Emergent Chaos in the Verge of Phase Transitions

Home , Double pendulum, Lyapunov exponent

UNIVERSIDADDELOS ANDES

BACHELOR THESIS

Emergent chaos in the verge of phase transitions

Author: Supervisor: Jerónimo VALENCIA Dr. Gabriel TÉLLEZ

A thesis submitted in fulﬁllment of the requirements for the degree of Bachelor in Physics in the

Department of Physics

January 14, 2019

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Declaration of Authorship

I, Jerónimo VALENCIA, declare that this thesis titled, “Emergent chaos in the verge of phase transitions” and the work presented in it are my own. I conﬁrm that:

• This work was done wholly or mainly while in candidature for a research degree at this University.

• Where any part of this thesis has previously been submitted for a degree or any other qualiﬁcation at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have con- tributed myself.

Signed:

Date:

“The fluttering of a butterfly’s wing in Rio de Janeiro, amplified by atmospheric currents, could cause a tornado in Texas two weeks later.”

Edward Norton Lorenz

“We avoid the gravest difﬁculties when, giving up the attempt to frame hypotheses con- cerning the constitution of matter, we pursue statistical inquiries as a branch of rational mechanics”

Josiah Willard Gibbs

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UNIVERSIDAD DE LOS ANDES Abstract Faculty of Sciences Department of Physics

Bachelor in Physics

Emergent chaos in the verge of phase transitions

by Jerónimo VALENCIA

The description of phase transitions on different physical systems is usually done using Yang and Lee, 1952, theory. In short, it describes a phase as a region of the space of conﬁgurations in which certain quantities, state functions, behave in a similar manner and where they vary smoothly. In this work, an attempt of discovering similar characteristics in quantities describing the dynamical system which arise from Hamilton equations was done. The Lyapunov exponents, Hausdorff dimension and Kolmogorov Sinai entropy gave interesting results. In fact, the macroscopical description of the system using statistical mechanics and the microscopical characterization as a dynamical system seems to coincide and this leads to interesting questions of how this relation could be exploited to diagnose critical behavior in a many-body system.

ix Acknowledgements

The first person to thank for the confidence and unconditional support in my whole life must be my mother. Although some mixed feelings come into play as my aunt has also play an important role in my academical development. In fact my family’s support has always been fundamental throughout this twenty one years, and they all shall be mentioned in a way or another. But my mother and aunt need special mentions. They taught me how to pursuit a dream and do whatever I like doing, and be confident in how everything, eventually, turns out with a good result when it is done with joy and good will. For all patience I want to thank them, and ask them to continue to be an cornerstone in my life, and to be in many occasions a reason to stop, take a deep breath and retake everything with a whole new perspective. I will be always grateful with them for all these years of training to face life responsibly. Regarding academical experiences I would like to thank my friends from Math and Physics department. People whom I have had the pleasure to meet from school and life, and those I met in my first semester in Universidad de los Andes deserve to be mentioned for all laughter and good times spent together. In addition, there are some professors which helped me become a better student and whose guidance I appreciate as it helped me to find an interesting topic to develop this work. In particular I want to acknowledge to my advisor for his time and tips which made this work better. If you are reading these acknowledgements, maybe you have already been listed before, or you correspond to those people that need not be mentioned to know this work is also their fault. Whether it is this case or not, I hope this work is as delectable for you to read as it was for me to write. And most importantly, I expect you to find at least one interesting idea to think beyond phase transitions, chaos theory or their possible relationships.

Contents

Declaration of Authorship iii

Abstract vii

Acknowledgements ix

1 Chaos Theory1 1.1 Generalities on dynamical systems...... 1 1.1.1 Stability and asymptotic behavior...... 3 1.2 Chaotic dynamics...... 5 1.2.1 Lyapunov exponents...... 7 Calculation of principal Lyapunov exponents...... 7 Calculation of higher dimensional exponents...... 8 1.2.2 Dimension...... 9 1.2.3 Kolmogorov-Sinai entropy...... 11 1.3 Examples...... 11 1.3.1 Lorenz System...... 11 Trajectories for the Lorenz system...... 13 Lyapunov exponents...... 15 Dimension...... 19 1.3.2 Double Pendulum...... 19 Trajectories for the double pendulum...... 21 Lyapunov exponents for the double pendulum...... 22

2 Statistical Mechanics 29 2.1 Phase Transitions...... 29 2.2 Fully Coupled Rotators Model...... 30 2.2.1 Canonical Partition Function...... 30 2.2.2 Free Energy...... 32 2.2.3 Magnetization...... 34 Ferromagnetic and paramagnetic behavior...... 35 2.2.4 Internal Energy...... 36 2.3 Numerical simulations for the Fully Coupled Rotators Model.. 37 2.3.1 Magnetization...... 38 2.3.2 Fluctuations...... 39 2.4 Chaos in the Fully Coupled Rotators Model...... 40 xii

2.4.1 Lyapunov exponents...... 40 2.4.2 Dimension...... 45 2.4.3 Kolmogorov-Sinai entropy...... 46 2.5 Conclusions, conjectures and further work...... 51

Bibliography 53 xiii

List of Figures

1.1 10 time steps for the canonical Hénon’s 2D Map with initial condition (0.8, 0.5) ...... 2 1.2 Numerical solution of equation 1.3 using a fourth-order Runge- Kutta algorithm. An ω-limit set and an attractor are present in this dynamical system...... 4 1.3 Lorenz system trajectory for r = 10.0...... 13 1.4 Lorenz system trajectory for r = 18.0...... 14 1.5 Lorenz system trajectory for r = 26.0...... 14 1.6 Lorenz system trajectory for r = 100.0...... 15 1.7 Convergence of Lyapunov exponents calculations for different initial conditions...... 16 1.8 Convergence of Lyapunov exponents calculations for different basis...... 17 1.9 Lyapunov exponents for the Lorenz system as a function of r ... 18 1.10 Dimension of the Lorenz system using Mori’s formula 1.12 as a function of the order parameter of the system...... 19 1.11 Dimension of the Lorenz system using Kaplan-Yorke’s formula 1.13 as a function of the order parameter of the system...... 20 1.12 Double pendulum coordinates and important variables for the system...... 20 = 1.13 Trajectories for the double pendulum with pθ2 0.1...... 23 = 1.14 Trajectories for the double pendulum with pθ2 1.0...... 24 = 1.15 Trajectories for the double pendulum with pθ2 2.7...... 25 = 1.16 Trajectories for the double pendulum with pθ2 3.5...... 26 = 1.17 Trajectories for the double pendulum with pθ2 5.0...... 27

1.18 Lyapunov exponents for the double pendulum as a function of pθ2 28 2.1 Graphical interpretation for the self-consistency equation for the Fully Coupled Rotators Model...... 33 2.2 Evolution of the rotators in the paramagnetic regime e = −5.... 35 2.3 Evolution of the rotators in the ferromagnetic regime e = 5.... 36 2.4 Energy evolution using the fourth order Runge-Kutta algorithm for different time steps...... 38 2.5 Fluctuations of the Fully Coupled Rotators Model as a function of internal energy...... 41 xiv

2.6 Lyapunov exponents of the Fully Coupled Rotators System as a function of energy for different number of particles...... 42 2.7 Value of L characterizing Lyapunov exponents for energies be- L yond the critical value which behave as λ1 ∼ (U − Uc) , varying the number of particles of the system...... 43 2.8 Correspondence between critical energies for different values of e and the emergent discontinuity in Lyapunov exponents curves for 400 particles in the system...... 44 2.9 Hausdorff dimension per degree of freedom estimations for the Fully Coupled Rotators Model as a function of internal energy U for different number of particles...... 47 2.10 Hausdorff dimension per degree of freedom estimations for the Fully Coupled Rotators Model as a function of the re-scaled energy U/e for varying values of the coupling e...... 48 2.11 Kolmogorov-Sinai entropy per particle for the Fully Coupled Ro- tators Model for varying number of particles...... 49 2.12 Kolmogorov-Sinai entropy per particle for the Fully Coupled Ro- tators Model as a function of U/e for different values of the interaction term...... 50 xv

Physical Constants

23 2 2 Boltzmann Constant kB = 1.3806 × 10 m kg/s K

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To my family and friends, and every other person which need not be mentioned.

