Statistical Physics Applied to Geophysics

--- Elective Course; 3 credits ---

Natural phenomena such as earthquakes, extreme weather events, and global warming pose significant risks to humanity. Due to the non-linear and long-range interactions between the various elements of terrestrial structures, the understanding and, in particular, the prediction of such disruptive events represent enormous challenges for the scientific community. During the last few years, the emergence and evolution of Earth system science have attracted much attention and produced new concepts and frameworks. In particular, new statistical physics techniques and techniques based on complex systems and networks were developed and implemented to advance the Earth system's knowledge substantially.

We present here a comprehensive review of recent scientific progress in the development and application of combined approaches from statistical physics and complex systems science (such as critical phenomena, network theory, fractals, and entropy) applied to geophysical systems. Remarkably, these integration tools and approaches provide new insights and perspectives for understanding the dynamics of Earth systems. Thus, the general objective of this course is to highlight how concepts and theories of statistical physics can be helpful in the field of Geophysics.

Topics:

1. Fundamentals of Statistical Mechanics

1.1- Elementary concepts of statistics
1.2- Notions of probability theory
1.3- Average values and moments calculations
1.4- Probability distributions
1.5- Boltzmann Distribution
1.6- Thermodynamic equilibrium
1.7- Entropy as: irreversibility, disorder, and uncertainty.

2- Complex Systems

2.1- Complex Systems and Complexity
2.2- Emergency and Self-Organization
2.3- Self-Organized Criticality
2.4- Power Law Distributions
2.5- Fractals and Scale Laws
2.6- Applications in Geophysics

3- Nonextensive Statistical Mechanics

3.1- Extensivity and Additivity
3.2- Limitations of Boltzmann Formalism in Complex Systems
3.3- Tsallis' entropy
3.4- q-exponential functions
3.5- Mathematical Properties
3.6- Self-similarity, memory effects, long-range correlations
3.6- Applications in Geophysics

4- Complex Networks

4.1- Graph Theory
4.2- Adjacency Matrix
4.3- Simple and Weighted Networks
4.4- Regular Networks
4.5- Random Networks
4.6- Small-world Phenomenon
4.7- Scale-free Networks
4.8- Measurements in Networks:
-    connectivity
-    average connectivity
-    connectivity distribution
-    clustering coefficient
-    average path length
-    small-world-ness
-    correlation properties
4.9- Communities Detection
4.10- Applications in Geophysics

5- Computational Applications

5.1- Models and algorithms for building networks from initial data
5.2- Complex network models: random, small-world, and scale-free
5.3- Extraction of patterns through community detection in complex networks
5.4- Cellular automaton applied to Geophysics

Bibliography

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