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## Mechanisms at the origin of PTs: Landau’s theory and Yang-Lee Theorems

As reviewed before, PTs have been characterized in mathematical terms by the loose of analyticity of the thermodynamic potentials of open systems.

However the thermodynamic potential F(a) is a smooth or even an analytic function in the » nite-size » case (meaning both a nite number of degrees of freedom N for discrete system and/or a nite accessible range for microscopical degree of freedom) for a very large class of systems. This suggests the idea that the loose of analyticity of the thermodynamic potentials is mathematically possible only in the so called in the sense of succession of functions: a suc-cession of analytic function does not necessarily converge to an analytic function. In order to accord the statistical mechanics with thermodynamic experimental evidence, this leads to the idea to take the thermodynamic limit where the number of degrees of freedom tends to in nity at a xed density: the succession of function is composed, in this case, by the thermodynamic potentials at di erent N. Historically, the rst major evidence of this facts relies in the Onsager’s exact solution for 2D-Ising model in canonical ensemble; when a nite lattice with N site is considered, the free energy for degree of freedom FN (T ) = N 1F (T; N) is analytic, while in the thermodynamic limit, the free energy punctually to a piece-wise analytic function.

A rst attempt to give a deeper insight to phase transitions is due to the Landau theory (1937) which relates phase transitions with a spontaneous symmetry breaking phenomena. Such a theory has been developed in the contest of canonical ensemble where energy uctuations are allowed. The main idea is that when a system undergoes a phase transition the set of sets acces-sible to the system is characterized by a di erent set of symmetries: this origin discontinuities in second order derivatives of thermodynamic potential respect to the control parameters of the systems. The maximal set of the possible symmetries that a physical system can have is repre-sented by all the symmetries of the Hamiltonian describing it. In general, at low temperatures the accessible states of a system can lack some of the symmetries of the Hamiltonian, so that the corresponding phase is the less symmetric one, whereas at higher temperatures the thermal uctuations allow the access to a wider range of energy states having more, and eventually all, the symmetries of the Hamiltonian. In the broken-symmetry phase, an extra variable is required to characterize the physical states belonging to it. Such a variable, of extensive nature, is called an order parameter. The order parameter vanishes in the more-symmetric phase and is di erent from zero in the less-symmetric phase. Under the hypothesis that the free energy F (P; T; f ig) has the same symmetries of the Hamiltonian and that it can be expressed as a function of order parameters f ig, the discontinuous behaviour of some order parameter at the transition point induce discontinuities in second order derivatives of free energy.

Landau’s theory of phase transition and the spontaneous symmetry breaking mechanism repre-sent a very important achievement in comprehension of phase transitions and critical phenom-ena giving a rst theoretical tools to understand in a wide class of phenomena from condensed matter to fundamental physics. Despite of this Landau’s theory is a mean- eld theory, as the order parameters are averages of some quantities over the degrees of freedoms of the system, and neglect the contribution of uctuations around the transition point. Moreover it requires the thermodynamic limit, so an in nite number of degrees of freedom to be exact, as for nite systems thermal uctuations always allow the system to explore a set of states invariant for the hamiltonian set of symmetries.

The idea that phase transition phenomena are strictly possible only in thermodynamic limit has been enforced in a rigours and mathematical sense by Yang-Lee theories, formulated in the framework of grand canonical ensemble. The main idea is that, for nite systems, the grand canonical potential FGCan(T; z; V) is an analytic function as a function of temperature and fu-gacity z = e as the Grand partition function ZGCan(T; z; V) can be written as a polynome in z with not real roots.

Nevertheless in the thermodynamic limit it is possible that some root of the Grand partition function tend to real axes on the fugacity complex plane. If this happens, it can be rigorously argued that the succession of analytic functions FGCan(T; z; V) for V ! +1 does not con-verge to an analytic function on the real axes of fugacity complex plane but only to a piecewise analytic function FGCan(T; z). Such result assumes a great relevance in statistical mechanics as for some physical relevant model it is possible to calculate the distribution of roots of grand partition function in the limit V ! +1 on complex fugacity plane.

This two important rigours results lead to the idea that phase transitions intended as a loss of analyticity of the thermodynamic potentials can be take place only in thermodynamic limit or, in other words, in in nite systems.

Although this idea attained a strong and coherent mathematical description of phase transitions as usually described in thermodynamics, it is not a well suited scheme to study self-organization or transitional phenomena (as nuclear fragmentation or polymerization, i.e.) in system with a nite number of degrees of freedom far from the « thermodynamic limit ».

