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Journal articles on the topic 'Continuum (Mathematics) Dynamics'

Author: Grafiati

Published: 4 June 2021

Last updated: 13 February 2022

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1

Delgado–Buscalioni, Rafael, and Peter V. Coveney. "Hybrid molecular–continuum fluid dynamics." Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 362, no. 1821 (June 3, 2004): 1639–54. http://dx.doi.org/10.1098/rsta.2004.1401.

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2

Hudson, Thomas, Patrick van Meurs, and Mark Peletier. "Atomistic origins of continuum dislocation dynamics." Mathematical Models and Methods in Applied Sciences 30, no. 13 (December 15, 2020): 2557–618. http://dx.doi.org/10.1142/s0218202520500505.

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This paper focuses on the connections between four stochastic and deterministic models for the motion of straight screw dislocations. Starting from a description of screw dislocation motion as interacting random walks on a lattice, we prove explicit estimates of the distance between solutions of this model, an SDE system for the dislocation positions, and two deterministic mean-field models describing the dislocation density. The proof of these estimates uses a collection of various techniques in analysis and probability theory, including a novel approach to establish propagation-of-chaos on a spatially discrete model. The estimates are non-asymptotic and explicit in terms of four parameters: the lattice spacing, the number of dislocations, the dislocation core size, and the temperature. This work is a first step in exploring this parameter space with the ultimate aim to connect and quantify the relationships between the many different dislocation models present in the literature.

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3

Kato, Hisao. "Continuum-Wise Expansive Homeomorphisms." Canadian Journal of Mathematics 45, no. 3 (June 1, 1993): 576–98. http://dx.doi.org/10.4153/cjm-1993-030-4.

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AbstractThe notion of expansive homeomorphism is important in topological dynamics and continuum theory. In this paper, a new kind of homeomorphism will be introduced and studied, namely the continuum-wise expansive homeomorphism. The class of continuum-wise expansive homeomorphisms is much larger than the one of expansive homeomorphisms. In fact, the class of continuum-wise expansive homeomorphisms contains many important homeomorphisms which often appear in "chaotic" topological dynamics and continuum theory, but which are not expansive homeomorphisms. For example, the shift maps of Knaster's indecomposable chainable continua are continuum-wise expansive homeomorphisms, but they are not expansive homeomorphisms. Also, there is a continuum-wise expansive homeomorphism on the pseudoarc. We study several properties of continuum-wise expansive homeomorphisms. Many theorems concerning expansive homeomorphisms will be generalized to the case of continuum-wise expansive homeomorphisms.

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4

Aidagulov, R. R., and M. V. Shamolin. "General spectral approach to the dynamics of continuum." Journal of Mathematical Sciences 154, no. 4 (October 2008): 502–22. http://dx.doi.org/10.1007/s10958-008-9192-2.

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5

Xiang, Yang, and Xiaohong Zhu. "A continuum model for the dynamics of dislocation arrays." Communications in Mathematical Sciences 10, no. 4 (2012): 1081–103. http://dx.doi.org/10.4310/cms.2012.v10.n4.a3.

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6

Finkelshtein, Dmitri L., Yuri G. Kondratiev, and Maria João Oliveira. "Glauber Dynamics in the Continuum via Generating Functionals Evolution." Complex Analysis and Operator Theory 6, no. 4 (July 14, 2011): 923–45. http://dx.doi.org/10.1007/s11785-011-0170-1.

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7

Wei, Chaozhen, Luchan Zhang, Jian Han, David J. Srolovitz, and Yang Xiang. "Grain Boundary Triple Junction Dynamics: A Continuum Disconnection Model." SIAM Journal on Applied Mathematics 80, no. 3 (January 2020): 1101–22. http://dx.doi.org/10.1137/19m1277722.

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8

XU, Z., R. C. PICU, and J. FISH. "HIGHER ORDER CONTINUUM WAVE EQUATION CALIBRATED ON LATTICE DYNAMICS." International Journal of Computational Engineering Science 05, no. 03 (September 2004): 557–73. http://dx.doi.org/10.1142/s1465876304002563.

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9

Campos, J., E. N. Dancer, and R. Ortega. "Dynamics in the neighbourhood. of a continuum of fixed points." Annali di Matematica Pura ed Applicata 180, no. 4 (January 2002): 483–92. http://dx.doi.org/10.1007/s102310100024.