Chapter 1

Chaos Theory

1.1 Generalities on dynamical systems

The study of dynamical systems becomes important in physics as they enclose a time evolution for a given phenomena. This evolution rule can be characterized as a one parameter map to a phase-space which contains possible states of the system (Meiss, 2007). According to the characteristics of the phase space, actually of the trajectories that this time evolution gives for an speciﬁc initial condition, of time inter- vals, being discrete or continuous, and the rules underneath the evolution itself, many dynamical system categories can be drawn. First of all, the evolution can be governed by a difference equation, such as Hénon’s special 2D map (Vladimir G. Ivancevic, 2008) given by

2 x + = yn + 1 − ax , n 1 n (1.1) yn+1 = bxn where a and b are the parameters of the map. Taking a = 1.4 and b = 0.3, this system of equations stand for the canonical Hénon map, for which 10 steps are plotted in 1.1 given an initial condition (x0, y0) = (0.8, 0.5), which is shown in a blue dot, and arrows show the next step in the iteration of the map. This kind of evolution regards time as being discrete. Taking this parameter to be continuous, the dynamical system is described by a differential equation; this work will focus on the ordinary case (ODE’s). Another important feature of this time evolution is concerned with how the system is responding in the next time-step. This can be either deterministic or stochastic. In a deterministic system, from a given point in the phase space a trajectory emerging from it is completely deﬁned for any time thereafter. How- ever, when a stochastic rule determines the next step, all that is possible to get is a set of probabilities associated to each point in the phase space which is acces- sible to the actual point. Random walks are well known examples of this kind of systems. As in this work the main systems to be studied are the ones arising from ODE’s, this means it sufﬁces to deal with deterministic rules of evolution. 2 Chapter 1. Chaos Theory

0.4

0.2

0.0

0.2

0.4 1.0 0.5 0.0 0.5 1.0

FIGURE 1.1: 10 time steps for the canonical Hénon’s 2D Map with initial condition (0.8, 0.5) 1.1. Generalities on dynamical systems 3

Restricting ourselves to deterministic systems, the evolution can be regarded as a map

U : R × M → M with a shorthand U(t, x) = Ut(x) where R represents time and M is the phase-space. If the dynamical system in d consideration is dt~x = f (~x) then the associated ﬂow satisﬁes

d Ut(~x) = f (~x) (1.2) dt t=0 where it is understood that Ut(x~0) = ~x(t) where x~0 ∈ M is a given initial condition. In addition to this, there are additional requirements for U to be a ﬂow (Meiss, 2007):

1. U must be a differentiable map, regarding M as a manifold,

2. U0(~x) = ~x for any ~x ∈ M and

3. Ut ◦ Us = Ut+s for any t, s ∈ R which are reasonable conditions to demand on a flow as it should vary smoothly (1.) and act in a consistent way as time goes on (2. and 3.). Having a flow which describes the system precisely for any time, the char- acterizations of the trajectories that the system draws inside the phase space is one of the most important tools for investigating properties of the intrinsic dynamics being studied. These orbits, as trajectories will be referred, are the set of states that the system visits starting at a given point in M; that is, an orbit O(x~0) = {Ut(x~0) | t ∈ R}. These sets in a system can be either equilibrium, O(~x) = {~x}, periodic, which implies that O(~x) is a finite set, or chaotic, which are the ones of greater interest in the current work.

1.1.1 Stability and asymptotic behavior There is an important distinction to be made between equilibrium and stability, even though they appear to mean the same. As stated before, equilibrium is a property regarding orbits, namely size-one orbits. Stability is based in the topology of the phase space itself. To define stability, lets refer to Aleksandr Lyapunov definition: an equilibrium (point) x~e of a flow Ut is stable if for every neighborhood N of x~e, there is a M ⊂ N such that ~x ∈ M implies Ut(~x) ∈ N for all time t ≥ 0 (Meiss, 2007). In other words, a stable point is a state in the phase space to which trajectories that have a near starting point stay close as time goes on. Physically, this means that a measurement uncertainty will be non increasing in time, which will regard the system as predictable. Therefore, for a stable point, the asymptotic behavior of a trajectory starting nearby will be almost trivial depending on the proximity. Moreover, Hartman-Grobman 4 Chapter 1. Chaos Theory

1.0

0.5

0.0

0.5

1.0

1.5 Starts at (0.8,0.0) Starts at (2.0,0.0) Starts at (-1.0,0.6) Starts at (-2.0,-2.0) 2.0 Starts at (0.05,0.0)

2 1 0 1 2

FIGURE 1.2: Numerical solution of equation 1.3 using a fourth- order Runge-Kutta algorithm. An ω-limit set and an attractor are present in this dynamical system.

Theorem (Meiss, 2007) states that a nicely behaved flow will be (topologically) identical to its linearization near a stable or equilibrium point. This means that locally, analyzing a dynamical system which satisfies reasonable assumptions of differentiability, reduces to finding eigenvalues of a given matrix (the Jacobian matrix) and classifying the possible outcomes (positive, negative or complex). In classical mechanics, dynamical systems arise from Euler-Lagrange or Hamil- ton equations, so Hartman-Grobman theorem applies and locally the time evolution may not be that complex. Nevertheless, global asymptotic behavior is a completely different scenario which requires analysis of the orbits as t → ∞. The characterization of orbits that will be useful considers their asymptotic behavior, i.e. the portions of phase space they reach as time increases. Once again, considerations about this will be strongly related to the topology of the space M as asymptotic properties are view in terms of limit points. To view this relationship more clearly, define an ω-limit set, ω(~x), as the set of all limit points of a trajectory that starts in ~x. For instance, consider the system

x˙ = y, x4 (1.3) y˙ = x − x3 − µy y2 − x2 + 2 1.2. Chaotic dynamics 5 with µ = 1.0. In figure 1.2 it is possible to see that this system has an ω-limit set which resembles an figure 8. Actually, most of nearby trajectories to this set are attracted to it; moreover, they converge asymptotically to it. With this in mind, it is useful to define one of the most important concepts that are used to characterize dynamical systems, mainly chaotic ones. An attractor is, roughly speaking, a set to which all nearby trajectories converge. However, some orbits that start far away from the attractor need not come close to it as t → ∞, so special attention may be taken with the definition of nearby that is used to introduce this concept. To be more rigorous with this term, consider a trapping region T to be a set that o o is compact, and satisfy Ut(T) ⊂ T for t > 0, where T denotes the interior of T. And given a trapping region in the phase space, the attracting set related to this T region is A(T) = t>0 Ut(T). Using these definitions it is possible to state that an attractor Λ is an attracting set such that there is a point in the phase space ~x with Λ = ω(~x). Note that there is a fundamental difference between attractors and ω-limit sets. While one of the former is always one of the latter, ω-limit sets need not be attractive in their neighborhood. For instance, the red line in figure 1.2 correspond to a point near (0.0, 0.0), which can be thought as a ω-limit set because the 8-figure contains it, but it does not stays near this point; in fact, it evolves and gets close to the real attractor of the system. Therefore, classification of the dynamical systems can rely on the possible ω-limit sets the system can achieve. For instance, in two dimensions, that is for flows in R2, a complete classification of such sets is given by Poincaré-Bendixson Theorem (Meiss, 2007): Theorem: Let S be a simply connected subset of R2 and ϕ be a flow on S. Suppose that the forward orbit of p ∈ S (i.e., the orbit starting at p for t > 0) is totally contained in a compact set and that ω(p) contains no equilibrium points. Then ω(p) is a periodic orbit. The proof for this result rely on the Jordan curve theorem and some technical lemmas which can be found in the reference mentioned before. Roughly speaking, this theorem states that there is no irregular or strange behavior or orbits in two dimensional flows as all limit sets are either equilibrium or periodic orbits. This can be seen for the system considered in equation 1.3 as both possible limiting cycles are (0, 0) or the 8-figure previously shown. In contrast to this regular behavior, there are systems in which this asymptotic evolution may be more complicated, or even unpredictable, and consequently more complicated to analyze.

1.2 Chaotic dynamics

Chaoticity is always referred as the impossibility of predicting the long term behavior of a given rule of evolution. It is not surprising that this kind of irregu- larity can be found in natural sciences. The phenomenon of sensitivity to initial 6 Chapter 1. Chaos Theory conditions was discovered by Henry Poincaré in his study of the n−body problem (Oestreicher, 2007), which was proposed by the King of Sweden in 1887, Oscar II. He offered a price for answering the question about stability of the So- lar system, and Poincaré found that the evolution of a 3-body system could be dramatically altered by a slight change in an initial position of one of the bodies. In his attempt to found or dismiss the stability of the Solar system, Poincaré pro- moted topology as a way to found a suitable solution (Vladimir G. Ivancevic, 2008). More recently, in the early 1960’s, the meteorologist Edward Lorenz studied a model for weather prediction, which will be discussed later in detail, and found out the same sensitivity to initial conditions described by Poincaré when he found that making calculations with rounding to 3 or 6 digits did not yield the same results (Oestreicher, 2007). Apart from physics, chaos can be applied to study biological systems; for instance, Huisman and Weissing, 1999, propose a solution for the plankton paradox based on chaos theory and oscillation of solutions of resource competition models as the number of limiting resources increase. As historical facts may suggest, the irregular behavior can be expressed in terms of how big an initial error becomes as time goes by, or more generally, how does the system become unpredictable while increasing some order parameter. To be more specific, given a deterministic continuous dynamical system described by a flow Ut : M → M, it exhibits sensitivity to initial conditions in an invariant set I ⊂ M (that is, Ut(I) ⊆ I for all times) if there is a fixed r such that for any ~x ∈ I, there is a nearby ~y ∈ N(x) (a neighborhood of x) such that |Ut(~x) − Ut(~y)| > r for some t > 0. However, this condition alone does not imply that a given system is unpredictable. For instance, take M = R t and Ut(x0) = x0e . Then, for y = x0 + e with e > 0, using the usual norm t |Ut(x0) − Ut(y)| = |e|e , thus the two points diverge exponentially. For a motion to be considered chaotic, it is needed to include a long term unpredictability condition regarding the asymptotic behavior of the flow. Thus, there is a requirement that the flow should be transitive, that is, if for any pair of non empty sets M and N in I, for a finite time t, Ut(N) ∩ M 6= ∅. In addition to this, in order to consider a flow to be chaotic, it may be defined over a compact subset of the phase space, in order to

1. keep the ﬂow inside a bounded set in order to restrict the possibility of escaping to inﬁnity and

2. guarantee chaos to be a topological invariant, that is, similarly behaved ﬂows may be both chaotic or non-chaotic.

