Generalizations of the ‘Linear Chain Trick’: incorporating more flexible dwell time distributions into mean field ODE models

Ordinary differential equations models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models to yield much more general ODEs with phase-type distributed delays or dwell times.

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Building mean field ODE models using the generalized linear chain trick & Markov chain theory

Journal of Biological Dynamics ◽

10.1080/17513758.2021.1912418 ◽

2021 ◽

pp. 1-25

Author(s):

Paul J. Hurtado ◽

Cameron Richards

Keyword(s):

Markov Chain ◽

Linear Chain ◽

Mean Field ◽

Ode Models ◽

Markov Chain Theory ◽

Chain Theory

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ABSENCE OF PHASE TRANSITIONS ON TREE STRUCTURES

Modern Physics Letters B ◽

10.1142/s0217984993001971 ◽

1993 ◽

Vol 07 (29n30) ◽

pp. 1947-1950 ◽

Cited By ~ 4

Author(s):

RAFFAELLA BURIONI ◽

DAVIDE CASSI

Keyword(s):

Long Range ◽

Nearest Neighbor ◽

Linear Chain ◽

Mean Field ◽

Range Order ◽

The Other ◽

Long Range Order ◽

Field Phase ◽

Tree Structures ◽

Bethe Lattices

We rigorously prove that the correlation functions of any statistical model having a compact transitive symmetry group and nearest-neighbor interactions on any tree structure are equal to the corresponding ones on a linear chain. The exponential decay of the latter implies the absence of long-range order on any tree. On the other hand, for trees with exponential growth such as Bethe lattices, one can show the existence of a particular kind of mean field phase transition without long-range order.

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Fast and slow dynamics for classical and quantum walks on mean-field small world networks

Scientific Reports ◽

10.1038/s41598-019-55580-2 ◽

2019 ◽

Vol 9 (1) ◽

Author(s):

Andre M. C. Souza ◽

Roberto F. S. Andrade

Keyword(s):

Transition Probabilities ◽

Linear Chain ◽

Mean Field ◽

Small World ◽

Quantum Walks ◽

Polynomial Decay ◽

Small World Networks ◽

Quantum Random Walks ◽

Local Maxima ◽

Mathematical Properties

AbstractThis work investigates the dynamical properties of classical and quantum random walks on mean-field small-world (MFSW) networks in the continuous time version. The adopted formalism profits from the large number of exact mathematical properties of their adjacency and Laplacian matrices. Exact expressions for both transition probabilities in terms of Bessel functions are derived. Results are compared to numerical results obtained by working directly the Hamiltonian of the model. For the classical evolution, any infinitesimal amount of disorder causes an exponential decay to the asymptotic equilibrium state, in contrast to the polynomial behavior for the homogeneous case. The typical quantum oscillatory evolution has been characterized by local maxima. It indicates polynomial decay to equilibrium for any degree of disorder. The main finding of the work is the identification of a faster classical spreading as compared to the quantum counterpart. It stays in opposition to the well known diffusive and ballistic for, respectively, the classical and quantum spreading in the linear chain.

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Spectroscopies of rod- and pear-shaped nuclei in covariant density functional theory

International Journal of Modern Physics E ◽

10.1142/s0218301318300072 ◽

2018 ◽

Vol 27 (10) ◽

pp. 1830007 ◽

Cited By ~ 17

Author(s):

Pengwei Zhao ◽

Zhipan Li

Keyword(s):

Density Functional Theory ◽

Density Functional ◽

Linear Chain ◽

Mean Field ◽

Spectroscopic Properties ◽

Chain Structure ◽

Functional Theory ◽

Covariant Density Functional Theory ◽

Collective Hamiltonian ◽

Covariant Density

The spectroscopic properties play a crucial role in understanding the structure of nuclei, in particular, the shape and shape transitions of nuclei. In recent years, the exotic shapes of nuclear systems, such as the rod and pear shapes, have attracted a lot of attention. Covariant density functional theory (CDFT) has become a standard tool for nuclear structure calculations, and it provides a global and accurate description of nuclear ground states and excitations. In the present paper, we briefly review the recent progress in covariant density functional theory (DFT) for spectroscopic properties of the rod- and pear-shaped nuclei with the cranking calculations in a rotating mean field and the collective Hamiltonian method beyond mean field. The novel linear-chain structure of alpha clustering is discussed with the cranking approach, and low lying spectra of pear-shaped nuclei are illustrated with the quadrupole–octupole collective Hamiltonian.

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Mean-Field Bounds and the Classical Nature of Thermal Conduction in Nanofluids

Heat Transfer: Volume 1 ◽

10.1115/ht2008-56089 ◽

2008 ◽

Cited By ~ 2

Author(s):

Jacob Eapen

Keyword(s):

Thermal Conductivity ◽

Thermal Conduction ◽

Conduction Mechanism ◽

Linear Chain ◽

Mean Field ◽

Liquid Mixtures ◽

Large Set ◽

Inhomogeneous Systems ◽

Conduction Behavior ◽

Intermediate Configuration

The initial promise of nanofluids as an advanced, nanoengineered coolant has been tempered in the recent years by a conspicuous lack of consensus on its thermal conduction mechanism. Several new mechanisms have been hypothesized in the recent years to characterize the thermal conduction behavior in nanofluids. In this presentation, we show that a large set of nanofluid thermal conductivity data is enveloped by the well-known Hashin and Shtrikman (H-S) mean-field bounds for inhomogeneous systems. The thermal conductivity in nanofluids, therefore, is largely dependent on whether the nanoparticles stays dispersed in the base fluid, form linear chain-like configurations, or assume an intermediate configuration. The experimental data, which is strikingly analogous to those in most solid composites and liquid mixtures, provides a strong evidence for the classical nature of thermal conduction in nanofluids.

