| 1. |
- Abdul Salam, Parveena Shamim, et al.
(författare)
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Disease contagion models coupled to crowd motion and mesh-free simulation
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 31:6, s. 1277-1295
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Tidskriftsartikel (refereegranskat)abstract
- Modeling and simulation of disease spreading in pedestrian crowds have recently become a topic of increasing relevance. In this paper, we consider the influence of the crowd motion in a complex dynamical environment on the course of infection of the pedestrians. To model the pedestrian dynamics, we consider a kinetic equation for multi-group pedestrian flow based on a social force model coupled with an Eikonal equation. This model is coupled with a non-local SEIS contagion model for disease spread, where besides the description of local contacts, the influence of contact times has also been modeled. Hydrodynamic approximations of the coupled system are derived. Finally, simulations of the hydrodynamic model are carried out using a mesh-free particle method. Different numerical test cases are investigated, including uni- and bi-directional flow in a passage with and without obstacles.
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| 2. |
- Abdulle, Assyr, et al.
(författare)
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A parabolic local problem with exponential decay of the resonance error for numerical homogenization
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 31:13
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Tidskriftsartikel (refereegranskat)abstract
- This paper aims at an accurate and efficient computation of effective quantities, e.g., the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro-macro coupling, where the macromodel describes the coarse scale behaviour, and the micro model is solved only locally to upscale the effective quantities, which are missing in the macro model. The fact that the micro problems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first order error in ε/δ, where ε < δ represents the characteristic length ofthe small scale oscillations and δ^d is the size of micro domain. This error dominates all other errors originating from the discretization of the macro and the micro problems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of the present work is to analyse a parabolic approach, first announced in [A. Abdulle,D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019], for computing the homogenized coefficients with arbitrarily high convergence rates in ε/δ. The analysis covers the setting of periodic microstructure,and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic micro structures.
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| 3. |
- Abdulle, Assyr, et al.
(författare)
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A parabolic local problem with exponential decay of the resonance error for numerical homogenization
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific Pub Co Pte Ltd. - 0218-2025 .- 1793-6314. ; 31:13, s. 2733-2772
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Tidskriftsartikel (refereegranskat)abstract
- This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro–macro-coupling, where the macromodel describes the coarse scale behavior, and the micromodel is solved only locally to upscale the effective quantities, which are missing in the macromodel. The fact that the microproblems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first-order error in ?/??/δ, where ?<??<δ represents the characteristic length of the small scale oscillations and ??δd is the size of microdomain. This error dominates all other errors originating from the discretization of the macro and the microproblems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of this work is to analyze a parabolic approach, first announced in A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019, for computing the homogenized coefficients with arbitrarily high convergence rates in ?/??/δ. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic microstructures.
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| 4. |
- Altmann, Robert, et al.
(författare)
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Quantitative Anderson localization of Schrodinger eigenstates under disorder potentials
- 2020
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 30:5, s. 917-955
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Tidskriftsartikel (refereegranskat)abstract
- This paper analyzes spectral properties of linear Schrodinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps among the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local operator preconditioning by domain decomposition.
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| 5. |
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| 6. |
- Babuska, I., et al.
(författare)
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Solving stochastic partial differential equations based on the experimental data
- 2003
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 13:3, s. 415-444
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Tidskriftsartikel (refereegranskat)abstract
- We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, omega) is expressed by a finite number N-KL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable N-KL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, omega) is completely known. When a random function a(x, omega) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.
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| 7. |
- Beilina, L., et al.
(författare)
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A posteriori error estimation in computational inverse scattering
- 2005
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 15:1, s. 23-35
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Tidskriftsartikel (refereegranskat)abstract
- We prove an a posteriori error estimate for an inverse acoustic scattering problem, where the objective is to reconstruct an unknown wave speed coefficient inside a body from measured wave reflection data in time on parts of the surface of the body. The inverse problem is formulated as a problem of finding a zero of a Jacobian of a Lagrangian. The a posterori error estimate couples residuals of the computed solution to weights the reconstruction reflecting the sensitivity of the reconstruction obtained by solving an associated linaerized problem for the Hessian of the Lagrangian. We show concrete examples of reconstrution including a posteriori error estimation.
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| 8. |
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| 9. |
- Carlen, E., et al.
(författare)
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Kinetic Limits for Pair-Interaction Driven Master Equations and Biological Swarm Models
- 2013
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 23:7, s. 1339-1376
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Tidskriftsartikel (refereegranskat)abstract
- We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair-interaction driven master equations. In the spatially homogeneous case, we prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well-known result for the Kac model in kinetic theory. We use this result to study kinetic limits for two biological swarm models. We show that propagation of chaos may be lost at large times and we exhibit an example where the invariant density is not chaotic.