It appears that the deﬁnition of a chaotic system would require so many conditions that make those systems difﬁcult to be present. Nevertheless, it will be shown that chaos is an intrinsic characteristic of some systems. Therefore some tools to diagnose this chaotic behavior may be explored and understood deeply to gain insight about the physics causing this unpredictability. 1.2. Chaotic dynamics 7

1.2.1 Lyapunov exponents As mentioned before, sensitivity to initial conditions for a given dynamical system is expressed in terms of continuous separation of nearby starting trajectories. However this may be a little vague and could include so many different types of divergences. In particular, for chaotic dynamics, the time dependence of this separation must be exponential. Nevertheless, as the flow is considered inside a compact set, points of the phase space itself cannot diverge exponentially. To fix this, the requirement of exponential splitting might be assumed on infinitesimally near trajectories. d To formalize the latter idea, consider a dynamical system described by dt~x = f (~x); lets associate a flow Ut to this system and take an initial point of the phase ~ space ~x. Take e > 0 and an initial deviation δ0, so that, to first order in e, ~ ~ Ut(~x + eδ0) = Ut(~x) + eD~xUt(~x)δ0 + o(e)

Therefore, the deviation at time t is given by ~ ~ δ(t) = D~xUt(~x)δ0

Inserting this in equation 1.2,

d U (~x) + e~δ(t) = f (U (~x + e~δ(t)) = f (U (~x)) + eD f (U (~x))~δ(t) + o(e) dt t t t t which implies that the time evolution of the deviation is d ~δ(t) = D f (U (~x))~δ(t) (1.4) dt t Thus, Hartman-Grobman’s theorem is evidenced here, as evolution of inﬁnitesi- mal deviations may be regarded as a linearization. Solutions to equation 1.4 can be found in many cases, and yield an exponential behavior in time. Therefore it is rather natural to deﬁne the principal Lyapunov exponent λ1 as a real number such that

~ λ1t ~ |δ(t)| = e |δ0| (1.5) It is called principal as it only takes into account a one-dimensional deviation. It will become clear that Lyapunov exponents can also quantify higher dimensional deviations when the method that is going to be used for their calculation is explained.

Calculation of principal Lyapunov exponents The process of calculation Lyapunov exponents will require numerical computations and is based in the work of Shimada and Nagashima, 1979. As studying 8 Chapter 1. Chaos Theory asymptotic behavior of the systems is the main goal, deﬁnition 1.5 can be rewritten as

1 |~δ(t)| 1 |~δ(N∆t)| λ1 = lim log = lim log (1.6) t→ ~ → ~ ∞ t |δ0| N ∞ N∆t |δ0| with log being the natural logarithm function. Numerically, a time step ∆t is taken, and then the above limit t → ∞ can be replaced with the later expression. Within the numerical calculation a sufﬁciently large N is chosen and then

! 1 |~δ(N∆t)| |~δ((N − 1)∆t)| |~δ(∆t)| λ ≈ log ... 1 ~ ~ ~ N∆t |δ((N − 1)∆t)| |δ((N − 2)∆t)| |δ0| (1.7) 1 N |~δ(i∆t)| 1 N |~δ | = log = log i ∑ ~ ∑ ~ N∆t i=1 |δ((i − 1)∆t)| N∆t i=1 |δi−1| ~ where δi is the error in the i-th time step. Moreover, as inﬁnitesimal variations ~ can be regarded as tangent vectors it is possible to consider δ0 as an element of a basis of a given one dimensional subspace. The election of this vector can be done arbitrarily. ~ When dealing with chaotic systems these δj evolve exponentially with time. Therefore some computational overﬂows can occur in the calculation of the log- arithms. To deal with this, after each time step a normalization of the given tan- ~ gent vector is needed, such as starting with a normalized δ0. This process does not affect equation 1.7, and then the expression that was used in the simulations below is obtained directly from the calculations above:

N 1 ~ λ1 ≈ ∑ log |δi| (1.8) N∆t i=1

Calculation of higher dimensional exponents

Generalizing λ1 to some new exponents, it is possible to measure how higher dimensional volumes in the phase space expand. For instance, the 2-dimensional exponent quantify area expansion, 3-dimensional stands for volume expansion and so on. The calculation setting will be the same of section 1.2.1. Given the exponential behavior of the evolution of deviations, it suffices to find one dimensional Lyapunov exponents for each vector of a given orthonormal basis for the tangent space of the phase space and then add this exponents up. Indeed, if λ1 ~ ~0 ~ ~ ~0 and λ2 are such exponents, and δ0 and δ0 are elements of the basis, |A| = |δ0||δ0| ~ λ1t ~ λ2t ~0 (λ1+λ2)t ~ ~0 ~ evolves as |A(t)| = e |δ0|e |δ0| → e |δ0| as t → ∞, if |δ0| < |δ0|. ~ n Therefore, if a basis {δ0,i}i=1 for the tangent space to any point in the phase ~ space is chosen, starting with δ0,1, after its evolution according to equation 1.4 1.2. Chaotic dynamics 9 in just one time step, using Gram-Schmidt orthogonalization process with the ~ evolved vector and δ0,1, the second vector is found in order to calculate the exponents using equation 1.8. Continuing this procedure inductively, it is possible n to find n one dimensional Lyapunov exponents {λi}i=1 which will be organized using the usual convention λi > λj when j > i. Finding the complete set of exponents for a given system will be very useful in order to characterize the underlying orbits of the system in the phase space, and therefore to gain insight about the microscopical motion of the system.

1.2.2 Dimension Lyapunov exponents are extremely powerful to diagnose chaotic behavior of a system but other kind of measurements about how the orbits in the phase space behave can give additional information. One of such possible measurements is fractal dimension. This suggests that for a chaotic motion, the attractor which arises for the dynamical system may have a fractal structure, that is, a self- similarity condition. This means that the set of points in the phase space which form the attractor are similar to a (strict) subset of itself, where similar means that there is a map which maintain distances (up to a constant factor) from the set to a proper subset. Although the attractor to be fractal is not needed for a flow to be chaotic (Meiss, 2007), fractal dimension provide a useful characterization of a flow, and strangeness of attractors for certain dynamical systems (Mori, 1980). However, lets define first what is meant with dimension of a set. Two important dimensions for a set can be defined. First, lets consider a set A ⊂ Rn and let B(e,~v) be a n-hypercube of side e centered at ~v. If it is suppose that N(e) cubes can cover the set completely, i.e. [ A ⊂ B(e, ~vi) i∈I with |I| = N(e), the box-counting dimension Dbox of the set is given by the behavior of the number of boxes needed when e → 0 (Meiss, 2007)

ln(N(e)) −Dbox N(e) ∼ e =⇒ Dbox = − lim (1.9) e→0 ln e Another possible definition of dimension was formulated by Felix Haus- dorff for metric spaces (Hausdorff, 1918). Given a metric d on the space sur- rounding the set of interest U, the diameter of the set is defined as diam(U) = supx,y∈U d(x, y); given a countable collection of sets which cover U, lets say {Ui}i∈N, such that each Ui has diameter less than some given e > 0, the m- dimensional Hausdorff measure of U is defined as (Meiss, 2007)

m m H (U) = lim inf (diam(Ui)) (1.10) e→0 ∑ i∈N 10 Chapter 1. Chaos Theory

Choosing the covering set as Ui = N(e) for all i, it is possible to estimate the above measure as Hm(U) ≤ lim em−Dbox e→0 using equations 1.9. Therefore, for some values of m, Hm(U) may be inﬁnite or zero, which implies that the Hausdorff dimension

m DH = inf{m : H = 0} (1.11) is well defined and given the previous remarks, it is clear that DH ≤ Dbox. Chaotic dynamics usually happen to express a fractal structure in their orbits, such that calculating dimensions may give additional information of the system. As it can be seen in the Lorenz system below, the change of behavior of the system can be noticed by analyzing dimension of the orbits (see figures 1.10 and 1.11). This comes from the heuristic understanding of dimension as a measurement of the internal complexity that the curve gains as the scale is reduced. According to Mori, 1980, the Hausdorff dimension of a flow can be calculated from the Lyapunov spectra, i.e. the complete set of Lyapunov exponents for the system, as Λ+ DMori = n0 + n+ 1 + (1.12) Λ− 1 where n0 is the number of zero Lyapunov exponents for the system , n+ is the number of positive exponents and Λ± is the average of positive/negative exponents for the system. Similarly, Kaplan and Yorke proposed that Hausdorff dimension was given by (Farmer, 1982; Grassberger and Procaccia, 1983)

λ1 + λ2 + ... + λj DKY = j + (1.13) |λj+1| where j is the largest integer such that the sum of the ﬁrst j Lyapunov exponents (ordered in the usual way) is positive. Nevertheless, for inﬁnite dimensional systems these three dimensions may not coincide because of the different behaviors of the Lyapunov spectra (Farmer, 1982). Even though, the inequality DH ≤ DKY holds and this makes both Mori and Kaplan-Yorke dimension estimates useful tools to analyze the local chaotic characteristics of the phase space orbits. 1Computationally, one will never get a fully zero Lyapunov exponent, so this must be considered up to a certain tolerance which will be assumed typically, of order 10−3 1.3. Examples 11

1.2.3 Kolmogorov-Sinai entropy As a chaotic flow will fill the compact subset K in which it is defined for t → ∞, it could be a clever idea to analyze how this fulfillment occurs. Following the description in Vladimir G. Ivancevic, 2008, lets consider a finite partition of the N compact set {Ki}i=1 and associate a symbolic dynamics sequence to a trajectory x0 → x1 → x2 → . . ., using a discretized time approach to the system, x0 → i1 → i2 → . . . where in ∈ {1, 2, . . . , N} means xn ∈ Kin at the time n. To a given symbolic word of length n, Wn = (i1, i2,..., in), let P(Wn) denote the probability of occurrence of this word in the dynamical system, and define the block entropy of the sequences of length n as ! Hn = sup − ∑ P(Wn) ln P(Wn) (1.14) partitions of K Wn Considering the asymptotic increase of this block entropy defines the Kolmogorov- Sinai entropy

SKS = lim Hn+1 − Hn (1.15) n→∞ Intuitively, this entropy quantify the amount of information obtained refining the partition. Connecting with the notion of dimension defined before, this entropy can give information about phase space orbits as for complex structures, refinement may give additional insight and then SKS should increase in the same way Dbox did. However, taking the supremum in equation 1.15 may not be that obvious, so to calculate this entropy Lyapunov spectra can be used due to Pesin’s entropy formula (Vladimir G. Ivancevic, 2008)

SKS = ∑ λi (1.16) λi>0 which states that the total change of entropy of the system may be considered as the total expansion of its volume in the phase space.