### Beyond the thermodynamic limit dogma: signature of PTs in microcanonical ensemble

In the last years there has been a growing interest in the emergence of cooperative behaviour in small or mesoscopic systems, where thermodynamic limit (and consequently a classical ther-modynamic interpretation) is meaningless.

This enlarged view on « phase transition » in nite systems leads also to new insight in the classical « thermodynamic » theory: it would be questionable if, for instance, is there in nite systems which entails a phase transition in thermodynamic limit, a signature of the asymptotic transitional behaviour.

All the di erent approach that try to answer to these questions have in common the framework where the problem is set, i.e. the microcanonical ensemble. This choice it seems quite natural for di erent reasons. First of all, as largely observed before, microcanonical ensemble constitute the foundation of statistical mechanics as all the other statistical ensemble; classical thermody-namic observables are not required to be known a priori to de ne the ensemble.

Moreover if the presence of self-organization and cooperative phenomena are assumed to be an essential features of phase transitions, then they are only due to the mutual interactions among the degrees of freedom: so the possibility for a system to undertake a phase transition could in principle be read in the Hamiltonian. This is coherent with the well known fact that the class of universality which a phase transition belongs to depends only on some properties of the Hamiltonian describing the « microscopic » degrees of freedom dynamics (as for instance the nature of the broken symmetries, range of the interactions and dimensionality of the system). From a statistical mechanics perspective, one of the simplest structure that encodes information on the system are the Hamiltonian level sets in phase space. The microcanonical entropy at a xed value E is the logarithm of the volume of this level sets E and consequently encodes a global information on the level sets: it is natural to wonder how the emergence of cooperative phenomena and organization between the degrees of freedom a ects the behaviour of the mi-crocanonical entropy as a function. This problem is equivalent to a pure statistical mechanics de nition of a phase transition in microcanonical ensemble.

Di erent approaches to this problem have been explored in the last thirty years. One of the more interesting has been proposed by D.H.E. Gross [GK05][Gro01] and it is kwown as Micro-canonical analysis: phase transitions in microcanonical systems are signalled by the presence of convex region of (microcanonical) entropy7 9Ec 2 [Emin; +1) such that @E 0 : (1.41) @SN E =Ec.

#### Necessity theorem for PTs and Pettini-Franzosi Theorem

A major leap forward of Topological Theory of phase transition is constituted by two the-orems claiming that that topological changes of equipotential hypersurfaces of con guration space|and of the regions of con guration space bounded by them are a necessary condition for the appearance of thermodynamic phase transitions. This is obtained for a wide class of potential functions of physical relevance, and for rst- and second-order phase transitions. How-ever, long-range interactions, nonsmooth potentials, unbound con guration spaces, \exotic » and higher-order phase transitions, are not encompassed by the actual formulation of the theory and are still open problems deserving further work.

In these standard approaches, a phase transition is seen as stemming from singular properties of the statistical measures, whereas the two theorems presented below show that these singular-ities are not \primitive » phenomena but are induced from a deeper level, that of con guration-space topology. In other words, once the microscopic interaction potential is given, the infor-mation about the existence of a phase transition is already contained in the topology of its level sets, prior to and independently of the de nition of any statistical measure.

Theorem 1.4.2 (Regularity under di eomorphicity). Let VN (q1; : : : ; qN ) : RN ! R, be a smooth, nonsingular, nite-range potential. Denote by v := VN 1(v), v 2 R, its level sets, or equipotential hypersurfaces, in con guration space.

Then let v = v=N be the potential energy per degree of freedom. If for any pair of values v and v0 belonging to a given interval Iv = [v0; v1] and for any N > N0, we have Nv Nv0; that is, Nv is di eomorphic to Nv0, then the sequence of the Helmholtz free energies fFN ( )gN2N| where = 1=T (T is the temperature) and 2 I = ( (v0); (v1))|is uniformly convergent at least in C2(I ), so that F1 2 C2(I ) and neither rst- nor second-order phase transitions can occur in the (inverse) temperature interval ( (v0); (v1)).

