On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

Bianca, Carlo; Menale, Marco

doi:10.1002/mma.5766

Cited by 8 publications

(4 citation statements)

References 21 publications

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“…As is known, the well-posedness in the Hadamard sense of a model ensures the development of numerical methods for the analysis of the numerical solutions and simulations. The proof of the existence and uniqueness of the nonequilibrium stationary state and the convergence results can be pursued by employing fixed-point arguments, measure theory and Fourier transform; see [76][77][78].…”

Section: A Critical Analysis and Research Perspectivesmentioning

confidence: 99%

On the Modeling of Energy-Multisource Networks by the Thermostatted Kinetic Theory Approach: A Review with Research Perspectives

Bianca

2022

Energies

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Recently, different mathematical frameworks of the thermostatted kinetic theory approach have been proposed for the modeling of complex systems. In particular, thermostatted kinetic frameworks have been employed for the modeling and time evolution of a hybrid energy-multisource network composed of renewable and nonrenewable energy sources, for the construction of the energy storage and for open networks. In the frameworks of the thermostatted kinetic theory approach, the evolution of an energy source and the interactions with other energy sources are modeled by introducing a distribution function and interaction rates. This paper is a survey of the recent proposed frameworks of the thermostatted kinetic theory for the modeling of a hybrid energy-multisource network and reviews the recent proposed models. The paper is not limited to review the existing frameworks, but it also generalizes the mathematical structures proposed in the pertinent literature and outlines future research perspectives and applications of this new approach proposed in 2012.

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Section: A Critical Analysis and Research Perspectivesmentioning

confidence: 99%

On the Modeling of Energy-Multisource Networks by the Thermostatted Kinetic Theory Approach: A Review with Research Perspectives

Bianca

2022

Energies

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“…The existence of solutions of the nonequilibrium stationary problem related to (1) is proved in [31]. A proof of the convergence of the solution of (3) to the nonequilibrium stationary solution as time goes to infinity is given in [32]. In many cases of interest, as for example in the description of the diffusion of epidemics η(u * , u * ) can be supposed to be constant, i.e., there exists η > 0 such that η(u * , u * ) = η, for all u * , u * ∈ D u .…”

Section: The Continuous Activity Frameworkmentioning

confidence: 99%

“…Proof. Bearing the (27) in mind, and by using the (32), straightforward calculations show, for i ∈ {1, 2, . .…”

Section: The Discrete Activity Frameworkmentioning

confidence: 99%

Towards the Dependence on Parameters for the Solution of the Thermostatted Kinetic Framework

Carbonaro

Menale

2021

Axioms

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A complex system is a system involving particles whose pairwise interactions cannot be composed in the same way as in classical Mechanics, i.e., the result of interaction of each particle with all the remaining ones cannot be expressed as a sum of its interactions with each of them (we cannot even know the functional dependence of the total interaction on the single interactions). Moreover, in view of the wide range of its applications to biologic, social, and economic problems, the variables describing the state of the system (i.e., the states of all of its particles) are not always (only) the usual mechanical variables (position and velocity), but (also) many additional variables describing e.g., health, wealth, social condition, social rôle ⋯, and so on. Thus, in order to achieve a mathematical description of the problems of everyday’s life of any human society, either at a microscopic or at a macroscpoic scale, a new mathematical theory (or, more precisely, a scheme of mathematical models), called KTAP, has been devised, which provides an equation which is a generalized version of the Boltzmann equation, to describe in terms of probability distributions the evolution of a non-mechanical complex system. In connection with applications, the classical problems about existence, uniqueness, continuous dependence, and stability of its solutions turn out to be particularly relevant. As far as we are aware, however, the problem of continuous dependence and stability of solutions with respect to perturbations of the parameters expressing the interaction rates of particles and the transition probability densities (see Section The Basic Equations has not been tackled yet). Accordingly, the present paper aims to give some initial results concerning these two basic problems. In particular, Theorem 2 reveals to be stable with respect to small perturbations of parameters, and, as far as instability of solutions with respect to perturbations of parameters is concerned, Theorem 3 shows that solutions are unstable with respect to “large” perturbations of interaction rates; these hints are illustrated by numerical simulations that point out how much solutions corresponding to different values of parameters stay away from each other as t→+∞.

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“…Bellomo and coworkers stressed in particular two concepts are of the utmost relevance in applying theory of active particles to living matter: (i) new agents that are generated can have an activity different than the one of their parent agent; (ii) non destructive interactions between two agents of the same or of different species (e.g., tumor cells and immune system effectors) can induce a change of activity level in both agents. Among the most recent developments of the Bellomo theories we cite: (i) the theory of thermostatted active particles, which allows to impose physically backgrounded constraints to the activity of individuals, developed by Bianca and Menale [21][22][23]; (ii) the stochastic evolutionary theory of tumor adaptation developed by Clairambault, Delitala, Lorenzi and coworkers [24][25][26][27][28][29]. Finally, it is worth mentioning the mathematical modeling of Darwinian species emergence by Volpert and colleagues that represent the evolution of active particles uniquely subject to a Brownian force and logistic non-local growth [30][31][32].…”

Section: Introductionmentioning

confidence: 99%

On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach

2021

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We consider an ensemble of active particles, i.e., of agents endowed by internal variables u(t). Namely, we assume that the nonlinear dynamics of u is perturbed by realistic bounded symmetric stochastic perturbations acting nonlinearly or linearly. In the absence of birth, death and interactions of the agents (BDIA) the system evolution is ruled by a multidimensional Hypo-Elliptical Fokker–Plank Equation (HEFPE). In presence of nonlocal BDIA, the resulting family of models is thus a Partial Integro-differential Equation with hypo-elliptical terms. In the numerical simulations we focus on a simple case where the unperturbed dynamics of the agents is of logistic type and the bounded perturbations are of the Doering–Cai–Lin noise or the Arctan bounded noise. We then find the evolution and the steady state of the HEFPE. The steady state density is, in some cases, multimodal due to noise-induced transitions. Then we assume the steady state density as the initial condition for the full system evolution. Namely we modeled the vital dynamics of the agents as logistic nonlocal, as it depends on the whole size of the population. Our simulations suggest that both the steady states density and the total population size strongly depends on the type of bounded noise. Phenomena as transitions to bimodality and to asymmetry also occur.

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On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

Cited by 8 publications

References 21 publications

On the Modeling of Energy-Multisource Networks by the Thermostatted Kinetic Theory Approach: A Review with Research Perspectives

On the Modeling of Energy-Multisource Networks by the Thermostatted Kinetic Theory Approach: A Review with Research Perspectives

Towards the Dependence on Parameters for the Solution of the Thermostatted Kinetic Framework

On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach

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