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10

Kondratiev, Yuri G., Oleksandr V. Kutoviy, and Eugene W. Lytvynov. "Diffusion approximation for equilibrium Kawasaki dynamics in continuum." Stochastic Processes and their Applications 118, no. 7 (July 2008): 1278–99. http://dx.doi.org/10.1016/j.spa.2007.09.001.

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11

Wilchinsky, Alexander V., and Daniel L. Feltham. "A continuum anisotropic model of sea-ice dynamics." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 460, no. 2047 (July 8, 2004): 2105–40. http://dx.doi.org/10.1098/rspa.2004.1282.

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12

Davydov, I. A., V. N. Piskunov, R. A. Veselov, B. L. Voronin, D. А. Demin, А. М. Petrov, N. V. Nevmerzhitskiy, and V. N. Sofronov. "Cluster dynamics method for simulation of dynamic processes of continuum mechanics." Computational Materials Science 49, no. 1 (July 2010): S32—S36. http://dx.doi.org/10.1016/j.commatsci.2010.02.043.

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13

Savu, Anamaria. "Hydrodynamic scaling limit of continuum solid-on-solid model." Journal of Applied Mathematics 2006 (2006): 1–37. http://dx.doi.org/10.1155/jam/2006/69101.

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A fourth-order nonlinear evolution equation is derived from a microscopic model for surface diffusion, namely, the continuum solid-on-solid model. We use the method developed by Varadhan for the computation of the hydrodynamic scaling limit of nongradient models. What distinguishes our model from other models discussed so far is the presence of two conservation laws for the dynamics in a nonperiodic box and the complex dynamics that is not nearest-neighbor interaction. Along the way, a few steps have to be adapted to our new context. As a byproduct of our main result, we also derive the hydrodynamic scaling limit of a perturbation of the continuum solid-on-solid model, a model that incorporates both surface diffusion and surface electromigration.

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14

Finkelshtein, Dmitri, Yuri Kondratiev, Oleksandr Kutoviy, and Elena Zhizhina. "An approximative approach for construction of the Glauber dynamics in continuum." Mathematische Nachrichten 285, no. 2-3 (October 31, 2011): 223–35. http://dx.doi.org/10.1002/mana.200910248.

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15

Bretti, Gabriella, Ciro D’Apice, Rosanna Manzo, and Benedetto Piccoli. "A continuum-discrete model for supply chains dynamics." Networks & Heterogeneous Media 2, no. 4 (2007): 661–94. http://dx.doi.org/10.3934/nhm.2007.2.661.

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16

Krause, Andrew L., Dmitry Beliaev, Robert A. Van Gorder, and Sarah L. Waters. "Bifurcations and Dynamics Emergent From Lattice and Continuum Models of Bioactive Porous Media." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1830037. http://dx.doi.org/10.1142/s0218127418300379.

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We study the dynamics emergent from a two-dimensional reaction–diffusion process modeled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with the evolution of the local cell density depending on a coupled quasi-static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long-time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction–diffusion mechanism mediated by quasi-static fluid flow.

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17

ALBEVERIO, SERGIO, and WOLFGANG ALT. "STOCHASTIC DYNAMICS OF VISCOELASTIC SKEINS: CONDENSATION WAVES AND CONTINUUM LIMITS." Mathematical Models and Methods in Applied Sciences 18, supp01 (August 2008): 1149–91. http://dx.doi.org/10.1142/s0218202508002991.

