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Modeling Complex Living SystemsMathematical Structures of the Kinetic Theory for Active Particles

Modeling Complex Living Systems: Mathematical Structures of the Kinetic Theory for Active Particles Mathematical Structures of the Kinetic Theory for Active Particles 2.1 Introduction The microscopic state of the active particles of large living systems in- cludes, in addition to geometric and mechanical variables, a set of variables, called the activity, suitable to describe their organized behavior. For in- stance activity variables can refer to their biological and social state. Modeling complex living systems characterized by these specific features requires mathematical methods suitable for capturing the relevant aspects of the phenomenology of the systems. This chapter focuses on this topic and specifically derives suitable evolution equations that constitute a gen- eral mathematical framework to be specialized towards modeling specific systems in life sciences. This framework consists of a system of integro-differential equations which defines the evolution of the probability distributions over the micro- scopic state of large systems of interacting active particles. The derivation of the evolution equation is obtained, by suitable developments of the meth- ods of mathematical kinetic theory, according to the following guidelines already introduced in Chapter 1: i) Assessment of the microscopic state of the active particles and of the probability distribution function over that state; ii) Modeling of microscopic interactions which may be localized in space or http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Modeling Complex Living SystemsMathematical Structures of the Kinetic Theory for Active Particles

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Publisher
Birkhäuser Boston
Copyright
© Birkhäuser Boston 2008
ISBN
978-0-8176-4510-6
Pages
27 –52
DOI
10.1007/978-0-8176-4600-4_2
Publisher site
See Chapter on Publisher Site

Abstract

Mathematical Structures of the Kinetic Theory for Active Particles 2.1 Introduction The microscopic state of the active particles of large living systems in- cludes, in addition to geometric and mechanical variables, a set of variables, called the activity, suitable to describe their organized behavior. For in- stance activity variables can refer to their biological and social state. Modeling complex living systems characterized by these specific features requires mathematical methods suitable for capturing the relevant aspects of the phenomenology of the systems. This chapter focuses on this topic and specifically derives suitable evolution equations that constitute a gen- eral mathematical framework to be specialized towards modeling specific systems in life sciences. This framework consists of a system of integro-differential equations which defines the evolution of the probability distributions over the micro- scopic state of large systems of interacting active particles. The derivation of the evolution equation is obtained, by suitable developments of the meth- ods of mathematical kinetic theory, according to the following guidelines already introduced in Chapter 1: i) Assessment of the microscopic state of the active particles and of the probability distribution function over that state; ii) Modeling of microscopic interactions which may be localized in space or

Published: Jan 1, 2008

Keywords: Kinetic Theory; Activity Variable; Velocity Variable; Active Particle; Range Interaction

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