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Table of contents
PART I Developments on Topological Theory of phase transitions
CHAPTER 1 State of art on theory of phase transitions in classical systems
1.1 Fundamentals of statistical mechanics
1.1.1 Mechanical foundations of statistical mechanics and statistical ensembles
1.1.2 Equivalence of statistical ensembles
1.2 Phase Transitions (PTs) at thermodynamic equilibrium: denition and classi- cation
1.2.1 General concepts on equilibrium Phase Transitions
1.2.2 Mechanisms at the origin of PTs: Landau’s theory and Yang-Lee Theorems
1.3 Beyond the thermodynamic limit dogma: signature of PTs in microcanonical ensemble
1.4 The Topological Theory of phase transitions
1.4.1 Motivations: From the dynamics of chaotic systems to the topological hypotheisis
1.4.2 Necessity theorem for PTs and Pettini-Franzosi Theorem
1.4.3 A counterexample to Topological Theory of PTs
CHAPTER 2 Developments of the Topological Theory of Phase Transitions
2.1 Preliminary results towards a generalization of the Topological Theory
2.1.1 Motivations
2.1.2 The model: ‘4-model on lattice
2.1.3 Numerical simulation of Hamiltonian dynamics
2.1.4 Monte Carlo simulation on equipotential level sets
2.1.5 Discussion of the numerical results
2.2 Geometry of regular potential energy level sets in conguration space
2.2.1 Microcanonical congurational statistical mechanics from dierential topology of regular equipotential level sets
2.3 Results of simulation on ‘4-model on 2D-lattice
2.3.1 From the numerical simulation to the revision of the Franzosi-Pettini necessity Theorem: outlooks and perspectives
2.4 Geometrization of thermodynamics through regular equipotential level sets
2.4.1 Regular equipotential surfaces as manifolds with density
2.4.2 Rescaled metric in conguration space
2.4.3 Geometry of Riemannian Manifolds (MV N [v0;v1]; ~g) and (V N v ; ~g V N v )
2.4.4 Geometrical interpretation of the congurational microcanonical statistical mechanics
2.5 Persistent homology: a method to « compute » topology
2.5.1 The Mean-Field XY Model
2.5.2 Topological analysis
2.5.3 Samples of the conguration space
2.5.4 Persistent Homology
2.5.5 Simplicial Complexes in conguration space
2.5.6 Results
2.5.7 Some remarks on the application of persistent homology to Topological Theory
PART II Self organization and out-of-thermal equilibrium PTs in biological systems
CHAPTER 3 Basics facts on the theory of long range interactions among biomolecules
3.1 Motivations
3.2 Intermolecular interactions
3.2.1 Electrostatic interactions
3.2.2 Dispersive interactions
3.3 Electrodynamic long range interactions among biomolecules
3.3.1 Why electrodynamic interactions can be long range in biological systems .
3.3.2 Electrodynamic interactions among by biomolecules
3.3.3 Frohlich condensation
3.3.4 Classical electrodynamic long range interactions two oscillating dipole .
3.4 Developments in research of long range interactions among biomolecules
CHAPTER 4 From theory to experiment and return: Frohlich condensation in classical systems
4.1 Looking for Frohlich condensation in classical open systems: motivations
4.2 Quantum Hamiltonian to describe Frohlich condensation: Wu and Austin model
4.3 Dequantization of Wu and Austin Hamiltonian by Time Dependent Variational Principle (TDVP)
4.4 Derivation of Frohlich-like rate equations using Koopman-Von Neumann (KvN) formalism
4.4.1 General considerations concerning KvN formalism
4.4.2 Liouvillian operator properties of Wu-Austin-like model
4.4.3 Derivation of rate equations for actions expectation values J!i
4.5 Discussion and properties of Frohlich-like rate equations (4.99)
4.5.1 Results of numerical simulation
4.6 Comments and conclusions
CHAPTER 5 Terahertz spectroscopy experiments for the observation of collective oscillations in biomolecules out-of-thermodynamic equilbrium .
5.1 Terahertz spectroscopy on biomolecules: motivations and methods
5.1.1 Set-up of the experiments
5.1.2 Experimental outcomes
5.2 Interpretation of experimental outcomes
5.2.1 A premise on methodology
5.2.2 Interpretation of the absorption peak frequency
5.2.3 Spectroscopic detection of the collective mode
5.3 Some remarks on THz spectroscopy measures
CHAPTER 6 Study of experimental strategies to detect long range interactions: Feasibility study
6.1 Motivations
6.2 Model and methods
6.2.1 Basic equations
6.2.2 Model potentials
6.2.3 Numerical algorithms
6.2.4 Long-time diusion coecient
6.2.5 Self-diusion coecient for interacting particles
6.2.6 Measuring chaos in dynamical systems with noise
6.3 Numerical Results
6.3.1 Excluded volume eects
6.3.2 Eects of long and short range electrostatic interactions at xed average intermolecular distance
6.3.3 Eects of long and short range electrostatic interactions at xed charge value
6.3.4 Long range attractive dipolar eects
6.4 Conclusions and perspectives
CHAPTER 7 Validation of Fluorescence Correlation Spectroscopy measures for detection of long-range interactions
7.1 Motivations
7.2 Experimental measures of self-diusion coecient of biomolecules interacting by Fluorescence Correlation Spectroscopy (FCS)
7.2.1 A brief review on FCS
7.2.2 How characterize long-range interaction with FCS experiments
7.2.3 FCS results
7.3 Numerical results for validation of the experimental tehcnique
7.3.1 Basic dynamical equations
7.3.2 Model potential
7.3.3 Long-time diusion coecient
7.3.4 Simulation Parameters
7.4 Concluding remarks
CHAPTER A Basic facts of dierential geometry and geometric measure theory
A.1 Brief review of Riemannian Geometry
A.1.1 The concept of dierentiable manifold
A.1.2 Tangent and cotangent space
A.2 Tensor elds, derivations, connections and curvatures
A.3 Dierential forms, exterior dierentiations, integration of forms
A.4 Riemannian structure
A.5 Riemmanian geometry of codimension one submanifolds (regular level sets)
CHAPTER B Details on the codes used in numerical simulations
B.1 A MonteCarlo code to explore regular level sets of potential energy
B.2 Derivatives of the Hirsch vector eld as function of potential
CHAPTER C Basic facts of homololgy
C.1 Simplicial Complexes
C.1.1 Simplicial Homology
C.1.2 Persistent Homology