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Skeins (one-dimensional queues) of migrating birds show typical fluctuations in swarm length and frequent events of "condensation waves" starting at the leading bird and traveling backward within the moving skein, similar to queuing traffic waves in car files but more smooth. These dynamical phenomena can be fairly reproduced by stochastic ordinary differential equations for a "multi-particle" system including the individual tendency of birds to attain a preferred speed as well as mutual interaction "forces" between neighbors, induced by distance-dependent attraction or repulsion as well as adjustment of velocities. Such a one-dimensional system constitutes a so-called "stochastic viscoelastic skein." For the simple case of nearest neighbor interactions we define the density between individualsu = u(t, x) as a step function inversely proportional to the neighbor distance, and the velocity function v = v(t, x) as a standard piecewise linear interpolation between individual velocities. Then, in the limit of infinitely many birds in a skein of finite length, with mean neighbor distance δ converging to zero and after a suitable scaling, we obtain continuum mass and force balance equations that constitute generalized nonlinear compressible Navier–Stokes equations. The resulting density-dependent stress functions and viscosity coefficients are directly derived from the parameter functions in the original model. We investigate two different sources of additive noise in the force balance equations: (1) independent stochastic accelerations of each bird and (2) exogenous stochastic noise arising from pressure perturbations in the interspace between them. Proper scaling of these noise terms leads, under suitable modeling assumptions, to their maintenance in the continuum limit δ → 0, where they appear as (1) uncorrelated spatiotemporal Gaussian noise or, respectively, (2) certain spatially correlated stochastic integrals. In both cases some a priori estimates are given which support convergence to the resulting stochastic Navier–Stokes system. Natural conditions at the moving swarm boundaries (along characteristics of the hyperbolic system) appear as singularly perturbed zero-tension Neumann conditions for the velocity function v. Numerical solutions of this free boundary value problem are compared to multi-particle simulations of the original discrete system. By analyzing its linearization around the constant swarm state, we can characterize several properties of swarm dynamics. In particular, we compute approximating values for the averaged speed and length of typical condensation waves.

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18

Zhang, Yan Ming, Hong Ling Ye, Yao Ming Li, and Yun Kang Sui. "The Research of Optimization Algorithm of Dynamic Topology Optimization Model of Continuum Structure Based on the ICM Method." Applied Mechanics and Materials 380-384 (August 2013): 1804–7. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1804.

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In this paper, we mainly focus on the structural optimal design of dynamics for continuum structures, and aim at constructing the topological optimal formulation by using the ICM (Independent, Continuum and Mapping) method, which is considering weight as objective function and fundamental eigenfrequency as constraint. The local model is removed by selecting suitable filter function. And two algorithms, dual sequential quadratic programming (DSQP) and global convergent method of moving asymptotes (GCMMA) algorithm, were used to solve the mathematic optimal model. Finally, numerical example is provided to demonstrate the validity and effectiveness of the ICM method and compare the optimization results of two optimization algorithms. The results show that both optimization algorithms can solve the mathematics optimization model effectively.

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19

DEVANEY, ROBERT L., XAVIER JARQUE, and MÓNICA MORENO ROCHA. "INDECOMPOSABLE CONTINUA AND MISIUREWICZ POINTS IN EXPONENTIAL DYNAMICS." International Journal of Bifurcation and Chaos 15, no. 10 (October 2005): 3281–93. http://dx.doi.org/10.1142/s0218127405013885.

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In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions Eλ(z) = λez where λ ∈ ℂ in the special case when λ is a Misiurewicz parameter, so that the Julia set of these maps is the entire complex plane. These invariant sets consist of points that share the same itinerary under iteration of Eλ. Previously, the only known types of such invariant sets were either simple hairs that extend from a definite endpoint to ∞ in the right half plane or else indecomposable continua for which a single hair accumulates everywhere upon itself. One new type of invariant set that we construct in this paper is an indecomposable continuum in which a pair of hairs accumulate upon each other, rather than a single hair having this property. The second type consists of an indecomposable continuum together with a completely separate hair that accumulates on this continuum.

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20

Kim, Jinkyu, Gary F. Dargush, and Young-Kyu Ju. "Extended framework of Hamilton’s principle for continuum dynamics." International Journal of Solids and Structures 50, no. 20-21 (October 2013): 3418–29. http://dx.doi.org/10.1016/j.ijsolstr.2013.06.015.

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21

Gilson, Michael K., J. Andrew McCammon, and Jeffry D. Madura. "Molecular dynamics simulation with a continuum electrostatic model of the solvent." Journal of Computational Chemistry 16, no. 9 (September 1995): 1081–95. http://dx.doi.org/10.1002/jcc.540160904.

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22

Fogolari, F., G. Esposito, P. Viglino, and H. Molinari. "Molecular mechanics and dynamics of biomolecules using a solvent continuum model." Journal of Computational Chemistry 22, no. 15 (2001): 1830–42. http://dx.doi.org/10.1002/jcc.1134.

