Alma Mater Studiorum Universita` di Bologna · DOTTORATO DI RICERCA IN MATEMATICA XXVIII Ciclo Settore Concorsuale di afferenza: 01/A4 Settore Scientifico disciplinare: MAT/07 STATISTICAL MECHANICS OF HARD-CORE PARTICLES WITH ATTRACTIVE INTERACTIONS Presentata da: DIEGO ALBERICI Coordinatore Dottorato: Relatore: Chiar.ma Prof.ssa Chiar.mo Prof. GIOVANNA PIERLUIGI CITTI CONTUCCI Esame finale anno 2016 Contents 1 Introduction 1 1.1 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Boltzmann-Gibbs measure . . . . . . . . . . . . . . . . . 2 1.1.2 Phase transitions . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Interactions and graphs. . . . . . . . . . . . . . . . . . . 5 1.1.4 A fundamental example: Ising models . . . . . . . . . . . 6 1.2 Monomer-dimer models . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Results obtained in this thesis . . . . . . . . . . . . . . . 10 2 Definitions and general results 13 2.1 Pure hard-core interaction . . . . . . . . . . . . . . . . . . . . . 15 2.2 Hard-core and imitative interactions . . . . . . . . . . . . . . . . 20 3 Hard-core interaction on the complete graph 25 3.1 Proof of the convergence . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Hard-core and imitative interactions on the complete graph 41 4.1 Solution of the model . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Study of the phase transition . . . . . . . . . . . . . . . . . . . 47 4.2.1 Analysis of the stationary points . . . . . . . . . . . . . 47 4.2.2 Coexistence curve . . . . . . . . . . . . . . . . . . . . . . 56 4.2.3 Critical exponents . . . . . . . . . . . . . . . . . . . . . . 62 CONTENTS 4.3 Appendix: properties of the function g . . . . . . . . . . . . . . 67 4.4 Appendix: critical exponents . . . . . . . . . . . . . . . . . . . . 69 5 Law of large numbers, central limit theorem and violations 77 5.1 Hard-core interaction on the complete graph . . . . . . . . . . . 82 5.2 Hard-core and imitative interactions on the complete graph . . . 85 5.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6 Hard-core interaction on locally tree-like random graphs 97 6.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2 Solution on trees . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3 From trees to graphs . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4 Numerical estimates . . . . . . . . . . . . . . . . . . . . . . . . 122 7 A liquid crystal model on the 2D-lattice 125 7.1 Polymer representation . . . . . . . . . . . . . . . . . . . . . . . 134 7.2 Convergence of the cluster expansion . . . . . . . . . . . . . . . 141 7.3 Proofs of the liquid crystal properties . . . . . . . . . . . . . . . 151 7.4 Appendix: 1D systems . . . . . . . . . . . . . . . . . . . . . . . 158 7.5 Appendix: cluster expansion . . . . . . . . . . . . . . . . . . . . 166 Acknowledgements 168 Bibliography 169 Chapter 1 Introduction 1.1 Statistical Mechanics Statistical Mechanics studies the collective behaviour of systems made up of a large number of elementary components. The interesting phenomena emerge from the interactions among the elementary components of the system, while if each elementary component acts independently of the other ones their collec- tive behaviour is a trivial superposition. The Statistical Mechanics formalism reveals that very simple microscopic interactions, when sufficiently strong, can produce critical phenomena at the macroscopic level (phase transitions, namely discontinuities of some physical observable in the thermodynamic limit). The classical examples come from Physics: the transition from water to ice is characterized by an abrupt change of the density of the system at 0 Celsius temperature; the transition from a paramagnetic to a ferromagnetic material is characterized by an abrupt change of the magnetisation at the Curie tem- perature. In both examples the elementary components of the system are the molecules constituting a certain material. But it can be interesting to apply the Statistical Mechanics models also to other fields. In Computer Science and Neuroscience, the(artificial) neural networks are systems of interconnected neu- rons that exchange messages [58]. In Biology the bird flocks are an example of 1 2 1.1. Statistical Mechanics collective behaviour [15]. A possible approach to Socio-Economic Sciences is to consider interconnected groups of people that exchange trends, information and opinions (see [22] for an example of application): the social networks actualize this concept. Bouchaud’s approach to study Economics and economical crisis is of this type [20,19] (in particular he affirms that a scientific revolution is needed: real data should be taken into account when they contradict classical economics assumptions). In a few words the Statistical Mechanics approach is actually multidisciplinary and can be interesting for a large number of applica- tions. 