Marina Dolfin

Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions

This paper deals with the modeling, qualitative and numerical analysis, of welfare dynamics in societies viewed as complex evolutive systems subject to different policies of wealth distribution. A nonlinear model of wealth distribution is presented. The state of a population is modeled by a probability distribution over wealth classes and the dynamic of interaction is parameterized by a threshold, whose dynamics depends on an internal competition related to the wealth distribution. Therefore, the model is a system of equations in which the threshold is one of the dynamic variables. The approach contains the whole path from modeling to simulations, through a qualitative analysis of the initial value problem.

Modeling Human Behavior in Economics and Social Science

The complex interactions between human behaviors and social economic sciences is critically analyzed in this paper in view of possible applications of mathematical modeling as an attainable interdisciplinary approach to understand and simulate the aforementioned dynamics. The quest is developed along three steps: Firstly an overall analysis of social and economic sciences indicates the main requirements that a contribution of mathematical modeling should bring to these sciences; subsequently the focus moves to an overview of mathematical tools and to the selection of those which appear, according to the authors bias, appropriate to the modeling; finally, a survey of applications is presented looking ahead to research perspectives.

MATERIAL ELEMENT MODEL AND THE GEOMETRY OF THE ENTROPY FORM

In this work we analyze and compare the model of the material (elastic) element and the entropy form developed by Coleman and Owen with that one obtained by localizing the balance equations of the continuum thermodynamics. This comparison allows one to determine the relation between the entropy function S of Coleman–Owen and that one imported from the continuum thermodynamics. We introduce the Extended Thermodynamical Phase Space (ETPS) and realize the energy and entropy balance expressions as 1-forms in this space. This allows us to realizes I and II laws of thermodynamics as conditions on these forms. We study the integrability (closure) conditions of the entropy form for the model of thermoelastic element and for the deformable ferroelectric crystal element. In both cases closure conditions are used to rewrite the dynamical system of the model in term of the entropy form potential and to determine the constitutive relations among the dynamical variables of the model. In a related study (to be published) these results will be used for the formulation of the dynamical model of a material element in the contact thermodynamical phase space of Caratheodory and Hermann similar to that of homogeneous thermodynamics.

Thermodynamic Transformations in Magnetically Polarizable Undeformable Media

Abstract In this paper the problem of magnetically undeformable polarizable media in the presence of an electromagnetic field is analyzed from a thermodynamic viewpoint. The thermodynamic transformations are explicited as functions in the state space in which one of the variables, characterizing the magnetic polarization, is given as an internal variable. The entropy form along the thermodynamic transformation is calculated for para- and diamagnetic media and the necessary conditions for its existence are given.

Uniform Materials and the Multiplicative Decomposition of the Deformation Gradient in Finite Elasto-Plasticity

Abstract In this work we analyze the relation between the multiplicative decomposition F = F e F p of the deformation gradient as a product of the elastic and plastic factors and the theory of uniform materials. We prove that postulating such a decomposition is equivalent to having a uniform material model with two configurations – total φ and the inelastic φ1. We introduce strain tensors characterizing different types of evolutions of the material and discuss the form of the internal energy and that of the dissipative potential. The evolution equations are obtained for the configurations (φ, φ1) and the material metric g. Finally, the dissipative inequality for the materials of this type is presented. It is shown that the conditions of positivity of the internal dissipation terms related to the processes of plastic and metric evolution provide the anisotropic yield criteria.

MATERIAL ELEMENT MODEL AND DISSIPATIVE PROCESSES FOR FERRIMAGNETIC CRYSTALS WITH A NON-EUCLIDEAN STRUCTURE

In a previous paper we outlined a geometric model for the thermodynamical description of ferrimagnetic crystals with a non-Euclidean structure. Applying a geometrization technique based on a model for magnetizable deformable media earlier introduced by Maugin, starting from an appropriate dynamical system on the fiber bundle of processes for simple material elements of these media, the expressions of the entropy function and the entropy 1-form were obtained. In this contribution we deepen the study of this geometrical model. We give a detailed description of the media under consideration, we introduce the transformation induced by the process and, applying the closure conditions for the entropy 1-form, we work out the necessary conditions for the existence of the entropy function. The derivation of the entropy 1-form is the starting point to introduce an extended thermodynamical phase space. Finally, from the exploitation of the Clausius–Duhem inequality, we derive, using Maugins techniques, the residual dissipation inequality and the heat equation in the first and second form.

A General Framework for Multiscale Modeling of Tumor–Immune System Interactions

In this paper we review methods that allow the construction of a consistent set of models that may describe the interactions between a tumor and the immune system on microscopic, mesoscopic, and macroscopic scales. The presented structures may be a basis for a description on the sub–cellular, cellular, and macroscopic levels. Important open problems are indicated.

A geometric perspective on Irreversible Thermodynamics. Part I: general concepts

A new geometrical formulation of the thermodynamics of irreversible processes revisiting Coleman-Owen material point model has been proposed by the authors (in collaboration with P. Rogolino) a decade ago and since then applied by dierent teams of researchers to many dierent physical models in continuum thermodynamics such as viscoanelastic media, deformable dielectrics and magnetically polarizable undeformable media. The geometrical tools of contact/symplectic geometry were applied to introduce the Extended Thermodynamic Phase Space (ETPS) with its contact structure; in this space Legendre surfaces of equilibrium and Gibbs bundle have been constructed and the relations between the constitutive properties of continuum systems and the class of the entropy form have been discussed together with the introduction of the Hamiltonian formalism. The basic features of this geometrical formulation is here reviewed leaving the illustrations of relevant applications to part II of the present paper. The review is linked to a critical analysis focused on various open problems.

Learning and dynamics in social systems: Comment on “Collective learning modeling based on the kinetic theory of active particles” by D. Burini et al.

The interesting novelty of the paper by Burini et al. [1] is that the authors present a survey and a new approach of collective learning based on suitable development of methods of the kinetic theory [2] and theoretical tools of evolutionary game theory [3]. Methods of statistical dynamics and kinetic theory lead naturally to stochastic and collective dynamics. Indeed, the authors propose the use of games where the state of the interacting entities is delivered by probability distributions. My attention has specifically focused on the influence of learning dynamics to models of social dynamics of large living systems [4–7], which is a field where I have been personally involved. It has been shown, starting from [4], that different types of social dynamics interact in social systems and that the interplay of different actions can even lead to not predictable events. Learning is an important engine of different types of social dynamics. On the other hand very simple models are proposed in the afore mentioned literature [4–7]. This rationale is also posed in the literature specialized in the field of economical and social sciences [8]. My comment, which might even be interpreted as a question, is that part of the literature on the mathematical theory of social systems should be revisited accruing to the contents of [1]. This comment might also refer to large dynamical systems of self-propelled particles, where social dynamics and learning play an important role on the overall dynamics [9–12]. Indeed the heterogeneous way of learning has an influence on the individual rules by which living entities interact and move. The paper by Burini et al. [1] does not focus on specific examples of interaction between learning and social dynamics. However, such a problem is enlightened for the benefit of readers and I feel that it is an interesting, however challenging, research perspective.

Boundary conditions for first order macroscopic models of vehicular traffic in the presence of tollgates

This paper presents a new approach to the modeling of boundary conditions for first order models of vehicular traffic in highways. The first step consists in deriving a model for the dynamics of the flow of vehicles. Simulations of the parameters lead to a detailed analysis of the qualitative properties of the model. Subsequently, for such model, the statement of initial-boundary value problems is deduced, with domain decomposition, for a tract of highway between tollgates.