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23

GIUPPONI, GIOVANNI, GIANNI DE FABRITIIS, and PETER V. COVENEY. "A COUPLED MOLECULAR-CONTINUUM HYBRID MODEL FOR THE SIMULATION OF MACROMOLECULAR DYNAMICS." International Journal of Modern Physics C 18, no. 04 (April 2007): 520–27. http://dx.doi.org/10.1142/s0129183107010759.

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We describe a hybrid simulation method that captures the combined effects of molecular and hydrodynamic forces which influence macromolecules in solution. In this method, the solvent contribution is accounted for implicitly as the Navier-Stokes equations are solved on a grid using a finite volume method, while we use coarse-grained molecular dynamics to describe the macromolecule. The two systems are coupled by a dissipative Stokesian force. We show that our method correctly captures the hydrodynamically enhanced self-diffusion of a single monomer for different fluids and grid sizes. Moreover, the monomer diffusion does not depend on the monomer mass for the mass range used, as postulated by polymer dynamics theories. We also show that the dynamical properties of the chain do not depend on the grid size a when the chain radius of gyration Rg ≫ a.

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24

Nejneru, Carmen, Anca Nicuţă, Boris Constantin, Liliana Rozemarie Manea, Mirela Teodorescu, and Maricel Agop. "Dynamics Control of the Complex Systems via Nondifferentiability." Journal of Applied Mathematics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/137056.

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A new topic in the analyses of complex systems dynamics, considering that the movements of complex system entities take place on continuum but nondifferentiable curves, is proposed. In this way, some properties of complex systems (barotropic-type behaviour, self-similarity behaviour, chaoticity through turbulence and stochasticization, etc.) are controlled through nondifferentiability of motion curves. These behaviours can simulate the standard properties of the complex systems (emergence, self-organization, adaptability, etc.).

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25

Wang, Min, and Jun Wang. "Multiscale volatility duration characteristics on financial multi-continuum percolation dynamics." International Journal of Modern Physics C 28, no. 05 (March 20, 2017): 1750067. http://dx.doi.org/10.1142/s012918311750067x.

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A random stock price model based on the multi-continuum percolation system is developed to investigate the nonlinear dynamics of stock price volatility duration, in an attempt to explain various statistical facts found in financial data, and have a deeper understanding of mechanisms in the financial market. The continuum percolation system is usually referred to be a random coverage process or a Boolean model, it is a member of a class of statistical physics systems. In this paper, the multi-continuum percolation (with different values of radius) is employed to model and reproduce the dispersal of information among the investors. To testify the rationality of the proposed model, the nonlinear analyses of return volatility duration series are preformed by multifractal detrending moving average analysis and Zipf analysis. The comparison empirical results indicate the similar nonlinear behaviors for the proposed model and the actual Chinese stock market.

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26

Burago, N. G., A. B. Zhuravlev, and I. S. Nikitin. "Continuum Model and Method of Calculating for Dynamics of Inelastic Layered Medium." Mathematical Models and Computer Simulations 11, no. 3 (May 2019): 488–98. http://dx.doi.org/10.1134/s2070048219030098.

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27

COSCIA, V., and C. CANAVESIO. "FIRST-ORDER MACROSCOPIC MODELLING OF HUMAN CROWD DYNAMICS." Mathematical Models and Methods in Applied Sciences 18, supp01 (August 2008): 1217–47. http://dx.doi.org/10.1142/s0218202508003017.

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This paper deals with the mathematical modelling of crowd dynamics within the framework of continuum mechanics. The method uses the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients. The closures take into account movement in more than one space dimension, presence of obstacles, pedestrian strategies, and modelling of panic conditions. Numerical simulations of the initial-boundary value problems visualize the ability of the models to predict several interesting phenomena related to the complex system under consideration.

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28

Delgado-Buscalioni, Rafael, Peter V. Coveney, Graham D. Riley, and Rupert W. Ford. "Hybrid molecular-continuum fluid models: implementation within a general coupling framework." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 363, no. 1833 (July 20, 2005): 1975–85. http://dx.doi.org/10.1098/rsta.2005.1623.

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Over the past three years we have been developing a new approach for the modelling and simulation of complex fluids. This approach is based on a multiscale hybrid scheme, in which two or more contiguous subdomains are dynamically coupled together. One subdomain is described by molecular dynamics while the other is described by continuum fluid dynamics; such coupled models are of considerable importance for the study of fluid dynamics problems in which only a restricted aspect requires a fully molecular representation. Our model is representative of the generic set of coupled models whose algorithmic structure presents interesting opportunities for deployment on a range of architectures including computational grids. Here we describe the implementation of our HybridMD code within a coupling framework that facilitates flexible deployment on such architectures.

