Course guide of Physics of Complex Systems (26711I2)

General knowledge of Mathematics and Physics (particularly, Mechanics) is required, as acquired, for example, in the basic and mandatory subjects of the first courses of the degree in Physics. It would also be advisable to have taken "Computational Physics", an optional third-year course, as well as third-year Statistical Physics.

In the case of using AI tools for the development of the course, the student must adopt an ethical and responsible use of them. The recommendations contained in the document 'Recommendations for the use of artificial intelligence at the UGR', published at this location, must be followed: https://ceprud.ugr.es/formacion-tic/inteligencia-artificial/recomendaciones-ia#contenido0

Introduction. Complexity. Chaos. Fractal geometry. Scale invariance. Collective or cooperative phenomena. Critical phenomena. Pattern formation.

By this semester, the student already knows the microscopic and macroscopic descriptions of physics as provided, respectively, by Classical and Quantum Mechanics as well as the basic concepts of Thermodynamics and Statistical Physics, which rigorously relate these descriptions in the case of systems in thermodynamic equilibrium. However, thermodynamic equilibrium is a special circumstance that usually does not occur in the cases that are currently of most interest to Science, such as, for example, when turbulent regimes are established in a fluid, or when chemical compounds grouped together to achieve the first hint of independent life, or the nervous system achieves high-level information processing functions through the coordination of individual groups of neurons, or epidemics spread through a network of interactions. The concept of a complex system, capable of showing a fascinating phenomenology due to cooperation between elements, is then relevant. The recent study in physics of these complex systems has led to the development of powerful analysis methods that rely on computing and has generated or renewed concepts, all of which transcend the boundaries of physics to invade the foundations of other sciences, including biology and sociology. This is the situation that the course aims to describe, while also intending to help the student to:

1. Develop their skills to analyze and capture the essentials in natural systems and processes through algorithms, thus learning to effectively and accurately solve various problems,
2. Use computers creatively in modeling situations of interest in science, technology, and management, and
3. Face, if they wish to do so, the challenges that current research in public or private centers can pose once they graduate.

1. Introduction. Complexity. Order and entropy in nature. Cooperative effects in statistical physics. Scales and levels of description. Non-linearity. Predictability. Measures of complexity.
2. Theory of dynamic systems. Introduction to the theory of dynamic systems and chaos (Poincaré. Lorenz. May. Feigenbaum). Non-linear maps. Fixed points, limit cycles, and strange attractors. Stability theory in one-dimensional and two-dimensional differential equations. Bifurcation and catastrophe theory. Lyapunov coefficients. Universality. Integrability and Hamiltonian chaos.
3. Scale invariance. Power laws. Mechanisms of power law generation. Fractal geometry. Regularity, randomness, and self-similarity. Hausdorff or fractal dimension. (Multifractality). Roughness and self-affine structures.
4. Theory of stochastic processes. Brief historical introduction. Brownian motion. Random walker. Einstein's theory. Perrin's experiments. Markov processes. Master equation. Stochastic equations: Langevin and Fokker Planck. Path integrals. Levy flights.
5. Phase transition theory (I): Percolation. Scale invariance at the critical point. Introduction to the renormalization group. Dynamic percolation (forest fires). Directed percolation and the contact process.
6. Phase transition theory (II): Guggenheim curve and universality. Lattice models (Heisenberg, XY, etc.). Ising model. Spontaneous symmetry breaking. Order parameter and control parameter. Correlations and fluctuations. Critical exponents and scaling laws. Mean field theory. Ginzburg Landau theory. Ginzburg's criterion. Kadanoff blocks and real space renormalization.
7. Self-organization and criticality. Sandpiles. Earthquakes. Criticality in biology.
8. Other concepts that will be taught in specialized seminars: Introduction to the theory of complex networks. Evolutionary game theory. Applications: Neuroscience, Theoretical and Evolutionary Ecology, Systems Biology, etc.

A supervised research project will be developed, related to some of the content covered in the theoretical syllabus. This project will allow students to delve deeper into a specific problem of interest within the field of complex systems, combining a literature review, the formulation of research questions, theoretical analysis and, where appropriate, numerical simulation or data study.

The main objective will be to foster the integration of the concepts covered in class through their application to a specific case. To this end, each student or group will select, with the guidance of the teaching staff, a topic linked to areas such as nonlinear dynamics, chaos, stochastic processes, scale invariance, fractals, percolation, phase transitions, self-organized criticality, complex networks, or applications in neuroscience, ecology, systems biology, or other complex systems.

The project will be developed progressively, including the definition of the problem, the search and critical analysis of scientific literature, the setting of objectives, the application of appropriate theoretical or computational tools, and the preparation of a final report. Where applicable, it may be complemented by an oral presentation outlining the motivation, the methodology used, the main results obtained, and the conclusions reached.

- D. Sornette, "Critical Phenomena in Natural Sciences", Springer 2009.

- J. Sethna, "Entropy, Order parameters and Complexity". Oxford 2015.

- J.J. Binney et al. "The theory of Critical Phenomena". Oxford. 1999.

- S.H. Strogatz, "Non-linear dynamics and Chaos", Addison Wisley 2012.

- A. Fuchs, "Nonlinear dynamics in complex systems", Springer 2013.

- K. Christensen and N.R. Moloney, "Complexity and Criticality", Imperial College, London 2005.

- R.J. Creswick et al., "Introduction to Renormalization Group Methods in Physics”, Wiley, NY 1992.

- J. Marro and R. Dickman, "Nonequilibrium Phase Transitions in Lattice Systems". Cambridge 2005.