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| 10. |
- Ciallellla, Alessandro, et al.
(författare)
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Free to move or trapped in your group : Mathematical modeling of information overload and coordination in crowded populations
- 2018
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Ingår i: Mathematical Models and Methods in Applied Sciences. - Singapore : World Scientific. - 0218-2025. ; 28:9, s. 1831-1856
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Tidskriftsartikel (refereegranskat)abstract
- We present modeling strategies that describe the motion and interaction of groups of pedestrians in obscured spaces. We start off with an approach based on balance equations in terms of measures and then we exploit the descriptive power of a probabilistic cellular automaton model. Based on a variation of the simple symmetric random walk on the square lattice, we test the interplay between population size and an interpersonal attraction parameter for the evacuation of confined and darkened spaces. We argue that information overload and coordination costs associated with information processing in small groups are two key processes that influence the evacuation rate. Our results show that substantial computational resources are necessary to compensate for incomplete information - the more individuals in (information processing) groups the higher the exit rate for low population size. For simple social systems, it is likely that the individual representations are not redundant and large group sizes ensure that this non-redundant information is actually available to a substantial number of individuals. For complex social systems, information redundancy makes information evaluation and transfer inefficient and, as such, group size becomes a drawback rather than a benefit. The effect of group sizes on outgoing fluxes, evacuation times and wall effects is carefully studied with a Monte Carlo framework accounting also for the presence of an internal obstacle.
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| 11. |
- Duong, G. K., et al.
(författare)
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Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025 .- 1793-6314. ; 31:07, s. 1469-1503
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary [Formula: see text] such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition [Formula: see text] we construct a solution which blows up in finite time and only at an interior point [Formula: see text] of [Formula: see text] i.e. [Formula: see text] where [Formula: see text] More precisely, we also give a description on the final asymptotic profile at the blowup point [Formula: see text] and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348].
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| 12. |
- Engström, Christian
(författare)
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On the spectrum of a holomorphic operator-valued function with applications to absorptive photonic crystals
- 2010
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 20:8, s. 1319-1341
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Tidskriftsartikel (refereegranskat)abstract
- We study electromagnetic wave propagation in a periodic and frequency dependent material characterized by a space- and frequency-dependent complex-valued permittivity. The spectral parameter relates to the time-frequency, leading to spectral analysis of a holomorphic operator-valued function. We apply the Floquet transform and show for a fixed quasi-momentum that the resulting family of spectral problems has a spectrum consisting of at most countably many isolated eigenvalues of finite multiplicity. These eigenvalues depend continuously on the quasi-momentum and no nonzero real eigenvalue exists when the material is absorptive. Moreover, we reformulate the special case of a rational operator-valued function in terms of a polynomial operator pencil and study two-component dispersive and absorptive crystals in detail.
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| 13. |
- Henning, Patrick, 1983-, et al.
(författare)
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Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials
- 2017
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific Publishing Co. Pte Ltd. - 0218-2025. ; 27:11, s. 2147-2184
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Tidskriftsartikel (refereegranskat)abstract
- This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in L∞(L2) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.
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| 14. |
- Hoffman, Johan
(författare)
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Subgrid Modeling for Convection-Diffusion-Reaction in Two Space Dimensions Using a Haar Multiresolution Analysis
- 2003
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 13:10, s. 1515-1536
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study a subgrid model based on extrapolation of a modeling residual, in the case of a linear convection-diffusion-reaction problem Lu=f in two dimensions. The solution u to the exact problem satisfies an equation Lhu=[f]h+Fh(u), where Lh is the operator used in the computation on the finest computational scale h, [f]h is the approximation of f on the scale h, and Fh(u) is a modeling residual, which needs to be modeled. The subgrid modeling problem is to compute approximations of Fh(u) without using finer scales than h. In this study we model Fh(u) by extrapolation from coarser scales than h, where Fh(u) is directly computed with the finest scale h as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a mesh of size less than h/4.
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| 15. |
- Hoffman, Johan, et al.