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29

Xu, Mei, and Ted Belytschko. "Conservation properties of the bridging domain method for coupled molecular/continuum dynamics." International Journal for Numerical Methods in Engineering 76, no. 3 (October 15, 2008): 278–94. http://dx.doi.org/10.1002/nme.2323.

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30

Simić, Srboljub S. "Shock structure in continuum models of gas dynamics: stability and bifurcation analysis." Nonlinearity 22, no. 6 (April 24, 2009): 1337–66. http://dx.doi.org/10.1088/0951-7715/22/6/005.

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31

Till, John, Vincent Aloi, and Caleb Rucker. "Real-time dynamics of soft and continuum robots based on Cosserat rod models." International Journal of Robotics Research 38, no. 6 (April 25, 2019): 723–46. http://dx.doi.org/10.1177/0278364919842269.

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The dynamic equations of many continuum and soft robot designs can be succinctly formulated as a set of partial differential equations (PDEs) based on classical Cosserat rod theory, which includes bending, torsion, shear, and extension. In this work we present a numerical approach for forward dynamics simulation of Cosserat-based robot models in real time. The approach implicitly discretizes the time derivatives in the PDEs and then solves the resulting ordinary differential equation (ODE) boundary value problem (BVP) in arc length at each timestep. We show that this strategy can encompass a wide variety of robot models and numerical schemes in both time and space, with minimal symbolic manipulation required. Computational efficiency is gained owing to the stability of implicit methods at large timesteps, and implementation is relatively simple, which we demonstrate by providing a short MATLAB-coded example. We investigate and quantify the tradeoffs associated with several numerical subroutines, and we validate accuracy compared with dynamic rod data gathered with a high-speed camera system. To demonstrate the method’s application to continuum and soft robots, we derive several Cosserat-based dynamic models for robots using various actuation schemes (extensible rods, tendons, and fluidic chambers) and apply our approach to achieve real-time simulation in each case, with additional experimental validation on a tendon robot. Results show that these models capture several important phenomena, such as stability transitions and the effect of a compressible working fluid.

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32

Hadjiconstantinou, Nicolas G., and Anthony T. Patera. "Heterogeneous Atomistic-Continuum Representations for Dense Fluid Systems." International Journal of Modern Physics C 08, no. 04 (August 1997): 967–76. http://dx.doi.org/10.1142/s0129183197000837.

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We present a formulation and numerical solution procedure for heterogeneous atomistic-continuum representations of fluid flows. The ingredients from atomistic and continuum perspectives are non-equilibrium molecular dynamics and spectral element, respectively; the matching is provided by a classical procedure, the Schwarz alternating method with overlapping subdomains. The technique is applied to microscale flow of a dense fluid (supercritical argon) in a complex two-dimensional channel.

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33

Alex Greaney, P., Lawrence H. Friedman, and D. C. Chrzan. "Continuum simulation of dislocation dynamics: predictions for internal friction response." Computational Materials Science 25, no. 3 (November 2002): 387–403. http://dx.doi.org/10.1016/s0927-0256(02)00242-2.

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34

Romeo, Maurizio. "Electromagnetic field-current coupling in rigid polarized conductors." Mathematics and Mechanics of Solids 23, no. 1 (September 13, 2016): 85–98. http://dx.doi.org/10.1177/1081286516666404.

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A continuum model, based on a theory of electromagnetic media with microstructure, is exploited to deal with rigid conductors endowed with polarization and magnetization. Charge carriers are considered as a continuum superimposed to the microstructured conductor where the density of bound charges depends on the internal degrees of freedom of the continuum particle. The non-linear dynamical model is formulated, deriving the mechanical balance laws that are coupled with the electromagnetic field equations. A reduction to the micropolar linear case is performed in order to analyze admissible solutions in the form of one-dimensional plane waves. Dispersion equations are derived for modes pertaining to longitudinal and transverse fields and the effects of conductivity and polarization are evidentiated. Polariton modes, arising from the dynamics of microdeformation, are also discussed.