1.3 Examples

1.3.1 Lorenz System In 1963, Edward Lorenz studied a simple model of a ﬂuid moving in a rectangu- lar box assumed to be two dimensional (Lorenz, 1963). In the lower end of the box, placed in the x − z plane, temperature T0 was ﬁxed and the top of it was kept at T0 − ∆T. According to Saltzman, 1962, the Boussinesq equations reduce, in the case for a box with no dependence in the y direction, to 12 Chapter 1. Chaos Theory

∂ ∂θ ∇2ψ + (~v · ∇)∇2ψ = ν∇4ψ + gα ∂t ∂x (1.17) ∂ ∆T ∂ψ θ + (~v · ∇)θ = + κ∇2θ ∂t H ∂x where ψ(x, z, t) is the stream function, ~v = yˆ × ∇ψ, θ(x, z, t) is the deviation of the temperature from the conducting state, which occur when ∆T is small and temperature behaves linearly decreasing, ν is the viscosity of the fluid, g is the gravitational acceleration, α the thermal expansivity and κ the thermal diffusivity. As for low values of ∆T the fluid is in a stationary phase, as this quantity increases, it will acquire some kind of motion. When a given amount of fluid is in the region near the hot reservoir it will loose density and rise; while that fluid approaches the cold part of the box, density is regained and then it falls down. Therefore, the motion is a convection roll. Lorenz ansatz for this convection was (Vladimir G. Ivancevic, 2008)

πx πz ψ(x, z, t) = A(t) sin sin L H πx πz 2πz (1.18) θ(x, z, t) = B(t) cos sin − C(t) sin L H H where H is the height and L the length of the 2D box. Inserting this ansatz in equation 1.17, and neglecting additional terms related to the advective term (~v · ∇), a system of ordinary differential equations arises for the time-depending amplitudes:

d πgα A = B − νk2 A dt Lk2 d π∆T 4π2 B = A − AC − κk2B (1.19) dt LH LH d π2 4π3κ C = AB − C dt LH H2 where k2 = π2(L−2 + H−2) is the wave number. Up to a certain re scaling, the Lorenz system which is studied below appears, described by new variables x(t), y(t) and z(t) which has nothing to do with position variables in the original weather model. 1.3. Examples 13

15.0 12.5 10.0 7.5 5.0 2.5 0.0

10 5 7.5 0 5.0 2.5 0.0 2.5 5 5.0 7.5 10.0 10

FIGURE 1.3: Lorenz system trajectory for r = 10.0

x˙ = σ(y − x) y˙ = rx − xz − y (1.20) z˙ = xy − bz

To state the relationship to the original model, σ represents the fraction between viscous and thermal diffusion, r is the Rayleigh number which quantiﬁes the applied heat and b is a geometric factor.

Trajectories for the Lorenz system To integrate equations 1.20 a fourth order Runge-Kutta algorithm was used, with 10.000 iterations and a time step of 0.001. The initial condition for all of the following graphs is (x0, y0, z0) = (10.0, 5.0, −2.0). For the parameters σ and b, the values of the canonical Lorenz system were used, i.e., 10.0 and 8/3 respec- tively. From figure 1.3 it is clear that the ω-limit set of the system is a point. As the parameter starts to become larger, the aperiodicity property of chaotic flows mentioned when transitivity of the flow as required becomes more relevant. And when r > 25, the system looses this localization, becomes unstable and starts to wander inside a (compact) part of the phase space which reminds of a 14 Chapter 1. Chaos Theory

30 25 20 15 10 5 0

20 15 10 10 5 5 0 0 5 5 10 15 10

FIGURE 1.4: Lorenz system trajectory for r = 18.0

30 20 10 15 0 10 5 0 10 5 10 15 20 20

FIGURE 1.5: Lorenz system trajectory for r = 26.0 1.3. Examples 15

175 150 125 100 75 50 25 0

100 75 50 25 20 0 0 25 20 50 40 75

FIGURE 1.6: Lorenz system trajectory for r = 100.0 butterfly. So, naively, as there is no full certainty of this flow being chaotic for r > 25, the complexity of the system can be coded via r as its increasing means a more structured flow graph. Returning to Lorenz original system, r was an indicator of the amount of heat the system was receiving. Therefore it is possible to establish a relationship between the energy that the system gains and the internal disorder it shows. In the next chapter another similar relation will be presented, in which a parameter that measures chaoticity, namely the Lyapunov exponent for the system and the fluctuations of internal energy for a many-body system seem to have similar behaviors near the phase transition point.

Lyapunov exponents The method described in section 1.2.1 was implemented using fourth order Runge- Kutta algorithm for integration of the Lorenz equations 1.20 and for the deviation rule of evolution 1.4. First, convergence of equation 1.8 was studied in terms of the amount of time steps N considered. The parameters for the system will be the canonical ones, i.e. (σ, r, b) = (10.0, 28.0, 8/3). For this system, the principal Lyapunov exponent is reported to be λ1 = 0.9051 ± 0.0005 (Bovy, 2004). Figure 1.7 shows the successive convergence for increasing amount of terms in the sum for three different initial conditions, (10.0, 10.0, 30.0), (1.0, 2.0, 3.0) and (−1.0, −1.0, −1.0), and the resulting Lyapunov exponents, for N ∼ 106, present an error of ∼ 0.01. 16 Chapter 1. Chaos Theory

0.905

0.900 1

0.895

0.890

0.885

200000 400000 600000 800000 1000000 N

FIGURE 1.7: Convergence of Lyapunov exponents calculations for different initial conditions. 1.3. Examples 17

0.906

0.904

0.902

0.900 1

0.898

0.896

0.894

0.892

200000 400000 600000 800000 1000000 N

FIGURE 1.8: Convergence of Lyapunov exponents calculations for different basis.

~ In 1.2.1 it was mentioned that considering δ0 as a vector in a one dimensional subspace of the tangent space could help clarify how the calculation is done. In figure 1.8 it can be checked that the choice of this basis element can be done arbitrarily as for three different vectors, √1 (1.0, 1.0, 1.0), (1.0, 0.0, 0.0) 3 and √1 (1.0, −1.0, 0.0), the Lyapunov exponent value lies in the same error men- 2 tioned before. For the given amount of steps needed for a reliable converge, the error for a change of vector is ∼ 0.02. Figure 1.9 helps to clarify how the behavior of the system is reflected in the Lyapunov exponents. These become positive when r > 24, and this corresponds to a chaotic trajectory such as figure 1.5.

Regarding higher dimensional exponents, for this system there are 3 one- dimensional exponents. As a proof that the calculations are precise, the sum of this exponents must be equal to the trace of the Jacobian matrix of the system which describes the evolution of the ﬁrst variation. This comes from the algorithms which were used because, roughly speaking, they are based in diag- onalizing this matrix in a way that divergence given the chaotic behavior does not imply large eigenvalues. To start the calculation, the derivative of equation 1.20 is taken, and it yields 18 Chapter 1. Chaos Theory

1.5

1.0

0.5 1

0.0

0.5

1.0

0 10 20 30 40 50 60 r

FIGURE 1.9: Lyapunov exponents for the Lorenz system as a function of r

r λ1 λ2 λ3 Sum 10 -0.595833 -0.595582 -12.4756 -13.667 18 -0.228581 -0.228449 -13.21 -13.667 25 0.821223 -0.000364817 -14.4878 -13.667 26 0.852421 -0.000125128 -14.5193 -13.667 30 0.953314 -0.000250187 -14.6201 -13.667 50 1.2925 -0.000577031 -14.9589 -13.667 100 -0.000242731 -0.0252016 -13.6415 -13.667

TABLE 1.1: Higher dimensional Lyapunov exponents for the Lorenz system for varying r

δx˙ = −σδx + σδy δy˙ = (r − z)δx − δy − xδz (1.21) δz˙ = yδx + xδy − bδz

From these equations is it clear that the trace of the Jacobian does not depend on the given point of evaluation and it is equal to −σ − b − 1. For the canonical 41 Lorenz system, this trace is 3 =≈ 13.667. In table 1.1 it is possible to check that, for different values of r, the calculations actually give rise to the desired values, and that the change of sign on the principal exponent remains as the most important tool to diagnose chaotic behavior. 1.3. Examples 19

2.0

1.5 i r o

M 1.0 D

0.5

0.0

15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 r

FIGURE 1.10: Dimension of the Lorenz system using Mori’s formula 1.12 as a function of the order parameter of the system.

Dimension Using equations relating dimensions and Lyapunov spectra, the estimates for Hausdorff dimension for the Lorenz system was obtained. In order to compare with the information given by the principal Lyapunov exponent, figures 1.10 and 1.11 are plots of D as a function of the order parameter r. Although D = 0 may not be well defined using the above remarks, a zero dimensional set may be understood using either Mori’s or Kaplan-Yorke’s formulas. With the former, a zero dimensional set arising from a flow implies the underlying dynamical system does not present chaotic characteristics as it implies n+ = n0 = 0 in equation 1.12. In the later picture, it is needed for j to be zero, and the meaning is exactly the same as mentioned before. Both figures showing dimension of the system show a clear discontinuity in the same value of r for which Lyapunov became positive. In addition, the value reached after this critical value coincides with the accepted value for the dimension of the Lorenz attractor D = 2.06 ± 0.01 (Grassberger and Procaccia, 1983) in the case of estimates formulas described above which give DMori = 2.0629 and DKY = 2.0628.

