2∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition) of continuous functions φt from X into X — such that Ttf(x) = f(φt (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating family satisfying Zf2 = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions are redundant. If the functions φt are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined by = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup and is equivalently given by N = {f ∈ D| f2∈ D and Zf2 = 2fZf} = {f ∈ D| Tt(f2)-(Ttf)2 is o(t2) in C(X)}. In the general case N is an algebra closed under composition with any C1-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as a derivation) is a special case of a theorem for C2-functions φ which holds more generally for all f in d, viz., Z(φ[f] = (Zf) φ′[f] + ½ φ″[f], Provided Z is a local operator and D is an algebra. In this case the form itself enjoys the relation = φ′ [f] ψ′[g] , for C2functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization of Markovian semigroups that are groups: Let Tt = etz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||Tt - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a2) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the self-adjoint elements of A. The etz , -∞ < t < ∞, is a group of Markovian operators.

Numerical analysis of diffusion of a quasiparticle in a dynamically fluctuating medium based on the path integral method III

A numerical study is developed on the motion of a quasiparticle in a dynamically fluctuating medium based on the path integral method. The diffusion constant of the quasiparticle is calculated for one-dimensional systems with both site-energy (diagonal) and transfer-energy (off-diagonal) fluctuations obeying a general stochastic process, which is a superposition of an arbitrary number of two-state-jump Markoff processes with asymmetric probability distribution of stochastic variables. This general stochastic process enables us to link a two-state-jump Markoff process and a Gaussian Markoff process and describe the intermediate process. Effects of the asymmetric distribution on the diffusion constant is also clarified. It is found that off-diagonal fluctuations bring qualitatively different effects on the diffusion of the quasiparticle compared with the diagonal fluctuation case; for example, the diffusion constant has minima as a function of the amplitude of fluctuating transfer energy and has a minimum as a function of the asymmetry parameter. Adding diagonal fluctuations, the former minimum remains, whereas the latter minimum disappears for off-diagonal fluctuations consisting of a single two-state-jump Markoff process. This behavior of the diffusion constant strongly depends on N 1, the number of the constituent two-state-jump Markoff process.

Numerical analysis of diffusion of a quasiparticle in a dynamically fluctuating medium based on the path integral method II

A numerical study of the density of states and the diffusion constant is developed based on the path integral method for a stochastic Hamiltonian describing the motion of a quasiparticle in a dynamically fluctuating medium. A general stochastic process consisting of an arbitrary number of asymmetric two-state-jump Markoff processes is introduced to describe the site-energy fluctuations. This general process enables us to make an interpolation between a single two-state-jump Markoff process and a Gaussian Markoff process and also describe the intermediate process. Transient behavior of the density of states and the diffusion constant is observed for slow fluctuations as N changes from one to infinity, where N is the number of the constituent two-state-jump Markoff process. Effects of the asymmetric distribution, which may be ascribed to the temperature of the fluctuating medium, on the diffusion constant have also been clarified for several values of N. The results of the CPA and those of the perturbational method are compared with our results.

Numerical analysis of diffusion of a quasiparticle in a dynamically fluctuating medium based on the path integral method I

The path integral method is applied to the study of a stochastic Hamiltonian describing the motion of a quasiparticle in a dynamically fluctuating medium. Formulas to calculate the density of states and the diffusion constant of the quasiparticle are derived on the basis of the path integral method, which enables us to compute their exact values numerically for arbitrary fluctuations. Numerical calculation is demonstrated for one-dimensional systems with site-energy fluctuations obeying an asymmetric two-state-jump Markoff process. As far as one-particle quantities, such as the density of states, are concerned, no marked discrepancy between our results and those of the coherent potential approximation are observed. As for the diffusion constant, on the other hand, the two results exhibit qualitatively different behavior which is related to the localization of a quasiparticle, particularly for slow fluctuations.

Exact solution of a stochastic dimer problem with single-site energy modulation

In the framework of the generalized master equation we have investigated the absorption line shape and the dynamics of excitation in a simple two-site system with stochastic modulation of local energy at just one site. We have obtained new exact results for a general type of stochastic process, which is composed of M independent dichotomic Markoff processes. Particularly, our analysis yields an exact solution for the two frequently used process, that is, for the dichotomic Markoff process ( M = 1) and for the Gaussian Markoff process ( M→∞). Rigorous expressions are derived in the form of continued fractions and they are discussed under various physical conditions and compared with commonly used approximations (such as the noninteracting blip approximation or the white noise approximation).

Stochastic theory of a quasiparticle propagating in a dynamically fluctuating medium

A stochastic theory describing properties of a quasiparticle propagating in a dynamically fluctuating medium is developed on the basis of the single-site dynamical coherent potential approximation and the time-convolution equation formalism involving the partial cumulant expansion. The optical absorption spectra I(ω), the density of states N(ω) and the diffusion constant D are calculated for the quasiparticle suffering site-energy (diagonal) fluctuations consisting of an arbitrary number of two-state-jump Markoff processes with an asymmetric probability distribution of the stochastic variable. With these fluctuations, we have interpolated a two-state-jump Markoff process and a Gaussian Markoff process, and described the behavior of I(ω), N(ω) and D related to the intermediate processes. Effects of the asymmetric distribution, which may be ascribed to the temperature of the fluctuating medium, on I(ω), N(ω) and D have also been clarified, including a tendency for D to increase with increasing asymmetry.