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35

BAKER, GEORGE A., and JAMES P. HAGUE. "RISE OF THE CENTRIST: FROM BINARY TO CONTINUOUS OPINION DYNAMICS." International Journal of Modern Physics C 19, no. 09 (September 2008): 1459–75. http://dx.doi.org/10.1142/s0129183108013023.

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We propose a model that extends the binary "united we stand, divided we fall" opinion dynamics of Sznajd-Weron to handle continuous and multi-state discrete opinions on a linear chain. Disagreement dynamics are often ignored in continuous extensions of the binary rules, so we make the most symmetric continuum extension of the binary model that can treat the consequences of agreement (debate) and disagreement (confrontation) within a population of agents. We use the continuum extension as an opportunity to develop rules for persistence of opinion (memory). Rules governing the propagation of centrist views are also examined. Monte Carlo simulations are carried out. We find that both memory effects and the type of centrist significantly modify the variance of average opinions in the large timescale limits of the models. Finally, we describe the limit of applicability for Sznajd-Weron's model of binary opinions as the continuum limit is approached. By comparing Monte Carlo results and long time-step limits, we find that the opinion dynamics of binary models are significantly different to those where agents are permitted more than 3 opinions.

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36

Ghiba, Ionel-Dumitrel, Patrizio Neff, Angela Madeo, Luca Placidi, and Giuseppe Rosi. "The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics." Mathematics and Mechanics of Solids 20, no. 10 (January 7, 2014): 1171–97. http://dx.doi.org/10.1177/1081286513516972.

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37

Field, Timothy R., and Robert J. A. Tough. "Coupled dynamics of populations supported by discrete sites and their continuum limit." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2121 (April 29, 2010): 2561–86. http://dx.doi.org/10.1098/rspa.2010.0049.

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The illumination of single population behaviour subject to the processes of birth, death and immigration has provided a basis for the discussion of the non-Gaussian statistical and temporal correlation properties of scattered radiation. As a first step towards the modelling of its spatial correlations, we consider the populations supported by an infinite chain of discrete sites, each subject to birth, death and immigration and coupled by migration between adjacent sites. To provide some motivation, and illustrate the techniques we will use, the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived. A certain continuum limit is identified and its properties studied via asymptotic analysis. This forms the basis of the multi-particle model of a coupled population subject to single site birth, death and immigration processes, in addition to inter-site migration. A discrete rate equation is formulated and its generating function dynamics derived. This facilitates derivation of the equations of motion for the first- and second-order cumulants, thus generalizing the earlier results of Bailey through the incorporation of immigration at each site. We present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter-site variances in the closed form. The results for the first two moments at a single time are used to derive expressions for the asymptotic time-delayed correlation functions, which relates to Glauber’s analysis of an Ising model. The paper concludes with an analysis of the continuum limit of the birth–death–immigration–migration process in terms of a path integral formalism. The continuum rate equation and evolution equation for the generating function are developed, from which the evolution equation of the mean occupancy is derived, in this limit. Its solution is provided in closed form.

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38

KONDRATIEV, YURI, EUGENE LYTVYNOV, and MICHAEL RÖCKNER. "EQUILIBRIUM KAWASAKI DYNAMICS OF CONTINUOUS PARTICLE SYSTEMS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, no. 02 (June 2007): 185–209. http://dx.doi.org/10.1142/s0219025707002695.

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We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

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39

Gonçalves, Paulo F. B., and Hubert Stassen. "New approach to free energy of solvation applying continuum models to molecular dynamics simulation." Journal of Computational Chemistry 23, no. 7 (March 26, 2002): 706–14. http://dx.doi.org/10.1002/jcc.10076.

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40

Fackeldey, Konstantin, and Rolf Krause. "Multiscale coupling in function space-weak coupling between molecular dynamics and continuum mechanics." International Journal for Numerical Methods in Engineering 79, no. 12 (September 17, 2009): 1517–35. http://dx.doi.org/10.1002/nme.2626.

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41

Wattis, Jonathan A. D., Qi Qi, and Helen M. Byrne. "Mathematical modelling of telomere length dynamics." Journal of Mathematical Biology 80, no. 4 (November 14, 2019): 1039–76. http://dx.doi.org/10.1007/s00285-019-01448-y.

