scholarly journals Network Structured Kinetic Models of Social Interactions

2021 ◽  
Author(s):  
Martin Burger
Keyword(s):  
Social Interactions ◽  
Kinetic Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Macroscopic Models ◽  
Interacting Agents ◽  
Microscopic Interactions ◽  
Mesoscopic Level ◽  
Nonlocal Reaction

AbstractThe aim of this paper is to study the derivation of appropriate meso- and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agents. The latter differs from interactions treated in classical statistical mechanics in the sense that the agents do not have symmetric roles, but there is rather an active and a passive agent. We will demonstrate how a certain form of kinetic equations can be obtained to describe such interactions at a mesoscopic level and moreover obtain macroscopic models from monokinetics solutions of those. The derivation naturally leads to systems of nonlocal reaction-diffusion equations (or in a suitable limit local versions thereof), which can explain spatial phase separation phenomena found to emerge from the microscopic interactions. We will highlight the approach in three examples, namely the evolution and coarsening of dialects in human language, the construction of social norms, and the spread of an epidemic.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

The Dynamics of Localized Solutions of Nonlocal Reaction-Diffusion Equations

2000 ◽  
Vol 43 (4) ◽  
pp. 477-495
Author(s):  
Michael J. Ward
Keyword(s):  
Materials Science ◽  
Principal Eigenvalue ◽  
Dynamical Behavior ◽  
Reaction Diffusion ◽  
Propagation Model ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlocal Term ◽  
Localized Solutions ◽  
Nonlocal Reaction

AbstractMany classes of singularly perturbed reaction-diffusion equations possess localized solutions where the gradient of the solution is large only in the vicinity of certain points or interfaces in the domain. The problems of this type that are considered are an interface propagation model from materials science and an activator-inhibitor model of morphogenesis. These two models are formulated as nonlocal partial differential equations. Results concerning the existence of equilibria, their stability, and the dynamical behavior of localized structures in the interior and on the boundary of the domain are surveyed for these two models. By examining the spectrum associated with the linearization of these problems around certain canonical solutions, it is shown that the nonlocal term can lead to the existence of an exponentially small principal eigenvalue for the linearized problem. This eigenvalue is then responsible for an exponentially slow, or metastable, motion of the localized structure.

scholarly journals Bistable travelling waves for nonlocal reaction diffusion equations

2014 ◽  
Vol 34 (5) ◽  
pp. 1775-1791 ◽  
Author(s):  
Matthieu Alfaro ◽  
◽  
Jérôme Coville ◽  
Gaël Raoul ◽  
◽  
...  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Fracture pattern formation in frictional, cohesive, granular material

2010 ◽  
Vol 368 (1910) ◽  
pp. 95-118 ◽  
Author(s):  
Alison Ord ◽  
Bruce E. Hobbs
Keyword(s):  
Pattern Formation ◽  
Granular Media ◽  
Large Scale ◽  
High Stress ◽  
Reaction Diffusion ◽  
Fracture Pattern ◽  
Reaction Diffusion Equations ◽  
Three Dimensions ◽  
Localized Deformation

Naturally, deformed rocks commonly contain crack arrays (joints) forming patterns with systematic relationships to the large-scale deformation. Kinematically, joints can be mode-1, -2 or -3 or combinations of these, but there is no overarching theory for the development of the patterns. We develop a model motivated by dislocation pattern formation in metals. The problem is formulated in one dimension in terms of coupled reaction–diffusion equations, based on computer simulations of crack development in deformed granular media with cohesion. The cracks are treated as interacting defects, and the densities of defects diffuse through the rock mass. Of particular importance is the formation of cracks at high stresses associated with force-chain buckling and variants of this configuration; these cracks play the role of ‘inhibitors’ in reaction–diffusion relationships. Cracks forming at lower stresses act as relatively mobile defects. Patterns of localized deformation result from (i) competition between the growth of the density of ‘mobile’ defects and the inhibition of these defects by crack configurations forming at high stress and (ii) the diffusion of damage arising from these two populations each characterized by a different diffusion coefficient. The extension of this work to two and three dimensions is discussed.

Computational modeling of radiation-induced segregation in concentrated binary alloys

2013 ◽  
Vol 1526 ◽  
Author(s):  
Santosh Dubey ◽  
Anter El-Azab
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Material Surface ◽  
Sharp Interface Model ◽  
Spatially Resolved ◽  
Radiation Induced ◽  
Cascade Damage ◽  
Defect Boundary

ABSTRACTA sharp-interface model to study radiation-induced segregation in binary alloy has been developed. This model is based on a set of reaction-diffusion equations for the point defect and atomic species concentrations, with a stochastic, spatially-resolved, discrete defect generation terms representing the cascade damage. An important feature of this model, which is significantly different from the way radiation-induced segregation has been studied in the past, is that the role of the boundaries as defect sinks has been ensured by defining defect-boundary interactions via a set of reaction boundary conditions. Defining defect-boundary interactions in this way makes it possible to capture the process of segregation as a consequence of boundary motion. The model is tested in 2D for Cu-Au solid solution with the material surface being free to move. The Gear method has been used to solve the reaction-diffusion equations. Enrichment of Cu and depletion of Au have been observed near to the boundaries.

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