In general, given a model described by a smooth, nonsingular, nite-range potential, it is a hard task to locate all its critical points and thus to ascertain whether Theorem ?? actually applies to it. Moreover, the requirement of the existence|at any N|of an energy density interval [v0; v1] free of critical values seems rather strong. Theorem 1.4.2 is very useful and crucial to prove Theorem 1.4.3 which establishes that the occurrence of a phase transition is necessarily driven by topological changes in con guration space. To do this we have to consider what happens to the entropy when a critical value of the potential is crossed. Taking just one critical value vc of the potential, and allowing an arbitrary growth with N of the number of critical points on vc , one can see that it is the energy variation of the volume only in the vicinity of critical points that can entail an unbounded growth with N of the third- or fourth-order derivative of the entropy. In other words, the breaking of uniform convergence of the entropy in C3 or in C2 can be originated only by a topological change of the v or, equivalently, of the Mv. To rule out any role|in the breaking of uniform convergence|of the part of con guration space volume which is free of critical points, one resorts to Theorem 1.4.2. Theorem Theorem 1.4.3 applies to all those systems whose potential is a good Morse function.9 But are there systems with only one critical value in an interval [v0; v1]? At present we can conjecture that the result expressed by Theorem Theorem 1.4.3 extends at least to those potential functions for which the number of critical values vcj contained in [v0; v1] grows at most linearly with N (thus encompassing a wide class of short-range interaction potentials). The basic case of only one critical value has a great conceptual meaning: it allows a direct proof of the role of critical points. Once we have proved that phase transitions can stem only from the neighbourhoods of critical points in the ideal case of one vc in [v0; v1], we can hardly imagine how the part of con guration space volume which is free of critical points could start playing any role when the number of critical values in the interval is let grow, although it is possible in principle. However, this possibility is ruled out by resorting to Theorem 1.4.2, though in the special case of one critical value in an interval [v0; v1]. Again, it seems very hard to imagine how this could change by simply allowing the existence of more critical values. As a consequence, topology changes are also necessary for the existence of phase transitions. Theorem 1.4.3, is enunciated as follows:

Theorem 1.4.3 (Entropy and topology). Let VN (q1; : : : ; qN ) : RN ! R, be a smooth, nonsin- gular, nite-range potential. Denote by Mv := VN 1(( ; v]), v 2 R, the generic submanifold of con guration space bounded by v. Let fqc(i) 2 RN gi2[1;N(v)] be the set of critical points of the potential, that is, such that rVN (qc(i)) = 0, and let N (v) be the number of critical points up to the potential energy value v. Let qc(i); « 0) be pseudocylindrical neighborhoods of the critical points, and i(Mv) the Morse indexes of Mv. Then there exist real numbers A(N; i; « 0), gi and real smooth functions B(N; i; v; « 0) such that the following equation for the microcanonical Z con gurational entropy SN( )(v) = (1=N) log dN q holds: V (q) v SN( )(v) = 1 Z Mv i=1 log N n S N(v) N Ncp(vX dN q + A(N; i; » ) g i (M v « 0 ) qc(i); »0) 0 i i=0 + )+1 B(N; i(n); v vc (v); « 0) n=1 X.

**Monte Carlo simulation on equipotential level sets**

To perform the analysis prospected at the end of previous section, we resort to a Monte Carlo algorithm constrained on any given Vv N of a general speci c potential function V N of the class considered in Theorem 1.4.2 and Theorem 1.4.3.

This is obtained by generating a Markov Chain with a Metropolis importance sampling of the microcanonical con guration weight appearing in (2.7) N = krV N kRN1 . The details of the Monte Carlo code are discussed in Appendix B.

**Geometry of regular potential energy level sets in con guration space**

As prospected at the end of the previous Section, the numerical simulations therein re-ported suggest that a suitably de ned concept of « asymptotic di eomorphicity » among the equipotential level sets for N ! +1 can lead to a generalization of the Theorem 1.4.2 (Neces-sity Theorem) by adding the assumption of asymptotic di eomorphicity to its hypotheses, thus excluding the 2d lattice ’4 model from the domain of validity of the theorem. This will x the problem by eliminating the counterexample.

A rst encouraging step in this direction is in the possibility of expressing the microcanonical (con gurational) Boltzmann entropy in terms of integrals on regular level sets HN ( Vv N ) of geometrical properties of the vector eld that generates the di eomorphism of eq.(2.8) among level sets.

**Microcanonical con gurational statistical mechanics from di erential topology of regular equipotential level sets**

We consider in what follows the con gurational microcanonical ensemble4 ( q; N (q; v)) where the constraint is obtained by xing the value of some speci c potential energy function V N : q ! R and the corresponding microcanonical con gurational density function is given by N (q; v) = N v dVolg : (2.15) V N (q) V N (q) v . where g is a natural metric structure in con guration space5 and dVolg the associated Rieman-nian volume form. The normalization constant in (2.15) is the microcanonical partition function according to Boltzmann’s de nition: N;Boltz(v) = @v N;Gibbs(v) = @v Z q (V N (q) v)dVolg = Z q V N (q) v dVolg (2.16) @ @ Where (x) is the Heaviside step function.

**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