Lecture Notes On The Mathematical Theory Of Generalized Boltzmann Models

by Nicola Bellomo, Mauro Lo Schiavo

Overview

This book is based on the idea that Boltzmann-like modelling methods can be developed to design, with special attention to applied sciences, kinetic-type models which are called generalized kinetic models. In particular, these models appear in evolution equations for the statistical distribution over the physical state of each individual of a large population. The evolution is determined both by interactions among individuals and by external actions.Considering that generalized kinetic models can play an important role in dealing with several interesting systems in applied sciences, the book provides a unified presentation of this topic with direct reference to modelling, mathematical statement of problems, qualitative and computational analysis, and applications. Models reported and proposed in the book refer to several fields of natural, applied and technological sciences. In particular, the following classes of models are discussed: population dynamics and socio-economic behaviours, models of aggregation and fragmentation phenomena, models of biology and immunology, traffic flow models, models of mixtures and particles undergoing classic and dissipative interactions.

Product Details

ISBN-13:	9789810240783
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/13/2000
Series:	Series On Advances In Mathematics For Applied Sciences , #51
Pages:	356
Product dimensions:	6.00(w) x 8.68(h) x 0.90(d)

Table of Contents

	Preface	xi
Chapter 1.	Generalized Kinetic Models	1
1.1	Introduction	1
1.2	Generalized Kinetic Models	3
1.3	Generalized Models and Plan of the Book	11
1.4	Aim of the Book	16
1.5	References	17
Chapter 2.	Mathematical Background: Measure, Integration, Topology	21
2.1	Introduction	21
2.2	Tools from Measure Theory	22
2.2.1	Physical connections	23
2.2.2	Basics on measure theory	25
2.2.3	Measures on the real line	30
2.2.4	Measures on IR[superscript n]	35
2.3	Tools from Integration Theory	40
2.3.1	Lebesgue integrals	40
2.3.2	Main theorems of integration theory	42
2.3.3	Distribution functionals	49
2.3.4	Some notable generalizations	56
2.3.5	Two conclusive examples	61
2.4	Tools from Topology	68
2.4.1	Basics on topologies	68
2.4.2	Function spaces	74
2.4.3	The linear spaces case	79
2.5	References	84
Chapter 3.	Models of Population Dynamics with Stochastic Interactions	87
3.1	Introduction	87
3.2	The Generalized Jager and Segel Model	89
3.3	On the Initial Value Problem	93
3.4	Stationary Points	97
3.5	Applications and Perspectives	100
3.5.1	Modelling [eta] and [psi]	100
3.5.2	A model of social behaviors	103
3.5.3	A model in epidemiology	106
3.5.4	Perspectives	108
3.6	References	110
Chapter 4.	Generalized Kinetic Models for Coagulation and Fragmentation	113
4.1	Introduction	113
4.2	Description of the Models	116
4.3	Mathematical Problems	120
4.3.1	Existence of solutions	121
4.3.2	Equilibrium solutions and stability	126
4.4	Critical Analysis and Perspectives	131
4.5	References	134
Chapter 5.	Kinetic Cellular Models in the Immune System Competition	137
5.1	Kinetic Models Towards Immunology	137
5.2	Scaling in Kinetic Cellular Models	140
5.3	Phenomenological System and Modelling	144
5.4	Kinetic Evolution Equations	149
5.5	Qualitative Analysis, Applications, and Perspectives	158
5.5.1	Qualitative analysis	159
5.5.2	Simulation problems	161
5.5.3	Perspectives	165
5.6	References	167
Chapter 6.	Kinetic Models for the Evolution of Antigens Generalized Shape	171
6.1	An Introduction to the Generalized Shape	171
6.2	The Mathematical Model	173
6.3	On the Initial Value Problem	177
6.4	Applications and Developments	183
6.5	References	186
Chapter 7.	The Boltzmann Model	189
7.1	Introduction	189
7.2	The Nonlinear Boltzmann Equation	195
7.3	Mathematical Problems	199
7.4	Analytic Treatment	205
7.4.1	The Cauchy problem for small initial data	206
7.4.2	The Cauchy problem for large initial data	211
7.4.3	The initial-boundary value problem	216
7.4.4	Open problems	216
7.4.5	Evolution problems in the presence of a force field	218
7.4.6	Shock waves	219
7.4.7	Asymptotic analysis	219
7.5	Computational Methods	220
7.6	References	224
Chapter 8.	Generalized Kinetic Models for Traffic Flow	235
8.1	Introduction	235
8.2	Traffic Flow and Hydrodynamics	236
8.2.1	Scalar hydrodynamic models	240
8.2.2	Vector hydrodynamic models	243
8.3	Kinetic Traffic Flow Models	246
8.3.1	From Prigogine's to Paveri Fontana's modelling	249
8.3.2.	Developments in kinetic modelling	254
8.3.3	Evolution problems	262
8.4	Perspectives	265
8.5	References	268
Chapter 9.	Dissipative Kinetic Models for Disparate Mixtures	273
9.1	Introduction	273
9.2	Dissipative Collision Dynamics	275
9.2.1	Cluster conservative collisions	276
9.2.2	Cluster destructive collisions	278
9.3	Kinetic Equations for Mixtures of Clusters	282
9.4	Mixtures with Continuous Mass Distribution	286
9.5	Mathematical Problems	288
9.6	Perspectives in Modelling	289
9.7	References	290
Chapter 10.	Research Perspectives	293
10.1	Introduction	293
10.2	Discrete Generalized Models	295
10.2.1	The discrete Boltzmann equation	297
10.2.2	Discrete models in immunology	302
10.3	Looking for a General Structure	315
10.4	Development of New Models	323
10.5	Closure	325
10.6	References	326
	Books and Review Papers	329

Editorial Reviews

This volume is based on the idea that Boltzmann-like modeling methods can be developed to design kinetic-type models consisting of evolution equations for the statistical distribution over the physical state of each individual of a large population. Models reported and proposed refer to several fields of natural, applied, and technological sciences, including population dynamics and socioeconomic behaviors and mixtures of particles undergoing classic and dissipative interactions. Includes a survey of open problems and research perspectives. Lacks an index. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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