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AbstractTelomeres are repetitive DNA sequences located at the ends of chromosomes. During cell division, an incomplete copy of each chromosome’s DNA is made, causing telomeres to shorten on successive generations. When a threshold length is reached replication ceases and the cell becomes ‘senescent’. In this paper, we consider populations of telomeres and, from discrete models, we derive partial differential equations which describe how the distribution of telomere lengths evolves over many generations. We initially consider a population of cells each containing just a single telomere. We use continuum models to compare the effects of various mechanisms of telomere shortening and rates of cell division during normal ageing. For example, the rate (or probability) of cell replication may be fixed or it may decrease as the telomeres shorten. Furthermore, the length of telomere lost on each replication may be constant, or may decrease as the telomeres shorten. Where possible, explicit solutions for the evolution of the distribution of telomere lengths are presented. In other cases, expressions for the mean of the distribution are derived. We extend the models to describe cell populations in which each cell contains a distinct subpopulation of chromosomes. As for the simpler models, constant telomere shortening leads to a linear reduction in telomere length over time, whereas length-dependent shortening results in initially rapid telomere length reduction, slowing at later times. Our analysis also reveals that constant telomere loss leads to a Gaussian (normal) distribution of telomere lengths, whereas length-dependent loss leads to a log-normal distribution. We show that stochastic models, which include a replication probability, also lead to telomere length distributions which are skewed.

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Andia, P. C., F. Costanzo, and G. L. Gray. "A Lagrangian-based continuum homogenization approach applicable to molecular dynamics simulations." International Journal of Solids and Structures 42, no. 24-25 (December 2005): 6409–32. http://dx.doi.org/10.1016/j.ijsolstr.2005.05.027.

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Niu, Hongli, and Jun Wang. "Multifractal and recurrence behaviors of continuum percolation-based financial price dynamics." Nonlinear Dynamics 83, no. 1-2 (August 29, 2015): 513–28. http://dx.doi.org/10.1007/s11071-015-2344-2.

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44

FATTLER, TORBEN, and MARTIN GROTHAUS. "TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 14, no. 01 (March 2011): 105–36. http://dx.doi.org/10.1142/s0219025711004328.

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We study the dynamics of a tagged particle in an environment of infinitely many Brownian particles in continuum. All the particles interact via the gradient of an interaction potential. Our strategy is to construct the infinite particle environment process at first and afterwards the process coupling the motion of the tagged particle and the motion of the environment. First we derive an integration by parts formula with respect to the standard gradient ∇Γ on configuration spaces Γ for a general class of grand canonical Gibbs measures μ, corresponding to pair potentials ϕ and intensity measures σ = z exp (-ϕ)dx, 0 < z < ∞, having correlation functions fulfilling a Ruelle bound. Combining this with a second integration by parts formula with respect to the gradient ∇Γ in direction γ ∈ Γ, by Dirichlet form techniques we can construct the environment process and the coupled process, respectively. Our results give the first mathematically rigorous and complete construction of the tagged particle process in continuum with interaction potential. In particular, we can treat interaction potentials which might have a singularity at the origin, nontrivial negative part and infinite range as e.g., the Lennard–Jones potential.

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45

Ariunaa, U., M. Dumbser, and Ts Sarantuya. "Complete Riemann Solvers for the Hyperbolic GPR Model of Continuum Mechanics." Bulletin of Irkutsk State University. Series Mathematics 35 (2021): 60–72. http://dx.doi.org/10.26516/1997-7670.2021.35.60.

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In this paper, complete Riemann solver of Osher-Solomon and the HLLEM Riemann solver for unified first order hyperbolic formulation of continuum mechanics, which describes both of fluid and solid dynamics, are presented. This is the first time that these types of Riemann solvers are applied to such a complex system of governing equations as the GPR model of continuum mechanics. The first order hyperbolic formulation of continuum mechanics recently proposed by Godunov S. K., Peshkov I. M. and Romenski E. I., further denoted as GPR model includes a hyperbolic formulation of heat conduction and an overdetermined system of PDE. Path-conservative schemes are essential in order to give a sense to the non-conservative terms in the weak solution framework since governing PDE system contains non-conservative products, too. New Riemann solvers are implemented and tested successfully, which means it certainly acts better than standard local Lax-Friedrichs-type or Rusanov-type Riemann solvers. Two simple computational examples are presented, but the obtained computational results clearly show that the complete Riemann solvers are less dissipative than the simple Rusanov method that was employed in previous work on the GPR model.

