Quantum Tools for Macroscopic Systems

This book describes how complex systems from a variety of fields can be modeled using quantum mechanical ideas; from biology and ecology, to sociology and decision-making. Quantum mechanics is traditionally associated with microscopic systems; however, quantum concepts have also been successfully applied to a wide range of macroscopic systems both within and outside physics. The mathematical basis of these models is covered in detail, providing a self-contained and consistent approach. This book provides unique insight into the dynamics of these macroscopic systems and opens new interdisciplinary research frontiers. The authors present an essential resource for researchers in applied mathematics or theoretical physics who are interested in applying quantum mechanics to complex systems in the social, biological or ecological sciences. * Describes how complex systems from a variety of fields can be modeled using quantum mechanical ideas * Provides insight into the dynamics of macroscopic systems and opens new interdisciplinary research frontiers * Introduces quantum tools needed for the analysis of the dynamical behavior of macroscopic systems Preface Introduction Organization of the Book References Contents 1 Preliminaries 1.1 The Bosonic Number Operator 1.2 The Fermionic Number Operator 1.3 Dynamics for a Quantum System 1.3.1 Schrödinger Representation 1.3.2 Heisenberg Representation 1.4 From Operators to Functions 1.5 A Comment on Large Time Behavior and Equilibria 1.6 The (H,ρ)-Induced Dynamics 1.6.1 The Rule ρ as a Map from mathcalH to mathcalH 1.6.2 The Rule ρ as a Map in the Space of the Parameters of H 1.7 A Short Comment on H 1.8 Tutorial 1: A Two-Mode System 1.9 Tutorial 2: A First View to Open Systems 2 Dynamics with Asymptotic Equilibria 2.1 Introduction 2.2 Two-Mode Fermionic System and (H,ρ)-Induced Dynamics 2.3 A Generalized Hamiltonian Leading to Asymptotic Equilibria 2.4 Asymptotic Steady States and Parameters 2.5 Conclusions and Perspectives 3 Epidemics: Some Preliminary Results 3.1 Introduction 3.2 The Model and Its Dynamics 3.3 Applications 3.3.1 Lockdown Measures 3.4 Conclusions 4 Spreading of Information in a Network 4.1 Introduction 4.2 The Model and Its Dynamics 4.2.1 The Hamiltonian and Its Effect 4.2.2 The Effect of the Rule 4.3 Applications 4.3.1 A Simple Network with 3 Agents 4.3.2 A Network with 7 Agents 4.4 Conclusions 5 Population Dynamics in Large Domains 5.1 Introduction 5.2 The Mathematical Framework 5.2.1 Transport Operators 5.3 The Hamiltonian Operator 5.4 Application to Large Domain: The 2D Model 5.4.1 First Application: Prey-Predator Dynamics 5.4.2 Second Application: Marine Dynamics 5.5 Conclusions 6 Political Dynamics 6.1 Introduction 6.2 The Basic Model 6.2.1 Different Behaviors Depending on the Cross Interactions 6.3 The Advanced Model: Rule Induced Dynamics 6.4 A Case Study: The Dynamics of Turncoats in the Italian XVII Legislature 6.5 Conclusions 7 Phase Transitions and Decision Making 7.1 Introduction 7.2 Introducing the Framework 7.3 The Model 7.4 Conclusions 8 Conclusions [DELETE]