1.1.1 Boltzmann-Gibbs measure In the Statistical Mechanics of equilibrium, a probability is assigned to each possible configuration of the elementary components. Then the statistical be- haviour of the physical observables is studied when the number of elementary components goes to infinity (thermodynamic limit). Definition 1.1. The elementary components of the system are indexed by a finite set Λ. The possible configurations of each elementary component are collected in a set (we assume finite). Therefore the possible microscopic S S configurations of the system are represented by the vectors of Ω = Λ. S AHamiltonian : Ω ( , ] isintroduced: itassigns anenergytoeach H → −∞ ∞ microscopic configuration, modelling all the interactions among the elementary components. At the equilibrium, the associated Boltzmann-Gibbs probability measure is considered on the space of microscopic configurations: 1 P ( ) = e β ( ) Ω , (1.1) BG − H C C ∀C ∈ Z where is the normalizing factor (called partition function) and β [0, ) Z ∈ ∞ is a parameter (called inverse temperature). We will usually absorb β in the Hamiltonian. In this framework any observable : Ω R is a random variable O → with respect to the Boltzmann-Gibbs probability measure. Chapter 1. Introduction 3 Becauseofthedefinition(1.1),themorelikelymicroscopicconfigurationsare thosewiththelowestenergy. Onthecontraryconfigurationswithinfiniteenergy have zero probability: they can be excluded from the space of configurations Ω at the beginning. We will usually do so in this thesis; but for this introductory discussion it is convenient to keep Ω = Λ. S A justification of the Boltzmann-Gibbs distribution is out of our purposes: for particle systems it is related to Boltzmann entropy, to the ergodic hypoth- esis and the equivalence of ensembles (see [42] for an analysis of foundations of Statistical Mechanics). However it is interesting to observe that theBoltzmann- Gibbs distribution maximizes the information entropy given the expected en- ergy, namely P realizes BG max P( ) logP( ) P( ) = 1, P( ) ( ) = U 0 − C C C C H C n XC (cid:12) XC XC o (cid:12) (cid:12) for any fixed energy U [min , 1 ( )] and a suitable β 0. In other 0 ∈ H Ω H C ≥ | | C words, imagine that the expected enePrgy of the system is measured: there are several probability distributions that are consistent with this expected energy, but the Boltzmann-Gibbs distribution describes the system as random as it can be satisfying only the energy constraint. 1.1.2 Phase transitions The most interesting phenomena in Statistical Mechanics are phase transi- tions. At finite volume Λ any expected observable ( )e β ( ) = ΛO C − H C hOiBG C∈S e β ( ) P Λ − H C C∈S P is obviously an analytic function of the inverse temperature β. But since the systems studied in Statistical Mechanics are made up of a large number of elementary components, one is interested in the thermodynamic limit, that is the limit as Λ . After having chosen a proper way to let Λ go to infinity, | | → ∞ 4 1.1. Statistical Mechanics thesystemissaidtoexhibitaphasetransition ifthelimitinggeneratingfunction (called pressure) 1 1 p(β) := lim log Λ(β) = lim log e−βHΛ(C) Λ Λ Z Λ Λ %∞ | | %∞ | | Λ CX∈S is not analytic at some critical inverse temperature β = β . Observe that the c derivatives of log (β) are the cumulants of minus the energy: Z ∂ log = , BG ∂β Z h−Hi ∂2 log = 2 2 , ... ∂β2 Z hH iBG −hHiBG Therefore there is a phase transition if in the thermodynamic limit some cu- mulant of the energy is not continuous with respect to β [0, ) (while the ∈ ∞ function p itself is always continuous being concave). In the case depends an- H alytically on