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46

Xing, J. T., and W. G. Price. "Correction for Xing and Price, A power-flow analysis based on continuum dynamics." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455, no. 1992 (December 8, 1999): 4385. http://dx.doi.org/10.1098/rspa.1999.1000.

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WOOD, J. C., D. C. SPEIRS, S. W. NAYLOR, G. GETTINBY, and I. J. MCKENDRICK. "A CONTINUUM MODEL OF THE WITHIN-ANIMAL POPULATION DYNAMICS OF E. COLI O157." Journal of Biological Systems 14, no. 03 (September 2006): 425–43. http://dx.doi.org/10.1142/s021833900600188x.

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The high level of human morbidity caused by E. coli O157:H7 necessitates an improved understanding of the infection dynamics of this bacterium within the bovine reservoir. Until recently, a degree of uncertainty surrounded the issue of whether these bacteria colonize the bovine gut and as yet, only incomplete in-vivo datasets are available. Such data typically consist of bacterial counts from fecal samples. The development of a deterministic model, which has been devised to make good use of such data, is presented. A partial differential equation, which includes advection, diffusion and growth terms, is used to model the (unobserved) passage of bacteria through the bovine gut. A set of experimentally-obtained fecal count data is used to parameterize the model. Between-animal variability is found to be greater than between-strain variability, with some results adding further weight to the hypothesis that E. coli O157:H7 can colonize the bovine gastrointestinal tract.

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BESSE, NICOLAS. "GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC WATERBAG CONTINUUM." Mathematical Models and Methods in Applied Sciences 22, no. 01 (January 2012): 1150001. http://dx.doi.org/10.1142/s0218202512005848.

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In this paper we consider the relativistic waterbag continuum which is a useful PDE for collisionless kinetic plasma modeling recently developed in Ref. 11. The waterbag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations (with the complexity of a multi-fluid model) while keeping its kinetic features (Landau damping and nonlinear resonant wave-particle interaction). These models are very promising because they are very useful for analytical theory and numerical simulations of laser-plasma and gyrokinetic physics.10–16, 56, 57 The relativistic waterbag continuum is derived from two phase-space variable reductions of the relativistic Vlasov–Maxwell equations through the existence of two underlying exact invariants, one coming from physics properties of the dynamics is the canonical transverse momentum, and the second, named the "water-bag" and coming from geometric property of the phase-space is just the direct consequence of the Liouville Theorem. In this paper we prove the existence and uniqueness of global weak entropy solutions of the relativistic waterbag continuum. Existence is based on vanishing viscosity method and bounded variations (BV) estimates to get compactness while proof of uniqueness relies on kinetic formulation of the relativistic waterbag continuum and the associated kinetic entropy defect measure.

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TSOUMANIS, A. C., C. I. SIETTOS, G. V. BAFAS, and I. G. KEVREKIDIS. "EQUATION-FREE MULTISCALE COMPUTATIONS IN SOCIAL NETWORKS: FROM AGENT-BASED MODELING TO COARSE-GRAINED STABILITY AND BIFURCATION ANALYSIS." International Journal of Bifurcation and Chaos 20, no. 11 (November 2010): 3673–88. http://dx.doi.org/10.1142/s0218127410027945.

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We focus on the "trijunction" between multiscale computations, bifurcation theory and social networks. In particular, we address how the Equation-Free approach, a recently developed computational framework, can be exploited to systematically extract coarse-grained, emergent dynamical information by bridging detailed, agent-based models of social interactions on networks, with macroscopic, systems-level, continuum numerical analysis tools. For our illustrations, we use a simple dynamic agent-based model describing the propagation of information between individuals interacting under mimesis in a social network with private and public information. We describe the rules governing the evolution of the agents' emotional state dynamics and discover, through simulation, multiple stable stationary states as a function of the network topology. Using the Equation-Free approach we track the dependence of these stationary solutions on network parameters and quantify their stability in the form of coarse-grained bifurcation diagrams.

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Descombes, Xavier, Robert Minlos, and Elena Zhizhina. "Object Extraction Using a Stochastic Birth-and-Death Dynamics in Continuum." Journal of Mathematical Imaging and Vision 33, no. 3 (October 9, 2008): 347–59. http://dx.doi.org/10.1007/s10851-008-0117-y.

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