- C.W. Gardiner, “Hanbdbook of stochastic methods”, Springer Verlag, 2000.

- N.G. van Kampen, “Stochastic processes in Physics and Chemistry”, Springer, 2004

- M. Newman, "Networks: An introduction", Oxford 2011.

- A. Pikovsky et al. Synchronization: A universal concept in nonlinear sciences. Cambridge 2003.

- E. Ott, Chaos and dynamical systems, Cambridge, 2012.

- H. J. Jensen, "Self-Organized Criticality", Cambridge Univ. Press 1998.

- P. Krapivsky, S. Redner, E. Ben-Naim, "A kinetic view of Statistical Physics", Cambridge 2010.

• MD01. Theoretical classes

Students must demonstrate a balanced and sufficient knowledge of the subject matter as a whole. This knowledge is acquired through active participation in face-to-face sessions, continuous tracking of the course, and the regular completion of problems and exercises. Therefore, continuous attendance, classroom involvement, and the submission of activities will form an essential part of the evaluation process.

Likewise, in order to encourage the progressive tracking of the course and verify the gradual acquisition of content, the teaching staff may conduct small written or oral tests at the end of each chapter or thematic block. These tests will be formative and evaluative in nature, allowing for the assessment of both the understanding of fundamental concepts and the students' ability to apply them to specific problems.

Furthermore, students must delve deeper into one of the characteristic topics of the course by undertaking a personal, supervised research project. This project will allow them to apply the concepts covered in the syllabus, develop critical analysis skills, and become familiar with the scientific literature related to complex systems.

Assessment will be continuous throughout the course and will be completed at its conclusion through oral and/or written presentations agreed upon between the students and the faculty.

The final grade will be distributed as follows:

• Supervised research project: 2/3 of the final grade.

• Submitted problems, exercises, participation, and continuous work: 1/3 of the final grade.

The teaching staff reserves the right to conduct an oral exam on the research project to verify the student's degree of mastery over the presented material.

By mutual agreement with the teaching staff, the supervised personal project may be replaced by a final exam, which will be weighted equally toward the final grade.

Assessment due to incidents: students who are unable to attend the final evaluation exams—whether ordinary, extraordinary, or single final assessment—as well as the tests scheduled in the Teaching Guide with an official date, may request an assessment due to incidents, provided that one of the circumstances set out in Article 9 of the Assessment and Grading Regulations for Students of the University of Granada applies. The application and processing must be carried out in accordance with the procedure established in said regulations.

Students must demonstrate a balanced and sufficient knowledge of the subject matter as a whole. To this end, the student will take a single final exam, preferably oral, which will account for 100% of the final grade for the course.

In accordance with the "Normativa de Aplicación y de Calificación de los Estudiantes" of the University of Granada, the possibility of a single final assessment is available. Students who are unable to comply with the continuous assessment system for any of the reasons set out in Article 8 of the said regulations may opt for this modality.

To apply for the single final assessment, the student must request it through the electronic office (sede electrónica) of the University of Granada during the first two weeks of the course; within the two weeks following their enrollment, if this occurred after the start of the course; or at a later date, if there is a duly justified, unforeseen circumstance. In all cases, the student must state and provide documentary evidence of the reasons that prevent them from following the continuous assessment system.

The single final assessment will consist of two parts: a written exam and an oral exam. The written exam will account for 2/3 of the final grade, while the oral exam will account for 1/3. Both tests will be designed to verify that the student has acquired all the learning outcomes planned for the course, allowing them to achieve 100% of the final grade through this modality.

Students with Specific Educational Support Needs (NEAE or “Necesidades específicas de apoyo educativo”).

Following the recommendations of the CRUE (Conference of Rectors of Spanish Universities) and the Secretariat for Inclusion and Diversity of the University of Granada, the teaching, learning, and competence assessment systems outlined in this course syllabus will be applied in accordance with the principle of universal design for all people. Therefore, efforts will be made to facilitate access to learning and the demonstration of acquired knowledge and skills, taking into account the specific needs and functional diversity of the students.

The teaching methodology and assessment procedures may be adapted for students with disabilities or other specific educational support needs, in accordance with the provisions of Article 11 of the Student Assessment and Grading Regulations of the University of Granada, published in the Official Bulletin of the UGR number 112, of November 9, 2016.

In these cases, the tutoring system will be adapted, when necessary, to the particular circumstances of the students, following the recommendations of the Inclusion Unit of the University of Granada. The responsible departments and centers will establish the appropriate measures to ensure that tutoring sessions take place in suitable conditions and, in particular, in accessible spaces.

Furthermore, when special methodological adaptations are required, the teaching staff may request support and advice from the competent unit of the University of Granada in order to ensure adequate attention that is compatible with the educational objectives of the course. Students with disabilities and/or specific educational support needs can consult information on the management of services and support on the University of Granada's webpage dedicated to social care and students with disabilities: https://ve.ugr.es/servicios/atencion-social/estudiantes-con-discapacidad

For any additional clarification regarding the specific application of these measures in this course, it is recommended to contact the responsible instructor.
Information of interest for students with disabilities and/or Specific Educational Support Needs (NEAE): Management of services and support (https://ve.ugr.es/servicios/atencion-social/estudiantes-con-discapacidad).

Información de interés para estudiantado con discapacidad y/o Necesidades Específicas de Apoyo Educativo (NEAE): Gestión de servicios y apoyos (https://ve.ugr.es/servicios/atencion-social/estudiantes-con-discapacidad).