1.3.2 Double Pendulum The main interest of the work is analyzing dynamics which arise from a Hamil- tonian formulation, so it will be useful to consider one example of such a system, in this case a prototypical chaotic physical phenomena: the double pendulum. 20 Chapter 1. Chaos Theory

2.0

1.5 Y K 1.0 D

0.5

0.0

15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 r

FIGURE 1.11: Dimension of the Lorenz system using Kaplan- Yorke’s formula 1.13 as a function of the order parameter of the system.

FIGURE 1.12: Double pendulum coordinates and important variables for the system. 1.3. Examples 21

Given the system variables deﬁned as in ﬁgure 1.12, the Lagrangian of this system is

1 1 L(θ , θ , θ˙ , θ˙ ) = (m + m )`2θ˙2 + m `2θ˙2 1 2 1 2 2 1 2 1 1 2 2 2 2 + m2`1`2θ˙1θ˙2 cos(θ1 − θ2) + (m1 + m2)g`1 cos(θ1) + m2g`2 cos(θ2) (1.22) and the canonical conjugated momenta would be

∂L = = ( + )`2 ˙ + ` ` ˙ ( − ) pθ1 m1 m2 1θ1 m2 1 2θ2 cos θ1 θ2 ∂θ˙1 ∂L = = `2 ˙ + ` ` ˙ ( − ) pθ2 m2 2θ2 m2 1 2θ1 cos θ1 θ2 ∂θ˙2 All of this implies that the Hamiltonian for the double pendulum can be written as

`2 2 + `2( + ) 2 − ` ` ( − ) 2m2 p m1 m2 p 2m2 1 2 pθ1 pθ2 cos θ1 θ2 H(θ , θ , p , p ) = θ1 1 θ2 1 2 θ1 θ2 2 2 2 2`1`2m2(m1 + m2 sin (θ1 − θ2)) (1.23) Hamilton equations will deﬁne the temporal evolution for this system, and as they form a coupled ODE’s system such as the Lorenz system, trajectories and Lyapunov exponents’ calculation will be very similar to the previous example. In this case, the momentum of the second mass m2 was selected to be the parameter which will give rise to chaos in the system while it grows.

Trajectories for the double pendulum As before, a fourth order Runge-Kutta was used to integrate Hamilton equations which arise from the Hamiltonian 1.23. The parameter for the system were cho- 2 sen to be g = 9.8 m/s , m1 = m2 = 0.5 kg and `1 = `2 = 1.0 m, and the equations were integrated in an interval of 50 seconds using 105 time steps. The original trajectories shown in cyan lines and starting in the green markers, begin with ( ) = ( ) = ( ) = initial angles θ1 0 θ2 0 0 and momenta pθ1 0 0 and pθ2 is going to be used in analogy of r in the Lorenz model. In order to show sensitive dependence to initial conditions, a deviated trajectory was built adding e = 0.01 to each of the initial parameters. The difference in evolution of these markers will provide information about the system. For instance, in figure 1.13 initial conditions in the x − y plane which start close remain close throughout the integration time, which can be evidenced for the proximity of cyan and silver curves. Increasing the momentum of the second particle, a separation of this initial conditions can be evidenced, as in figure 1.15. Nevertheless, this figure shows only a portion of 22 Chapter 1. Chaos Theory the phase space, which also includes momenta, so the exponential divergence may not be completely evident in this range of momentum. However in figures 1.16 and 1.17 a notorious difference in both trajectories can be evidenced. It is = interesting to point out that for pθ2 3.5 it appears to be an attractor whose structure is very similar to the Lorenz system attractor (see 1.5 and 1.6) but with increasing values the order parameter this structure seems to vanish.

Lyapunov exponents for the double pendulum

Figure 1.18 shows that the critical value for pθ2 is 1.0 as from this point on the system behaves chaotically. This can be verified in Figures 1.13 and 1.14, while in Figures 1.15, 1.16 and 1.17 is clear that near initial conditions create trajectories that are different, and initial conditions which are close tend to diverge. Comparing the results in figure 1.9 and 1.18, it is interesting to mention how the transition to chaos in both systems occur. In the Lorenz system it is a discontinuity while for the double pendulum it is smoothly. This seems to be a conse- quence of a temporal evolution which comes from a Hamiltonian, which implies the flow associated is going to be differentiable. Moreover, in both systems λ1 appears to tend to infinity as the order parameter grows, that is an expected result coming from the heuristic definition of chaos. What is interesting about this behavior is that is a characteristic which won’t be present when the statistical mechanics’ model is analyzed. 1.3. Examples 23

1.9980

1.9985 y

1.9990

1.9995

Original With small deviation Initial (Orig.) Initial (Dev.) 2.0000 Final (Orig.) Final (Dev.)

0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 x

FIGURE 1.13: Trajectories for the double pendulum with pθ2 = 0.1 24 Chapter 1. Chaos Theory

1.80

1.85

y 1.90

1.95

Original With small deviation Initial (Orig.) Initial (Dev.) 2.00 Final (Orig.) Final (Dev.)

0.6 0.4 0.2 0.0 0.2 0.4 0.6 x

FIGURE 1.14: Trajectories for the double pendulum with pθ2 = 1.0 1.3. Examples 25

0.4

0.6

0.8

1.0

1.2 y

1.4

1.6

1.8 Original With small deviation Initial (Orig.) Initial (Dev.) 2.0 Final (Orig.) Final (Dev.)

1.5 1.0 0.5 0.0 0.5 1.0 1.5 x

FIGURE 1.15: Trajectories for the double pendulum with pθ2 = 2.7 26 Chapter 1. Chaos Theory

0.0

0.5 y

1.0

1.5

Original With small deviation Initial (Orig.) Initial (Dev.) 2.0 Final (Orig.) Final (Dev.)

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 x

FIGURE 1.16: Trajectories for the double pendulum with pθ2 = 3.5 1.3. Examples 27

Original With small deviation Initial (Orig.) Initial (Dev.) 1.0 Final (Orig.) Final (Dev.)

0.5

0.0 y 0.5

1.0

1.5

2.0

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 x

FIGURE 1.17: Trajectories for the double pendulum with pθ2 = 5.0 28 Chapter 1. Chaos Theory

1.0

0.5

0.0 1

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

p 2

FIGURE 1.18: Lyapunov exponents for the double pendulum as a

function of pθ2 29

Chapter 2

Statistical Mechanics

2.1 Phase Transitions

A fascinating phenomena which appears in systems describing many interacting particles is the existence of phases. In order to have a rigorous deﬁnition of phase the work done by Yang and Lee, 1952 shall be taken into account. Con- sider the grand canonical partition function of a system with chemical potential µ, volume V, at temperature T described by a canonical partition function for N particles Zc(N, V, T)

∞ ζ N Ξ(µ, V, T) = ∑ Zc(N, V, T) (2.1) N=0 N! as a function of fugacity ζ = eµβ. Yang and Lee considered ζ ∈ C and proved a theorem which defines a phase in a systematical way: Theorem (Yang and Lee, 1952): If in the ζ complex plane there is a region, containing a segment of real axis, free of roots for 2.1, then all logarithmic k ∂ 1 derivatives, i.e. ∂ log(ζ) for every integer k, of V log Ξ exist in the thermodynamic limit and are analytic functions of the fugacity. As the logarithmic derivatives of the grand canonical potential log Ξ give expressions for observables such as pressure, density and internal energy, it is possible to define a phase to be a subset of states of the system in which physical observables can vary in a infinitely differentiable way. Moreover, a change of phase will require a discontinuity in one of such derivatives of the potential. However, equation 2.1 is just a polynomial for a system of finite number of particles, thus it is an analytic function in the complex plane; so it is clear that for a phase transition to occur it is necessary to be in the limit N → ∞. Neverthe- less, some phase transitions such as condensation or evaporation of water can be seen apart from this limit. This is because although the discontinuity is not actually in the systems observables, an smooth road from some states functions to other happen, and macroscopic characteristics of the system may drastically change. 30 Chapter 2. Statistical Mechanics

To put the latter idea in a more concrete and useful context, given the scope of this work, lets consider a magnetic system of spins which interact through a potential of the form ~si · ~sj. Such as in the Ising Model (for a detailed description Huang, 1987), the system may have two different phases, namely, ferromagnetic and paramagnetic. The former is characterized for a non vanishing value of the magnetization, understood as the statistical average of ~si; micro- scopically, the system requires a collective alignment in order to maintain initial magnetization, or to respond to an external magnetic ﬁeld. The paramagnetic phase is completely opposite, with a zero average magnetization and disordered spins. Therefore, each phase implies a different microscopic behavior of the system, which will be shown to be chaotic in principle for the ferromagnetic part and emergent, i.e. in the thermodynamic limit, non chaotic in the paramagnetic phase for a system in the Mean Field universality class.

2.2 Fully Coupled Rotators Model

Consider a one-dimensional system of N classical particles with mass equal to one, described by coordinates θi ∈ [0, 2π) using periodic boundary conditions and canonical momenta pi = θ˙i. In analogy to a Heisenberg model (Antoni and Ruffo, 1995; Latora, Rapisarda, and Ruffo, 1998), the Hamiltonian of the system would be

N 2 N pi e H = ∑ + ∑ (1 − cos(θi − θj)) (2.2) i=1 2 2N i,j=1 where self interacting particles are not considered in the sum by subtracting one, and the potential energy is normalized by N to ensure the correct extensivity of the free energy in the thermodynamic limit as β f ∝ N . Therefore, to consider intensive quantities in this model all energies and magnetization will be taken as per particle properties. Note that this model can be realized as a collection of particles moving on a circle, without loss of generality, of radius 1 described by a spin vector m~ i = (cos θi, sin θi). This realization of the model will be useful to understand its underlying magnetic characteristics.

