UNIVERSIDAD DE LOS 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 iii 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: v “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 vii 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 re- gion 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 dy- namical system which arise from Hamilton equations was done. The Lyapunov exponents, Hausdorff dimension and Kolmogorov Sinai entropy gave interest- ing 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. xi 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 con- dition (0.8, 0.5) ..............................2 1.2 Numerical solution of equation 1.3 using a fourth-order Runge- Kutta algorithm. An w-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 ini- tial conditions............................... 16 1.8 Convergence of Lyapunov exponents calculations for different ba- sis...................................... 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 pq2 0.1........ 23 = 1.14 Trajectories for the double pendulum with pq2 1.0........ 24 = 1.15 Trajectories for the double pendulum with pq2 2.7........ 25 = 1.16 Trajectories for the double pendulum with pq2 3.5........ 26 = 1.17 Trajectories for the double pendulum with pq2 5.0........ 27 1.18 Lyapunov exponents for the double pendulum as a function of pq2 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 l1 ∼ (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 en- ergy 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