From two-state jump to Gaussian stochastic processes

A stochastic process, which bridges the two-state jump Markoff process and the Gaussian process, is introduced and fundamental properties of the process are fully analyzed. The resulting expressions are combined with the “partial cumulant” expansion formula to give an exact power spectrum in the form of a continued fraction. A rigorous expression is also obtained for a relaxation function in an analytic form.

Effective text compression with simultaneous digram and trigram encoding

A new approach to data compression based on digram encoding is presented. Statistical analysis of the linguistic property of digram co-occurrence in textual material leads to a new, more effective simultaneous digram and trigram encoding technique. A combined list of digrams and trigrams eligible for encoding is constructed by an original scheme that replaces some of the frequently co-occurring encodable digrams with so-called master trigrams. Using Shannon's second order Markoff process approximation to English, conditions which the master trigrams should satisfy are formulated and then tested on four representative textual files. The performance of the proposed technique is experimentally examined and com pared with the performance of the conventional digram encod ing. It is found that the simultaneous digram and trigram encoding produces better results with regard to the measured compression ratios and use of processing time for encoding.

Exciton migration in a fluctuating medium II: Two-state jump markoff process

A statistical theory is developed on exciton migration in a stochastic environment. The theory is based on the time-convolution equation formalism with the partial cumulant expansion. For the two-state jump Markoff process, an expansion method from the atomic limit is also presented. The theory is further extended to include itinerant motion of excitons. Our formalism is valid for any time scale and can be used to treat the off-diagonal as well as the diagonal fluctuations. The resulting equations for the self-energy of excitons are solved to give absorption spectra and the density of states.

Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk

This paper considers the null hypothesis that the errors on a regression equation form a random walk. By using the standard Durbin-Watson assumptions, we derive three test statistics that are uniformly most powerful against the alternative hypothesis that the errors are being generated by the stationary first order Markoff process. Unfortunately, the tabulated lower and upper bounds are too wide apart and so we compare the powers of the three tests using simulated as well as economic data. It is then recommended that the Imhof routine should be attached to standard regression programs to calculate the exact limit of the Berenblut-Webb statistic.

Analysis of strain hardening creep under random uniaxial loading

Significant advances have taken place in developing methods of analysis of creep and creep rupture under varying stress conditions. However, the attention so far has been focussed on studying creep under deterministic stress histories. Time dependent stress and environmental history of a typical structure element is often definitely not known and it can, at best be described in statistical terms. Idealizing creep as a Markoff process and considering strain hardening theory, a simplified theory is proposed to characterize creep strain under random loading. The random loading refers to statically determinate uniaxial stressing, changing in a stepwise manner with time only and forms an ergodic process. In the light of the feasibility of characterizing the strain distribution under the random loading, an indication is given how a reliability based design approach to creep problems can be developed. The theory and its application are illustrated by a numerical example.

Stochastic theory for clustering of quenched-in vacancies—1. General mathematical properties

The problem of clustering of quenched-in vacancies into various types of extended defects is considered. A master equation for the evolution of the concentration of clusters of various sizes is written down with general transition rates. It is shown that this model represents a continuous time non-stationary Markoff process. A particular choice of transition rates corresponding to the formation of vacancy loops and stacking fault tetrahedra is considered in some detail. It is shown that this choice of transition rates allows us to obtain the solution for the concentration of the single vacancy units, and hence yields some information on the nucleation time. Further, the transition matrix becomes stationary and doubly stochastic due to the short time constant of the concentration of single vacancy units. This in turn leads to an unphysical stationary state. Finally we show how the rate equations for the irradiated situation can be written down and derive the phenomenological rate equations that are conventionally used.

Efficiency of Least-Squares Estimation of Linear Trend when Residuals Are Autocorrelated

first-order stationary Markoff process with zero mean and autocorrelation coefficient p, - 1 < p < 1, the greatest lower bound for the efficiency of the least-squares estimator of 8 (relative to the Gauss-Markoff estimator) over the interval 0;p < 1 is .753763. This compares with a greatest lower bound of .535898 for the relative efficiency of the Cochrane-Orcutt estimator of 38.

Recognition of generalized markoff processes and application to electrocardiography

The observed sequence of electrocardiographic R-R intervals have been analyzed by appropriate linear and nonlinear Markoff processes. Recent investigations by the authors have shown that appropriate comprehensive descriptions of the models are provided only by generalized or multi-step Markoff processes. In this paper the problem of recognizing sequences of states in a generalized Markoff process model is discussed and solutions provided.