2.2.1 Canonical Partition Function −1 Fixing an inverse temperature β = (kBT) , it is possible to write the canonical partition function Z Z = dp1dp2 ... dpNdθ1dθ2 ... dθN exp(−βH) 2.2. Fully Coupled Rotators Model 31

As the momenta part can be factorized and integrated explicitly such that

2π N/2 Z = Z β V where ZV is the conﬁguration partition function

N ! Z eβ ZV = dθ1dθ2 ... dθN exp − ∑ 1 − cos(θi − θj) (2.3) 2N i,j=1 Using the realization of the model as particles on a circle, equation 2.3 becomes

βeN Z N βe ZV = exp − dθ1dθ2 ... dθN ∏ exp cos(θi − θj) 2 i,j=1 2N  !2 βeN Z βe N βeN = exp − dθ1dθ2 ... dθN exp  ∑ m~ i  ≡ exp − J 2 2N i=1 2 where J includes only spin-spin interactions in the system. Using Hubbard- Stratonovich transformation for µ > 0,

µ 1 Z ∞ Z ∞ p exp ~x2 = d~y exp −~y2 + 2µ~x ·~y (2.4) 2 π −∞ −∞ J converts to

! ! Z Z ∞ Z ∞ r N 1 2 2βe J = dθ1dθ2 ... dθN d~y exp −~y + ∑ m~ i ·~y π −∞ −∞ N i=1 ! ! Z Z ∞ Z ∞ Z r N 1 2 2βe = dθ1dθ2 ... dθN d~y exp(−~y ) dθ1dθ2 ... dθN exp ∑ m~ i ·~y π −∞ −∞ N i=1

q N Now, re scaling y 7→ 2βe y the above integral can be rewritten

! Z Z ∞ Z ∞ Z N 1 N N 2 J = dθ1dθ2 ... dθN d~y exp − ~y dθ1dθ2 ... dθN exp ∑ m~ i ·~y π −∞ −∞ 2βe 2βe i=1

The second integral can be done explicitly aligning the axis of integration to the direction from which the spin vector was deﬁned, and such that m~ i · ~y = miy cos θi = y cos θi as spin vectors have norm 1; consequently 32 Chapter 2. Statistical Mechanics

! Z N N Z π N dθ1dθ2 ... dθN exp ∑ m~ i ·~y = ∏ dθi exp (y cos θi) = (2πI0(y)) i=1 i=1 −π where it was used the integral representation for the modiﬁed Bessel function of the ﬁrst kind I0 (Téllez, 2004). Therefore, the integral J reduces to

1 N Z ~y2 J = d~y exp −N − ln (2πI (y)) (2.5) π 2βe 2βe 0 and the complete partition function for the system is

2π N/2 βeN 1 N Z ~y2 Z = exp − d~y exp −N − ln (2πI (y)) β 2 π 2βe 2βe 0 (2.6)

2.2.2 Free Energy Consider a free energy per particle in the thermodynamic limit given by

ln(Z) β f ≡ − lim (2.7) N→∞ N In order to get an analytic expression, a saddle point approximation was applied to the Gaussian integral remaining in 2.6

Z ~y2 d~y exp −N − ln (2πI (y)) 2βe 0 " # ~y2 ≈ K exp sup −N − ln (2πI0(y)) y∈R 2βe where K is a constant depending on N as K ∼ N−1/2. Then

1 2π βe ~y2 β f = − ln + − sup − ln (2πI0(y)) (2.8) 2 β 2 y∈R 2βe In order to obtain the supremmum in equation 2.8 it is required that

d ~y2 − ln (2πI (y)) = 0 dx 2βe 0 2.2. Fully Coupled Rotators Model 33

2.00 g(y) f(y, 2) 1.75 f(y, 4) f(y, 1) 1.50

1.25

1.00

0.75

0.50

0.25

0.00

0 1 2 3 4 5 y

FIGURE 2.1: Graphical interpretation for the self-consistency equation for the Fully Coupled Rotators Model.

From the integral representation of the Bessel function I0(x) it is possible to d check that dx I0(x) = I1(x), and then the equation above reads ∗ ∗ y I1(y ) − ∗ = 0 (2.9) βe I0(y ) which becomes the self-consistency equation for this model. The existence of a non trivial solution y∗ for this equation relies on the prod- uct βe and it can be deducted from the derivative in y = 0 of f (y, βe) = y/βe and g(y) = I1(y)/I0(y) as it can be noticed in figure 2.1. Indeed, given βe > 0, f (y, βe) and g(y) are both increasing functions of y ∈ R+. However, all modi- y p fied Bessel functions satisfy In(y) ∼ e / 2πy as y → ∞ (Téllez, 2004) so g(y) tends to 1 in this same limit. Thus, for a fixed value of βe, having

f 0(y = 0, βe) < g0(y = 0) (2.10) is equivalent to the existence of a non-zero solution for equation 2.9. Explicitly, 0 0 1 using I0(y = 0) = I1(y = 0) = 0, I0(y = 0) = 1 and I1(y = 0) = 2 (I0(y = 34 Chapter 2. Statistical Mechanics

1 0) − I2(y = 0)) = 2 . Replacing in equation 2.10 I0 (y = 0)I (y = 0) − I (y = 0)I0 (y = 0) 1 0( = ) = 1 0 1 0 = g y 0 2 (I0(y = 0)) 2 and equation 2.10 yields the condition

βe > 2 (2.11) in order to have a non-zero value of y∗.

2.2.3 Magnetization

∂ f Recall that the magnetization per particle of the system is defined as m = − h ∂ h=0 ~ where h is an external field acting on the system. Given a field h = (hx, hy) acting on our system, the modified Hamiltonian of the system would be

N 2 N N pi e ~ H = ∑ + ∑ (1 − m~ i · m~ j) − h · ∑ m~ i (2.12) i=1 2 2N i,j=1 i=1

The partition function calculation follows the same steps done before, but ZV includes an extra term

 2  Z N ! N βeN βe ~ ZfV = exp − dθ1dθ2 ... dθN exp  ∑ m~ i + βh · ∑ m~ i 2 2N i=1 i=1 βeN ≡ exp − eJ 2

However, using the Hubbard-Stratonovich transformation to decouple the particles results in

! ! ! Z N Z ∞ Z ∞ r N 1 ~ 2 2βe eJ = dθ1dθ2 ... dθN exp βh · ∑ m~ i d~y exp −~y + ∑ m~ i ·~y π i=1 −∞ −∞ N i=1 and after performing the suitable change of variables and grouping similar terms in the exponential,

1 N Z ~y2 eJ = d~y exp −N − ln (2πI (y + βh)) (2.13) π 2βe 2βe 0 2.2. Fully Coupled Rotators Model 35

Initial distribution Final distribution

1.00 1.00

0.75 0.75

0.50 0.50

0.25 0.25

0.00 0.00 y y

0.25 0.25

0.50 0.50

0.75 0.75

1.00 1.00

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 x x

FIGURE 2.2: Evolution of the rotators in the paramagnetic regime e = −5. where h is the norm of the external ﬁeld vector. Thus, free energy becomes

2 1 2π βe y~∗ β f˜ = − ln + − − ln (2πI (y∗ + βh)) (2.14) 2 β 2 2βe 0 and magnetization of the system is

∗ ∗ 1 ∂ ˜ I1(y ) y m = (β f ) = ∗ = (2.15) β ∂h I0(y ) βe

Ferromagnetic and paramagnetic behavior As y∗ depends implicitly on βe, equation 2.15 relates the magnetization of the system and the existence of a non-zero solution of equation 2.9. In particular, for e < 0 there is only a trivial solution for the self-consistency equation and the system is always in a paramagnetic regime. On the other hand, for e > 0, there exists a a critical temperature deﬁned by

βe = 2 (2.16) in which the system presents a phase transition from a ferromagnetic phase at low temperature to a paramagnetic phase in the high temperature limit. To investigate further the phase transition, consider the Taylor expansion of the Bessel functions x2 I (x) = 1 + + O(x4) 0 4 x x3 I (x) = + + O(x4) 1 2 16 36 Chapter 2. Statistical Mechanics

Initial distribution Final distribution

1.00 1.00

0.75 0.75

0.50 0.50

0.25 0.25

0.00 0.00 y y

0.25 0.25

0.50 0.50

0.75 0.75

1.00 1.00

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 x x

FIGURE 2.3: Evolution of the rotators in the ferromagnetic regime e = 5. such that

x x3 4 3 2 3 I1(x) 2 + 16 + O(x ) x x 4 x 4 x x 4 = 2 = + + O(x ) 1 − + O(x ) = − + O(x ) I0(x) x 4 2 16 4 2 16 1 + 4 + O(x ) and the self-consistency equation becomes (up to order 4 in y) s y∗ y∗ y∗3 1 1 = − → y∗ = 4 − βe 2 16 2 βe and the magnetization behaves as s 4 1 1 m ≈ − (2.17) βe 2 βe

1 Thus, equation 2.17 shows that m vanishes with a mean ﬁeld 2 exponent near the critical temperature.

2.2.4 Internal Energy The internal energy per particle of the system can be obtained directly from equation 2.8 as

∂ 1 e y∗2 1 e U = (β f ) = + − = + (1 − m2) (2.18) ∂β 2β 2 2β2e 2β 2 using equation 2.17. Near the critical temperature 2.16, 2.3. Numerical simulations for the Fully Coupled Rotators Model 37

s !2 1 e e 4 1 1 1 e 16e(βe − 2) U ≈ + − − ≈ + − (2.19) 2β 2 2 βe 2 βe 2β 2 4β3e3 e 1 8(βe − 2) e k T ≈ + 1 − ≈ + B 1 − 8e2(ek T − 2k2 T2) (2.20) 2 2β β2e2 2 2 B B

Recall the critical temperature 2.16 Tc = e/2kB and βe = 2 so the above expression becomes e e 3e U = + = (2.21) c 2 4 4 Taking derivative of 2.19 with respect to T the speciﬁc heat is

∂U k k T C = = B 1 − 8e2(ek T − 2k2 T2) − B 8e2(ek − 4k2 T) v ∂T 2 B B 2 B B and for the critical temperature, taking the derivative from the left such that equation 2.17 yields a real non-zero number

k e2 2e2 e C (T−) = B 1 − 8e2 − − 8e2(ek − 2ek ) (2.22) v c 2 2 4 4 B B k = B + 2e4k (2.23) 2 B Taking it from the right k C (T+) = B (2.24) v c 2 α which coincides with the critical exponents for a mean ﬁeld model Cv ∝ |T − Tc| with α = 0.

