Differential Equations (18.03)


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. Problems of interest include: river networks, drainage basins, and the shape of topography; percolation theory and transport in disordered media; fractals, scaling, and universality; the organization of microbial communities; ecological dynamics, metabolic scaling, and the structure of ecosystems; food webs and other natural networks; and kinetics of biogeochemical cycles.


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 the metabolic foundations of ecological organization. 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.


There are 5 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 exams.


  1. Introduction

    1.1 What is environmental complexity?
    1.2 Why study complexity?
    1.3 Complexity can emerge from simple interactions
    1.4 Objectives
    1.5 Requirements

  2. From microdynamics to macrodynamics

    2.1 Random walks

    2.1.1 One-dimension, discrete time and space
    2.1.2 Higher dimensions
    2.1.3 The binomial distribution and the Gaussian limit
    2.1.4 Central-limit theorem
    2.1.5 Macrodynamics: the diffusion equation

    2.2 The lattice gas

    2.2.1 Microdynamical equations
    2.2.2 Macrodynamical equations of the lattice gas
    2.2.3 Symmetry
    2.2.4 Separation of scales

  3. River networks

    3.1 Scale invariance of random walks
    3.2 Allometric scaling
    3.3 Size distribution of river basins
    3.4 Scaling relation
    3.5 Universality classes
    3.6 Sandpiles and self-organized criticality

    3.6.1 Avalanches
    3.6.2 Earthquakes
    3.6.3 Self-organized criticality
    3.6.4 Directed sandpiles

    3.7 The lesson learned

  4. Anomalous diffusion

    4.1 Beyond the central limit theorem
    4.2 Large fluctuations and scale invariance
    4.3 Lévy flights
    4.4 Continuous time random walk
    4.5 Diffusion on a comb

    4.5.1 Infinite L
    4.5.2 Finite L

    4.6 Lévy walks
    4.7 The lessons learned

  5. Statistical topography

    5.1 Self-affine surfaces

    5.1.1 Width and power spectra
    5.1.2 Wiener-Kintchine theorem
    5.1.3 Examples

    5.2 Discrete models of growing self-affine surface

    5.2.1 Random deposition
    5.2.2 Random deposition with surface diffusion
    5.2.3 Cluster aggregation

    5.3 Continuum models I: random deposition
    5.4 Continuum models II: noisy diffusion

    5.4.1 Fourier representation
    5.4.2 The width W (L, t)
    5.4.3 Evolution of the power spectrum
    5.4.4 Two dimensions

    5.5 Continuum models III: the KPZ equation

    5.5.1 Origin of the nonlinearity
    5.5.2 Qualitative behavior
    5.5.3 Roughness exponent α via the Fokker-Planck equation
    5.5.4 Scaling argument for time dependent roughening
    5.5.5 Summary and applications

    5.6 Gaussian surfaces [1, 2]

    5.6.1 Preliminaries
    5.6.2 Random phases imply Gaussian heights
    5.6.3 Distribution of gradients
    5.6.4 Slopes are uncorrelated to heights
    5.6.5 Slope-area relations [3, 4]
    5.6.6 Quantitative null hypothesis

  6. Percolation theory

    6.1 Introduction

    6.1.1 What's percolation?
    6.1.2 Examples

    6.2 Percolation in one dimension
    6.3 Clusters in two dimensions
    6.4 Percolation on the Bethe lattice

    6.4.1 Infinite dimensionality
    6.4.2 Percolation threshold
    6.4.3 The strength P
    6.4.4 Mean cluster size S
    6.4.5 Cluster numbers ns (p)

    6.5 Scaling laws in d-dimensions

    6.5.1 The strength P
    6.5.2 The mean cluster size S

    6.6 Fractals

    6.6.1 Cluster radius
    6.6.2 Correlation length
    6.6.3 The fractal dimension D
    6.6.4 Scaling of the infinite cluster at p = pc
    6.6.5 Relating D to d via the correlation length ξ

    6.7 Finite-size scaling

    6.8 Renormalization

    6.8.1 Self-similarity
    6.8.2 Real-space renormalization
    6.8.3 Calculation of the correlation-length exponent ν
    6.8.4 One dimension
    6.8.5 Triangular lattice

  7. Origin of biogeochemical cycles

    7.1 The carbon cycle

    7.1.1 The biological cycle
    7.1.2 The rock cycle

    7.2 Energy flow
    7.3 Two-reservoir model
    7.4 Reactive species

    7.4.1 No equilibrium solution
    7.4.2 Cycles

    7.5 Cycles and the breaking of detailed balance
    7.6 Summary

  8. Disordered Kinetics

    8.1 Relaxation in the carbon cycle
    8.2 Relaxation rate constants

    8.2.1 Arrhenius kinetics
    8.2.2 First order decay
    8.2.3 Aging
    8.2.4 Rationalizing the aging effect

    8.3 Disordered kinetics
    8.4 Random rate models

    8.4.1 Preservation of static disorder
    8.4.2 Continuous superposition

    8.5 Random channel model

    8.5.1 Fixed rates
    8.5.2 Fluctuating rates
    8.5.3 Decay function

    8.6 Relation between random rates and random channels
    8.7 Universal random rate distribution