### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Prerequisites

Students should have completed *18.03SC Differential Equations* or its equivalent and have some familiarity with partial differential equations.

### Description

This course provides an introduction to the study of environmental phenomena that exhibit both organized structure and wide variability—i.e., complexity. Emphasis is on the development of quantitative theoretical models, with special attention given to macroscopic continuum or statistical descriptions of microscopic dynamics. Concepts and problems include the microdynamics and macrodynamics of random walks and fluid flow; extreme deviations and anomalous diffusion; geological and ecological networks; percolation theory; dynamical origin of fractals and scale invariance; the origin and complex kinetics of biogeochemical cycles.

### Objectives

Through focused study of a variety of physical, biological, and chemical problems in conjunction with theoretical models, students learn a series of lessons with wide applicability to understanding the structure and organization of the natural world. Such lessons include: How complexity can derive from simple dynamics; why fractals are ubiquitous in the natural world; and generic consequences of complex biogeochemical kinetics. Students will also acquire specific skills, including: The statistical analysis of data with wide variability; how to use computer simulations to reveal fundamental phenomena; and how to construct a minimal model of a complex system that provides informative answers to precise questions. A unifying theme is the relation of macroscopic complexity to microscopic dynamics.

### Topics

- Introduction.
- What is environmental complexity?
- Why study it?
- Complexity can emerge from simple interactions.
- Objectives; course overview.

- From microdynamics to macrodynamics I: Random walks.
- From random walks to diffusion.
- The central limit theorem, Gaussian fluctuations, and the concept of universality.
- Lesson: Temporal and spatial averaging of random movements yields diffusion; Gaussian fluctuations are ubiquitous.

- From microdynamics to macrodynamics II: The lattice gas.
- The microdynamical square dance: Cellular automata.
- The macrodynamical fluid: Navier-stokes equations.
- Conservation laws, symmetry, and the separation of scales.
- Lesson: Complex macrodynamics—e.g., turbulence—can arise from simple microdynamics; identification of appropriate symmetries and conservation laws can suffice for correctness.

- The geometry of aggregating networks, especially rivers.
- The laws of Horton and Hack: Scale invariance, fractals, and allometric scaling.
- Scheidegger’s model of river networks.
- Universality classes.
- Lesson: The aggregation of random walks yields network geometries quantitatively consistent with real river networks.

- Self-organized criticality.
- Physical cartoons: Sandpiles, avalanches, and earthquakes.
- Relation to Scheidegger’s rivers.
- Equilibrium critical phenomena and non-equilibrium steady states.
- Lesson: Scale-invariant fluctuations can arise generically, without “tuning” to a critical point.

- Large deviations and anomalous diffusion.
- Beyond the central limit theorem: Long-tailed distributions and scale invariance.
- Levy flights; continuous time random walk.
- Diffusion in disordered media; first passage times.
- Possible relations to the movement of animals, people, and groundwater.
- Lesson: Long tails beget long tails, but transport through disordered media provides them for free.

- Percolation theory.
- Examples: Transport in porous media, forest fires, and epidemics.
- One-dimensional representation.
- Percolation on the Bethe lattice.
- Scaling laws in d-dimensions.
- Fractals, scaling, and universality at the percolation threshold.
- Finite-size scaling and renormalization.
- Lesson: Critical points generate scale-invariant phenomena.

- The geometry of random networks, including food webs.
- Connectivity, clusters, and the Erdos-renyi random network.
- Preferential attachment and the Barabasi-albert scale-free network.
- Food webs.
- Lesson: Mechanisms of connectivity can strongly influence network geometry.

- Global metabolism: The origin and structure of biogeochemical cycles.
- The biological and geological carbon cycles.
- Energy sources and sinks.
- Cycles and irreversible thermodynamics.
- Lesson: Cycles are expected in open systems held out of equilibrium.

- Disordered kinetics. 1. Averaging over first-order kinetics. 2. Aging and the 1 / t decay of apparent rate constants. 3. Random rate and random channel models. 4. The lognormal distribution and its ubiquity in natural systems. 5. Lesson: Complex kinetics can derive from aggregated first-order kinetics.

### Requirements

There are 3 project-oriented problem sets. A final, independent project on a topic of the student’s choice is due at the end of the term, including a written report and an oral presentation. There is no exam.

### Grading

ACTIVITIES | PERCENTAGES |
---|---|

Assignments | 40% |

Final Project | 40% |

Class Participation | 20% |