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Topological changes in telechelic micelles: flowers versus stars

10.1101/2021.08.13.455870 ◽

2021 ◽

Author(s):

Vladimir Baulin

Keyword(s):

Smart Materials ◽

Linear Chain ◽

Mean Field Theory ◽

Mean Field ◽

Polymer Chains ◽

Linear Polymers ◽

Stimuli Responsive ◽

Single Chain ◽

Low Concentrations ◽

Effective Attraction

Micellization and morphology of spherical telechelic micelles formed by tri-block copolymers with short solvophobic end blocks at low concentrations is discussed within scaling arguments and Single Chain Mean Field Theory (SCMFT). In ultra-dilute regime, individual telechelic polymer chains can exist in solution in two distinct states: open linear chain conformation with two free ends and closed loop conformation, when two ends are connected by the effective attraction between two solvophobic ends. At concentrations below gelation point, closed loops tend to form micelles comprised mostly of loops in flower-like micelles, while linear polymers in open conformations tend to form star-shaped aggregates with one hydrophobic dangling end. Resulting two kinds of micelles have remarkably different topology and dimensions, while the transition between them can be driven by the entropy, namely conformation changes between domination of the looped and linear conformations. As a result, the transition between two types of micelles lies in a narrow interaction parameters range. Thus, these topological micelles are very sensitive to the changes in external environment and they can serve as a very sensitive stimuli-responsive smart materials.

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From exact stochastic to mean-field ODE models: a new approach to prove convergence results

IMA Journal of Applied Mathematics ◽

10.1093/imamat/hxs001 ◽

2012 ◽

Vol 78 (5) ◽

pp. 945-964 ◽

Cited By ~ 10

Author(s):

P. L. Simon ◽

I. Z. Kiss

Keyword(s):

Mean Field ◽

New Approach ◽

Ode Models ◽

Convergence Results

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Fractionation of polymethylene chains: An electron crystallographic investigation

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100145315 ◽

1986 ◽

Vol 44 ◽

pp. 794-795

Author(s):

Douglas L. Dorset

Keyword(s):

Cross Section ◽

Binary Systems ◽

Linear Chain ◽

Pure Component ◽

Organic Substrates ◽

Crystallographic Investigation ◽

X Ray Diffraction ◽

X Ray ◽

Molecular Sizes ◽

The Stability

A variety of linear chain materials exist as polydisperse systems which are difficultly purified. The stability of continuous binary solid solutions assume that the Gibbs free energy of the solution is lower than that of either crystal component, a condition which includes such factors as relative molecular sizes and shapes and perhaps the symmetry of the pure component crystal structures.Although extensive studies of n-alkane miscibility have been carried out via powder X-ray diffraction of bulk samples we have begun to examine binary systems as single crystals, taking advantage of the well-known enhanced scattering cross section of matter for electrons and also the favorable projection of a paraffin crystal structure posited by epitaxial crystallization of such samples on organic substrates such as benzoic acid.

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Field ion microscope investigations of atomic processes at surfaces

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100153117 ◽

1989 ◽

Vol 47 ◽

pp. 228-229

Author(s):

G. L. Kellogg ◽

P. R. Schwoebel

Keyword(s):

Single Crystal ◽

Crystal Surface ◽

Linear Chain ◽

Atom Probe ◽

Nucleation And Growth ◽

Single Atom ◽

Initial Structure ◽

Atomic Processes ◽

Single Crystal Surfaces ◽

Field Ion Microscope

Although no longer unique in its ability to resolve individual single atoms on surfaces, the field ion microscope remains a powerful tool for the quantitative characterization of atomic processes on single-crystal surfaces. Investigations of single-atom surface diffusion, adatom-adatom interactions, surface reconstructions, cluster nucleation and growth, and a variety of surface chemical reactions have provided new insights to the atomic nature of surfaces. Moreover, the ability to determine the chemical identity of selected atoms seen in the field ion microscope image by atom-probe mass spectroscopy has increased or even changed our understanding of solid-state-reaction processes such as ordering, clustering, precipitation and segregation in alloys. This presentation focuses on the operational principles of the field-ion microscope and atom-probe mass spectrometer and some very recent applications of the field ion microscope to the nucleation and growth of metal clusters on metal surfaces.The structure assumed by clusters of atoms on a single-crystal surface yields fundamental information on the adatom-adatom interactions important in crystal growth. It was discovered in previous investigations with the field ion microscope that, contrary to intuition, the initial structure of clusters of Pt, Pd, Ir and Ni atoms on W(110) is a linear chain oriented in the <111> direction of the substrate.

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