Resonance fluorescence in modulated laser fields

On the basis of a stochastic model of Burshtein et al. (1967) which assumes the phases, frequencies and/or amplitudes of the radiation field to constitute a randomly time-varying jump-like Markoff process, a generalisation of Mollow's theory (1975) of the dynamic Stark effect is formulated leading to predictions for the resonant scattering spectrum for a two-level atom quite similar to those published in the work of Agarwal (1976), of Eberly (1976) and of Kimble et al. (1977) who use a different approach. Apparently, the general form of the resonant scattering spectrum in a modulated radiation field depends less strongly on the particular stochastic model of the laser light than might be expected.

On the Construction of a Time-Reversed Markoff Process

For a given Markoff process characterized by a set of transition probability densities there exists another process with time reversed (the retrodictive vs predictive process in the theory of measurements) such that any one of them multiplied by a single-event density may be symmetric with respect to an interchange of the events expressed as space-time variables, yielding a joint probability density. It is shown how this time-reversed process can be constructed by means of the generating operator of the associated evolution equation, and the basic properties with explicit applications to master equations and Fokker-Planck equations. Onsager's microscopic reversibility is reformulated on this basis. Possible sym. metries concerning time-correlation functions under the Markoffian law is summarized in comparion with the Kubo formula. In the application to Fokker-Planck equations, the Onsager-Machlup most probable paths are extended to general type of diffusion processes, and it is shown that the present method corresponds to a gauge transformation in dynamics of a charged particle which leaves its paths invariant. Reciprocity has been a fundamental subject in the theory of irreversible proces­ ses, since Onsager initiated the approach to the problem based on the consideration of microscopic reversibility.!) The Onsager relations for a linear dissipative sys­ tem in an external magnetic field @, given by L~,( -8) =L,~(@), have since been discussed in a number of papers. Kubo's general linear response theory•> among them provided an accurate statistical-mechanical foundation of these relations, ex­ pressing the microscopic reversibility in the time correlation functions between Hamiltonian-driven dynamical variables. Recently, the interests have been revised in connection with the statistical mechanics for non-equilibrium or open systems far from equilibrium conditions. Van Kampen discussed a possibility of extending the relations straightforwardly to the nonlinear regime of flux-force equations. 3> Another systematic approach has been developed by using equations of evolutions for probability densities based on the theory of Markoff processes. 4> An interesting finding in the latter approach has been that the microscopic reversibility as represented by a form of detailed balance condition (or its equivalent) is a situation rather restrictive for such non­ equilibrium states: There exist a number of important examples of violation such as complex optical systems and chemical reactions. A typical non-trivial system, a single-mode laser, is a special example for which the potential condition equiva­ lent to the reversibility is well satisfied, as discussed by Graham and Haken.

Interacting relativistic boson fields in the De Sitter universe with two space-time dimensions

The positive temperature Gibbs state of a scalar boson field with a relativistic local self-interaction in two space-time dimensional Minkowski universe as constructed in [1] is not relativistic invariant. We prove in this paper that the corresponding state in the De Sitter universe is actually relativistic invariant if the temperature is given byT=1/2πR whereR is the constant radius of curvature of the De Sitter universe. Moreover the construction gives that the Schwinger functions or imaginary time Wightman functions are the moments of a generalized Markoff process on the sphere of radiusR.

Relaxation theory and the stochastic Liouville equation

A new formalism of the theory of magnetic relaxation is given for systems where the random molecular motions are describable as a Markoff process. The master equation for the density matrix is derived form the stochastic Liouville equation. This method gives some new insight into the mechanism of relaxation and has several advantages over the conventional theories. The reason for this is that the correlation between the states of the lattice and the spins is taken into account. It is also possible to obtain the expression for T 1 under slow motion, while the conditions for its validity can be formulated. The description of the approach to a finite temperature is easily obtained by the use of the modified stochastic Liouville equation. This equation also simplifies the treatment of relaxation in the presence of a time-dependent field.

Statistical-dynamical model of wind-generated air/water interfaces

A new statistical-physical model is constructed for deep-ocean surface gravity waves in fully developed, directional wind-generated seas and analogous air/water interfaces. Basic assumptions are that surface elevation and velocity jointly form a Markoff process and that the Langevin equation describing surface motion at a point is that of a (linear) under-damped harmonic oscillator driven by a Gaussian wind-forcing term. Surface elevation and velocity are then jointly normal random processes, with explicit point covariance and spectra. From these, directional covariances and surface wavenumber spectra are obtained. Although the model is linear, and thus approximate at the high wavenumbers (breaking waves, etc.), agreement with experiment to date is good to excellent for both the second-order moments and (first-order) probability distributions. Furthermore, the model exhibits the required ν−4 falloff of directional spectra with high surface wavenumbers, and permits “dc” wave impulses, e.g., tsunamis, tidal bores, etc. Swell and capillary waves are handled similarly. These results are important in underwater acoustic scattering from water/air interfaces (e.g., sonar, communications applications), as well as for EM and acoustic scattering from air/water surfaces and oceanography. [Work supported by the Office of Naval Research, Code 412.]

Equilibrium states for classical systems

It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ϱ, and for the spin correlation functions σ, are essentially equivalent for all ϱ, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoff process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.