(författare)
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Unified Continuum modeling of fluid-structure interaction
- 2011
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 21:3, s. 491-513
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we describe an incompressible Unified Continuum(UC) model in Euler (laboratory) coordinates with a moving mesh for tracking the fluid-structure interface as part of the discretization, allowing simple and general formulation and efficient computation. The model consists of conservation equations for mass and momentum, a phase convection equation and a Cauchy stress and phase variable theta as data for defining material properties and constitutive laws. We target realistic 3D turbulent fluid-structure interaction (FSI) applications, where we show simulation results of a flexible flag mounted in the turbulent wake behind a cube as a qualitative test of the method, and quantitative results for 2D benchmarks, leaving adaptive error control for future work. We compute piecewise linear continuous discrete solutions in space and time by a general Galerkin (G2) finite element method (FEM). We introduce and compensate for mesh motion by defining a local arbitrary Euler-Lagrange (ALE) map on each space-time slab as part of the discretization, allowing a sharp phase interface given by theta on cell facets. The Unicorn implementation is published as part of the FEniCS Free Software system for automation of computational mathematical modeling. Simulation results are given for a 2D stationary convergence test, indicating quadratic convergence of the displacement, a simple 2D Poiseuille test for verifying correct treatment of the fluid-structure interface, showing quadratic convergence to the exact drag value, an established 2D dynamic flag benchmark test, showing a good match to published reference solutions and a 3D turbulent flag test as indicated above.
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| 16. |
- Jaffre, Jerome, et al.
(författare)
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Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws
- 1995
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Ingår i: Mathematical Models and Methods in Applied Sciences. - SINGAPORE : World Scientific. - 0218-2025. ; 5:3, s. 367-386
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Tidskriftsartikel (refereegranskat)abstract
- We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q greater than or equal to 0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.
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| 17. |
- Jansson, Johan, 1978, et al.
(författare)
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Computational modeling of dynamical systems
- 2005
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 15:3, s. 471-481
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Tidskriftsartikel (refereegranskat)abstract
- In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10-9 computed over the time interval [0, 100], and a lattice consisting of large and small point masses. © World Scientific Publishing Company.
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| 18. |
- Larson, Mats G, et al.
(författare)
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A mixed adaptive variational multiscale method with applications in oil reservoir simulation
- 2009
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 19:7, s. 1017-1042
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Tidskriftsartikel (refereegranskat)abstract
- We present a mixed adaptive variational multiscale method for solving elliptic second-order problems. This work is an extension of the adaptive variational multiscale method (AVMS), introduced by Larson and Malqvist,(15-17) to a mixed formulation. The method is based on a particular splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part based on solution of decoupled localized subgrid problems. We present the mixed AVMS method and derive a posteriori error estimates both for linear functionals and the energy norm. Based on the estimates we propose adaptive algorithms for automatic tuning of critical discretization parameters. Finally, we present numerical examples on a two-dimensional slice of an oil reservoir.
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| 19. |
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| 20. |
- Szepessy, Anders, 1960-
(författare)
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Langevin molecular dynamics derived from Ehrenfest dynamics
- 2011
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 21:11, s. 2289-2334
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Tidskriftsartikel (refereegranskat)abstract
- Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a KacZwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M-1/2) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small O(M -1/2) friction and o(M-1/2) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuationdissipation relation.
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| 21. |
- Willander, Magnus, 1948, et al.
(författare)
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Modelling living fluids with the subdivision into the components in terms of probability distributions
- 2004
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Ingår i: Math. Models Methods Appl. Sci.. - : World Scientific. - 0218-2025. ; 14:10, s. 1495-1520
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Tidskriftsartikel (refereegranskat)abstract
- As it follows from the results of C. H. Waddigton, F. E. Yates, A. S. Iberall, and other well-known bio-physicists, living fluids cannot be modelled within the frames of the fundamental assumptions of the statistical-mechanics formalism. One has to go beyond them. The present work does it by means of the generalized kinetics (GK), the theory enabling one to allow for the complex stochasticity of internal properties and parameters of the fluid particles. This is one of the key features which distinguish living fluids from the nonliving ones. It creates the disparity of the particles and hence breaks the each-fluid-component-uniformity requirement underlying statistical mechanics. The work deals with the corresponding modification of common kinetic equations which is in line with the GK theory and is the complement to the latter. This complement allows a subdivision of a fluid into the fluid components in terms of nondiscrete probability distributions. The treatment leads to one more equationthat describes the above internal parameters. The resulting model is the system of these two equations. It appears to be always nonlinear in case of living fluids. In case of nonliving fluids, the model can be linear. Moreover, the living-fluid model, as a whole, cannot have the thermodynamic equilibrium, only partial equilibriums (such as the motional one) are possible. In contrast to this, in case of nonliving fluids, the thermodynamic equilibrium is, of course, possible. The number of the fluid components is treated as the number of the modes of the particle-characteristic probability density. In so doing, a fairly general extension of the notion of the mode from the one-dimensional case to the multidimensional case is proposed. The work also discusses the variety of the time-scales in a living fluid, the simplest quantum-mechanical equation relevant to living fluids, and the non-equilibrium nonlinear stochastic hydrodynamics option. The latter is simpler than, but conceptually comparable to, stochastickinetic equations. A few directions for future research are suggested. The work notes a cohesion of mathematical physics and fluid mechanics with the living-fluid-related fields as a complex interdisciplinary problem.
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