2.3 Numerical simulations for the Fully Coupled Ro- tators Model

The underlying dynamical system in the rotators model will appear from Hamil- ton equations

∂H θ˙i = = pi (2.25) ∂pi ∂H e N p˙i = − = − ∑ sin(θi − θj) (2.26) ∂θi 2N j=1 38 Chapter 2. Statistical Mechanics

1.26

1.24

1.22 Energy

1.20 h=0.5 h=0.3 h=0.01 1.18

0 200 400 600 800 1000 Time

FIGURE 2.4: Energy evolution using the fourth order Runge-Kutta algorithm for different time steps. where 1 ≤ i ≤ N. By means of numerical integration of this dynamical system, thermodynamic properties such as heat capacity and magnetization of the system were wanted to be computed in order to show explicitly the phase transition. However, there are some important remarks worth mentioning about the methods used. First of all, the time step in the numerical integration was taken to be h = 0.01 so that the energy was conserved up to an error of 0.02 (fig. 2.4). Because of this fluctuations in the value of the system parameters which arise naturally from the numerical method, it was needed to take averages of quantities as energy and magnetization to account for the finite time it takes the system to reach an equilibrium state. Thus, initial distribution of the parameters where chosen as Gaussian distributions, because Latora, Rapisarda, and Ruffo, 1998, prove this condition ensures a small relaxation time.

2.3.1 Magnetization The different phases of the system were explored by ranging the initial energy of the rotators before and after U/e = 3/4 . According to equation 2.2, regarding a less energetic systems is equivalent to small kinetic energy, and closer particle positions in order to balance the fixed energy Uf = e/2 = 0.5 and get energies below this value. Therefore, ferromagnetic phase refers to the capacity of the system to maintain an initial magnetization. This is equivalent to applying an 2.3. Numerical simulations for the Fully Coupled Rotators Model 39 external magnetic field for a short time and then turning it off, thus it makes sense to associate both phenomena to ferromagnetism. In figure 2.6 there is a comparison of theoretical magnetization and the numerical estimation made for 100, 200, 400 and 1000 particles (P). Although the theoretical curve is apart from the data obtained in the simulation, the desired relations on the different phases can still be extracted based on a simulated critical energy that exceeds the value predicted by 2.19. This should come from a finite size effect or some remainder magnetization from the initial conditions given to the particles combined with the finite relaxation time of the system. In figure 2.8, the system for different interaction terms e was analyzed. Tak- ing into account equation 2.19, the graph |M| vs. U/e should present a critical value of 0.75. For e > 1, magnetization curves tend to be around this value, but curve tend to move to the left while increasing e. On the other hand, when the coupling parameter does not exceed 1, the magnetization curve moves rapidly to the right. This may come from the simulation parameters, in which kinetic energy was fixed to sweep different energy values, and this causes little e to be almost irrelevant to overcome the internal fluctuations of the particles in order to impose a magnetization on the system.

2.3.2 Fluctuations Kinetic energy fluctuations are important to gain understanding about the system response to time evolution. Actually, the way energy varies in the system is also an indicator of different phases of the system as it can checked in figure 2.5. Such as in figure 2.6, the simulated system behaves different from the theoretical prediction, but near the simulation critical energy, fluctuations reach a maximum value and suddenly decrease. This is expected for this system as in the ferromagnetic phase interactions are more relevant and energy conservation becomes a matter of changing particles momenta. As in simulations it is needed to deal with the system in a microcanonical way, because energy may remain fixed throughout the numerical integration, it is important to point out that, according to Lebowitz, Percus, and Verlet, 1967, there is an important dependence of extensive observables averages on q 2 the ensemble considered. Therefore, in order to compute Σµ ≡ σµ/P, that is microcanonical kinetic energy fluctuations standard deviation per particle, it is needed to use 2 σµ 1 1 = 1 − (2.27) P 2β2 2c(β) where P is the number of particles and the inverse temperature β ≡ 1/kBTµ is taken from equipartition of energy Tµ = 2hKiµ/N (Lebowitz, Percus, and Verlet, 1967). Using the latter the microcanonical heat capacity can be explicitly calculated (here, lowercase variables will mean per particle) 40 Chapter 2. Statistical Mechanics

1 2 dhHiµ 1 dhHi P d(hkiµ + (1 − m )) C(β) = = = 2 (2.28) dTµ 2 dhkiµ 2 dhkiµ P dm P dm = 1 − m = 1 − 2m (2.29) 2 dhkiµ 2 dTµ

1 2 as the potential energy of the system satisfies hViµ = V = 2 (1 − m ) which coincides with the mean field universality class of the system. Therefore, replacing in equation 2.28 s 1 1 Σµ = √ 1 − (2.30) 2β 2 1 − 2m(dm/dTµ) Using figure 2.6 data, the derivative involved in the latter calculation can be found explicitly with numerical differentiation and then figure 2.5 is obtained. That is the reason for the small deviation from the theoretical prediction, and also for the nonzero value after the critical energy, as system fluctuations depend on interactions given the finite size effects. However, from the different number of particles it can be predicted that as P → ∞, the fluctuations will tend to decrease even more sudden, according to thermodynamic limit prediction.

2.4 Chaos in the Fully Coupled Rotators Model

As it was checked for the Lorenz model in1, Lyapunov exponents and fractal dimension give a description of a dynamical system which becomes chaotic changing an internal parameter. For the Fully Coupled Rotators Model a similar study can be done in order to analyze the underlying time evolution of the microscopic system. In addition, Hausdorff dimension estimates using equations 1.12 and 1.13, and Kolmogorov-Sinai entropy were calculated from the Lyapunov spectra and reveal similar change of characteristics depending on the phase in which they are computed.

2.4.1 Lyapunov exponents Given the dynamical system described by 2.25, Lyapunov exponents were calculated using the approximation explained in 1.8. For different energies, ﬁgure 2.6 contains the results of the simulations; it is possible to check that there is a correspondence between the phase transition of the rotators from a ferromagnetic to a paramagnetic phase and the sudden decrease of the Lyapunov exponents. Moreover, beyond the critical energy the exponents decrease monotonically and tend to 0 as U → ∞, which is expected as in the high energy limit the model reduces to a set of moving particles which do not interact. Moreover, all 2.4. Chaos in the Fully Coupled Rotators Model 41

Theoretical 0.35 P=100 P=400 0.30 P=800 P=1000

0.25

0.20

0.15

0.10

0.05

0.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 U

FIGURE 2.5: Fluctuations of the Fully Coupled Rotators Model as a function of internal energy.

Lyapunov exponents curves show a region (0 < U < 0.5) in which exponents also decrease, corresponding to an energetic regime in which the system exits a low energy limit corresponding to strongly coupled oscillators, which is an integrable system described using normal modes. Examining the behavior of the Lyapunov exponents curve for U > Uc, it can be checked that the dependence L behaves as λ1 ∼ (U − Uc) where L is an exponent depending on the number of particles. In figure 2.7 is evident that L is a decreasing function, therefore it is expected a thermodynamic limit behaviour as a step function, i.e. acquiring a + − positive value for U → Uc and having a zero value for U → Uc . In figure 2.8 it can be seen explicitly that the phase transition of the system studied statistically corresponds with an internal behavior of the dynamical system underneath. It is worth noticing that Lyapunov exponents and fluctuations (fig. 2.6 and fig. 2.5 respect.) behave very similarly. In fact, as it was mentioned before, the decreasing of λ1 exceeding the critical energy for the system is more evident in figure 2.8 when, for a fixed number of particles, the interaction term e varies. In this figure the independent was chosen to be U/e in order to compare systematically results in view of equation 2.19. In terms of phases, the principal Lyapunov exponents behave differently in the ferromagnetic or the paramagnetic system. When the overall magnetization of the system tends to zero, λ1 also decreases rapidly to zero (see 2.7); this was 42 Chapter 2. Statistical Mechanics

1.0 Theoretical P = 100 P = 400 0.8 P = 800 P = 1000

0.6 |M| 0.4

0.2

0.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.175

0.150

0.125 1 0.100

0.075

0.050

0.025

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 U

FIGURE 2.6: Lyapunov exponents of the Fully Coupled Rotators System as a function of energy for different number of particles. 2.4. Chaos in the Fully Coupled Rotators Model 43