some parameters (e.g. magnetic field, chemical potential, ...), also the analyticity of the generating function p with respect to those parameters can be investigated. AbeautifullinkbetweenAlgebra,ComplexAnalysisandStatisticalMechan- icsisgivenbythefactthatphasetransitionsarestrictlyrelatedtocomplexzeros of the partition function (the main example is the Lee-Yang theorem [71,88]). To get an idea of this fact, assume that the Hamiltonian takes non-negative integer values up to N = O( Λ ) and rewrite the partition function as a poly- | | nomial: N (z) = C (k)zk , Λ Λ Z k=0 X bysettingz := e β andC (k) = card Ω ( ) = k (assumemax logC (k) = − Λ k Λ {C ∈ |H C } O( Λ )). By the Fundamental Theorem of Algebra (z) has N complex zeros Λ | | Z z and rewrites as Λ,p N (z) = C (N) (z z ) . Λ Λ Λ,p Z − p=1 Y Chapter 1. Introduction 5 If a complex stripe centred along a positive real interval (a,b)+i( δ,δ) is free − of zeros of for any Λ, then 1 log (z) is a uniformly bounded sequence ZΛ Λ ZΛ | | of analytic functions on the compact subsets of the stripe, therefore by the Vitali-Porter theorem [91] its limit p(z) is analytic for z (a,b). ∈ 1.1.3 Interactions and graphs When the space of configurations is a product space (Ω = Λ), and the S Hamiltonian writes as a sum on the elementary components ( ) = ( ) = ( ) Ω , i i i i Λ H C H C ∀C C ∈ ∈ i Λ X∈ then the system is called non-interacting, since its elementary components are independent according to the Boltzmann-Gibbs measure. An elementary but meaningful observation is that phase transitions do not occur in non-interacting systems: 1 p(β) = lim p (β) , i Λ Λ %∞ | | i Λ X∈ where each pi(β) = log e−βHi(C1) is an analytic function, hence p is ana- 1 C ∈S lytic provided that the CPesaro-limit can be interchanged with the series. Often systems with pairwise interactions are considered, namely ( ) = ( ) + ( , ) = ( ) Ω . i i i,j i j i i Λ H C H C H C C ∀C C ∈ ∈ i Λ i,j Λ X∈ X∈ It is natural to represent these systems on graphs: Λ is the vertex set, while the pairs (i,j) such that is not identically zero are the edges corresponding to i,j H the interactions. In Physics 2 or 3-dimensional regular lattices are usually con- sidered, since the particles interact according to their distance in the Euclidean space. For other collective phenomena instead sparse random graphs are more suitable: there are different methods to build random graphs [59] that have characteristic features observed in real-world networks [9,77], as the presence of hubs and the small-word properties. Models on the complete graph (namely 6 1.1. Statistical Mechanics the graph where each pair of vertices is connected) are usually considered as a first approximation in Physics, while for the emerging applications they play an important role since the behaviour on the complete graph is often similar to the behaviour on sparse random graphs. 1.1.4 A fundamental example: Ising models OneofthemostpopularexamplesinStatisticalMechanicsistheIsingmodel. Consider a system made up of spin variables that can take only two opposite values: the space of configurations is Ω = 1,+1 Λ, where Λ indexes the {− } spins. It can be shown that the most general Hamiltonian : Ω R writes, H → in a unique way, as (σ) = J σ σ Ω , X i H − ∀ ∈ X Λ i X X⊆ Y∈ with J R. In the case of pairwise interactions, the Hamiltonian reduces to X ∈ (σ) = h σ J σ σ σ Ω , i i i,j i j H − − ∀ ∈ i Λ i,j Λ X∈ Xi=∈j 6 with h , J R. It is clear from the definition of the Hamiltonian that if i i,j ∈ J > 0 the configurations with σ = σ are favoured, while if J < 0 the ij i j ij configurations with σ = σ are favoured. i j − Assume zero external field: h = 0 for all i Λ. When J 0 for all pairs i ij ∈ ≥ (i,j), the system is called a ferromagnet: all the spins tend to imitate with one another; when J 0 for allpairs (i,j) the system is called an antiferromagnet; ij ≤ when the J follow a symmetric distribution around 0 the system is called a ij spin glass. In ferromagnets a phase transition can occur [84,47] and it is characterized by the divergence of the second derivative of p(β,h) with respect to the field h; precisely below the critical temperature two states coexist, one characterized by most positive spins and the other one by most negative spins. Many results