Université Libre de Bruxelles Faculté des Sciences
Service de Physique Statistique et Phénomènes non Linéaires
Thermodynamic and Kinetic Aspects of Interaction Networks
Thèse presentée en vue de l’obtention du grade legal de Docteur en Sciences.
Anselmo García Cantú Ros Octobre 2007
Contents
Introduction………1
Chapter I : Macroscopic description 1.1State variables and mean field approach………4
1.2Determinsitic dynamics……….5
1.2.1Phase space……….6
1.2.2Dissipative systems……….7
1.3Linear stability analysis……….8
1.4Poincaré map………11
1.5Nonlinear behavior and bifurcation analysis………...13
1.6Selected Models………...17
1.6.1 Chemical reactions: The Schlögl model………...17
1.6.2 A continuous time chaotic system: The Rössler model…………....20
1.6.3 Chaotic maps……….23
1.7 Thermodynamic approach………...24
1.7.1 Local Equilibrium formulation……….26
1.7.2 Entropy balance………27
Chapter II: Stochastic Description 2.1Fluctuations………..30
2.2Markov Processes………31
2.2.1Markov chains and random walk process……….32
2.2.2Continuous time Markov processes………..34
2.2.3Langevin equation and the fluctuation-dissipation theorem……….36
2.3Probabilistic description of deterministic systems: The Perron-Frobenius equation………39
2.4Evolution of the entropy density in mesoscopic scale reaction systems…….43
Chapter III: Network description of dynamical systems 3.1 Basic definitions in network theory……….46
3.1.1 Local description………...46
3.1.2 Global description……….47
3.2.1 Random networks……….47
3.2.2 Small world networks………...49
3.2.3 Scale free networks………...52
3.2 Dynamical systems as networks………..54
3.3.1 Fully developed chaos………..56
3.3.2 Intermittent chaos……….59
3.3.3 Networks generated by chaotic flows……….. 64
3.4 Networks generated by stochastic processes………...67
Conclusions……….72
Chapter IV: Diffusion processes on networks
4.1 General formulation……….74
4.2 The trapping problem in one dimension as absorbing Markov chain………. 76
4.2.1 The trapping problem on a ring: the simplest case……….. 76
4.2.2 Extended trapping problem in ring lattices: role of non-nearest neighbor jumps...78
4.3 Beyond the trapping problem: Two diffusing walkers………82
4.3.1 Lattice with first nearest neighbor connections………82
4.3.2 Lattice with first and second neighbor connections………..90
4.4 The trapping problem in disordered lattices: Small world networks………...97
Conclusions……….102
Chapter V : Thermodynamics of reaction networks 5.1 Formulation………105
5.2 Regular Networks………..107
5.2.1 Linear chains………...106
5.2.2 Parallel chains……….109
5.2.3 Branching chains……….110
5.3 Feedbak dynamics………..113
5.4 Complex network models………..117
5.4.1 Small world network……….. 117
5.4.2 Hierarchical network………...120
5.5 Response of first order chemical reactions to stochastic perturbations…….121
5.5.1 Evolution of fluctuations……… 121
5.5.2 Entropy Balance in presence of fluctuations………..126
Conclusions…...128
References………...131