0.3

0.4 c U > 0.5 U

r o f

t n e

n 0.6 o p x E

0.7

0.8 200 400 600 800 1000 Number of Particles

FIGURE 2.7: Value of L characterizing Lyapunov exponents for en- L ergies beyond the critical value which behave as λ1 ∼ (U − Uc) , varying the number of particles of the system. 44 Chapter 2. Statistical Mechanics

1.0 = 0.5 = 1.5 0.8 = 2.0 = 2.5 = 3.0 0.6 |M| 0.4

0.2

0.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.8

0.6 1

0.4

0.2

0.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 U/

FIGURE 2.8: Correspondence between critical energies for different values of e and the emergent discontinuity in Lyapunov exponents curves for 400 particles in the system. 2.4. Chaos in the Fully Coupled Rotators Model 45 expected because particles in this phase need not interact so much, so the system may tend to a free particle model, which is obviously integrable. In the other phase, Lyapunov exponents behave in a non trivial way. For the low energy regime it was expected λ1 → 0 because the system reduces to a coupled oscillators model in which angles of motion are small due to little energy, which is well described described by normal modes. With an increase of energy there is a corresponding growth of the exponent, that means the system starts to be disordered, such as in the Lorenz model. But in the vicinity of the critical energy a maximum value is achieved followed by the sudden decrease in the paramagnetic phase, contrary to the behavior in Lorenz system. Here it is worth noting that this maximum of the Lyapunov exponent curve depends strongly on e, and this would imply that chaos in the system is mediated by the strength of the interaction between particles; that is, for a greater value of e the system becomes more unpredictable, but only in the ferromagnetic phase. Moreover, the structure gained in the phase space orbits because of the raise of λ1 is maintained even though Lyapunov exponents do not show it. This is where Hausdorff dimension becomes a useful and interesting tool to analyze chaos in the system.

2.4.2 Dimension To analyze dimension according to equations 1.12 and 1.13 the Lyapunov spectra of the system is needed. This becomes a computational difficulty as the degrees of freedom for the model grow linearly with the number of particles and the calculation of higher order Lyapunov exponents involves a Gram-Schmidt orthonormalization process of a basis of vectors which are labeled by these degrees of freedom. All of this implies that dimension analysis of the system may be done in low number of particles (up to P = 100) in order to keep a prudent simulation time. In figure 2.9 the number of particles for the system was varied. In the magnetization curves finite size effects are evident again for the critical energy, understood as the point in which magnetization stabilizes in a value near 0, does not coincide with equation 2.19. However, dimension estimates are not affected by this size effects in terms of the different characteristics the curve acquire depending on the energetic regime in which they are found. Increasing the value of the internal energy, the system tends to a dimension which is equal to the number of degrees of freedom 2P (recall 2.9 and 2.10 show dimension per degree of freedom to normalize), which is a non-chaotic condition. Nevertheless, in the ferromagnetic regime the dimension is less than the number of degrees of freedom, yielding a chaotic condition that coincide with the results obtained with the principal Lyapunov exponent. To analyze this dimension physically Grassberger and Procaccia, 1983, theory can be considered. They showed that dimension D of the system is also an estimate for the correlation integral exponential law. This correlation integral is 46 Chapter 2. Statistical Mechanics defined as 1 N C(r) = lim θ(r − |~xi − ~xj|) (2.31) N→∞ 2 ∑ N i,j=1 where ~xi is a point in the phase space and θ is the Heaviside function. This is a correlation expression as it counts the density of particles which are separated by a distance lesser or equal to r. For small values of r, this correlation behaves as C(r) ∼ r−ν (2.32) and this exponent satisfy ν ≤ D. Therefore dimension contains information about how, in average, two close points in the phase space of the system are related (spatially). So figure 2.9 shows that particles in the different phases of the system tend to correlate differently, and that for paramagnetic regimes this correlation is independent of the internal energy of the system. An interesting question would be to check if there is some kind of relationship between this dimension for the dynamical system and the characteristic correlation length of the system ξ.

2.4.3 Kolmogorov-Sinai entropy The behavior of the Kolmogorov-Sinai entropy is related to the Lyapunov exponents one because of equation 1.16. Nevertheless, the physical interpretation of these quantities is very different. while Lyapunov exponents give information about unpredictability for the system evolution, the entropy can be related to the complexity of the orbits on the phase space in a manner similar to the dimension. But there is a subtle difference between these two. While the latter quanti- ﬁes the way orbits cover the phase space, the former is more sensitive about the self-similarity condition of the fractal structure arising in chaotic dynamics. For instance, in the Lorenz model, in the chaotic regime of r’s, SKS = λ1 as the other two Lyapunov exponents were always negative, and also D = 2.06. However, comparing 1.9 and 1.10 it is possible to notice that while D remains constant, SKS increases monotonically, as the Lorenz attractor gains structure for greater values of the order parameter. In this case, the entropy tends to decrease reﬂecting a loose of fractal structure in the orbits of the system while this approaches the non-chaotic behavior predicted with Lyapunov exponents. Lets point out this is a non-trivial result as it was shown that the structure of the increasing chaos remained in a dimensional characterization, but its fractal arrangement tend to vanish with energy. 2.4. Chaos in the Fully Coupled Rotators Model 47

1.0 P = 20 P = 30 P = 40 0.8 P = 50 P = 60 0.6 | P = 70

M P = 100 | 0.4

0.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

1.1

1.0 i r 0.9 o M

D 0.8

0.7

0.6

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

1.05

1.00

0.95

Y 0.90 K

D 0.85

0.80

0.75

0.70

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 U

FIGURE 2.9: Hausdorff dimension per degree of freedom estimations for the Fully Coupled Rotators Model as a function of internal energy U for different number of particles. 48 Chapter 2. Statistical Mechanics

1.0 = 0.5 = 1.0 = 1.5 0.8 = 2.0 = 3.0

| 0.6 = 5.0 M |

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.4

1.2

i 1.0 r o M

D 0.8

0.6

0.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.0

0.9 Y K 0.8 D

0.7

0.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 U/

FIGURE 2.10: Hausdorff dimension per degree of freedom estimations for the Fully Coupled Rotators Model as a function of the re-scaled energy U/e for varying values of the coupling e. 2.4. Chaos in the Fully Coupled Rotators Model 49

1.0 P = 20 P = 30 P = 40 0.8 P = 50 P = 60 P = 70 0.6 P = 100 | M |

0.4

0.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.10

0.08

P 0.06 / S K S 0.04

0.02

0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 U

FIGURE 2.11: Kolmogorov-Sinai entropy per particle for the Fully Coupled Rotators Model for varying number of particles. 50 Chapter 2. Statistical Mechanics

1.0 = 0.5 = 1.0 = 1.5 0.8 = 2.0 = 3.0 = 5.0

| 0.6 M |

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.25

0.20

P 0.15 / S K

S 0.10

0.05

0.00

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 U/

FIGURE 2.12: Kolmogorov-Sinai entropy per particle for the Fully Coupled Rotators Model as a function of U/e for different values of the interaction term. 2.5. Conclusions, conjectures and further work 51

2.5 Conclusions, conjectures and further work

The main goal of this work was to exhibit how statistical mechanics’ characteristics of a system may have a dynamical correspondence in terms of different behaviors of somehow equivalents to state functions for a dynamical system arising from Hamilton equations. For this end, Lyapunov exponents, Hausdorff dimension and Kolmogorov -Sinai entropy turned out to be useful and showed interesting similarities and differences in the ferromagnetic and paramagnetic phases of a mean-ﬁeld model. Arising from the analysis made about this chaotic indicators, there are some loose ends and possible conjectures which are worth mentioning. Perhaps the most important feature that is possible to extract from this relationship could be a determination of the critical energy/temperature for a many- body system by using a dynamical system approach. From the present graphs it may not be that easy to state a critical value for the energy due to the limited number of particles and time steps taken in the simulations, a trade off between computation time and precision in averages which mainly affects magnetization curves. Now, regarding Lyapunov exponents, an initial conjecture about the behavior of the curve λ1 vs. U in the thermodynamic limit was already mentioned and tested partially using a raw regression on the vanishing exponent for λ1. It is also unclear if there is some clever scaling for the curves in 2.8 to be properly normalized. This could enlight some additional property connecting the statistical mechanics point of view and the dynamical system which could also clarify the presence of a emergent discontinuity near the phase transition. An interesting feature of the dimension curves (see ﬁg. 2.9, 2.10) may be pointed out. Near the expected critical parameter Uc = 3e/4 the dimension seems to stabilize, which could also be used as a signal for the phase transition. However, contrasting with Lyapunov exponents discontinuity, this approaching of the dimension seems continuous and, at least, one time differentiable. Testing for more particles could give more information about this curves and the behavior of their derivatives near the phase transition point. This may recall Yang and Lee, 1952, work, and therefore the different characteristic of this curve, and the Lyapunov exponents one, could depend on the universality class that is considered; this also raises a question of inverse engineering in which maybe some aspects about critical exponents for the system can be extracted from this curves. These questions can be explored using different models and approaches to many-body systems. For instance, studying Potts Models or two dimensional systems could give some insight about the dependence of Lyapunov exponents and Hausdorff dimensions on the type of phase transition. Of course extend- ing number of particles and computation time can give exactness to a possible determination of critical parameters. Also, other approaches as renormaliza- tion group or Landau-Ginzburg theory could be useful to explore criticallity and phase space correlations in view of the fractal structure which is present in 52 Chapter 2. Statistical Mechanics the ferromagnetic phase (at least of the Fully Coupled Rotators Model). So there may be a whole spectra of possibilities to be explored to get the most out of a emergent correspondence of statistical mechanics and chaotic dynamics. 53

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