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| Preface | |
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| Percolation | |
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| Introduction | |
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| Definition of site percolation | |
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| Quantities of interest | |
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| Percolation in d = 1 | |
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| Cluster number density | |
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| Average cluster size | |
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| Transition to percolation | |
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| Correlation function | |
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| Critical occupation probability | |
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| Percolation on the Bethe Lattice | |
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| Definition of the Bethe lattice | |
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| Critical occupation probability | |
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| Average cluster size | |
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| Transition to percolation | |
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| Cluster number density | |
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| Correlation function | |
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| Percolation in d = 2 | |
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| Transition to percolation | |
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| Average cluster size | |
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| Cluster number density - exact | |
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| Cluster number density - numerical | |
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| Cluster Number Density - Scaling Ansatz | |
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| Scaling function and data collapse | |
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| Scaling function and data collapse in d = 1 | |
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| Scaling function and data collapse on the Bethe lattice | |
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| Scaling function and data collapse in d = 2 | |
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| Scaling Relations | |
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| Geometric Properties of Clusters | |
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| Self-similarity and fractal dimension | |
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| Mass of a large but finite cluster at p = p[subscript c] | |
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| Correlation length | |
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| Mass of the percolating cluster for p > p[subscript c] | |
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| Finite-Size Scaling | |
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| Order parameter | |
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| Average cluster size and higher moments | |
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| Cluster number density | |
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| Non-Universal Critical Occupation Probabilities | |
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| Universal Critical Exponents | |
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| Real-Space Renormalisation | |
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| Self-similarity and the correlation length | |
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| Self-similarity and fixed points | |
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| Coarse graining and rescaling | |
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| Real-space renormalisation group procedure | |
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| Renormalisation in d = 1 | |
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| Renormalisation in d = 2 on a triangular lattice | |
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| Renormalisation in d = 2 on a square lattice | |
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| Approximation via the truncation of parameter space | |
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| Summary | |
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| Exercises | |
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| Ising Model | |
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| Introduction | |
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| Definition of the Ising model | |
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| Review of equilibrium statistical mechanics | |
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| Thermodynamic limit | |
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| System of Non-Interacting Spins | |
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| Partition function and free energy | |
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| Magnetisation and susceptibility | |
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| Energy and specific heat | |
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| Quantities of Interest | |
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| Magnetisation | |
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| Response functions | |
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| Correlation length and spin-spin correlation function | |
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| Critical temperature and external field | |
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| Symmetry breaking | |
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| Ising Model in d = 1 | |
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| Partition function | |
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| Free energy | |
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| Magnetisation and susceptibility | |
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| Energy and specific heat | |
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| Correlation function | |
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| Critical temperature | |
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| Mean-Field Theory of the Ising Model | |
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| Partition function and free energy | |
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| Magnetisation and susceptibility | |
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| Energy and specific heat | |
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| Landau Theory of the Ising Model | |
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| Free energy | |
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| Magnetisation and susceptibility | |
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| Specific heat | |
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| Landau Theory of Continuous Phase Transitions | |
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| Ising Model in d = 2 | |
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| Partition function | |
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| Magnetisation and susceptibility | |
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| Energy and specific heat | |
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| Critical temperature | |
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| Widom Scaling Ansatz | |
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| Scaling ansatz for the free energy | |
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| Scaling ansatz for the specific heat | |
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| Scaling ansatz for the magnetisation | |
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| Scaling ansatz for the susceptibility | |
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| Scaling ansatz for the spin-spin correlation function | |
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| Scaling Relations | |
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| Widom Scaling Form and Critical Exponents in d = 1 | |
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| Non-Universal Critical Temperatures | |
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| Universal Critical Exponents | |
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| Ginzburg Criterion | |
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| Real-Space Renormalisation | |
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| Kadanoff's block spin transformation | |
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| Kadanoff's block spin and the free energy | |
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| Kadanoff's block spin and the correlation function | |
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| Renormalisation in d = 1 | |
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| Renormalisation in d = 2 on a square lattice | |
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| Wilson's Renormalisation Group Theory | |
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| Coupling space and renormalisation group flow | |
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| Self-similarity and fixed points | |
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| Basin of attraction of fixed points | |
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| RG flow in coupling and configurational space | |
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| Universality and RG flow near fixed point | |
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| Widom scaling form | |
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| Summary | |
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| Exercises | |
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| Self-Organised Criticality | |
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| Introduction | |
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| Sandpile metaphor | |
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| BTW Model in d = 1 | |
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| Algorithm of the BTW model in d = 1 | |
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| Transient and recurrent configurations | |
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| Avalanche time series | |
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| Avalanche-size probability | |
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| Mean-Field Theory of the BTW Model | |
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| Random neighbour BTW model | |
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| Algorithm of the random neighbour BTW model | |
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| Steady state and the average avalanche size | |
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| Branching Process | |
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| Branching ratio | |
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| Avalanche-size probability - exact | |
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| Avalanche-size probability - scaling form | |
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| Avalanche-Size Probability - Scaling Ansatz | |
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| Scaling Relations | |
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| Moment Analysis of Avalanche-Size Probability | |
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| BTW Model in d = 2 | |
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| Algorithm of the BTW model in d = 2 | |
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| Steady state and the average avalanche size | |
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| Avalanche time series | |
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| Avalanche-size probability | |
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| Ricepile Experiment and the Oslo Model | |
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| Ricepile experiment | |
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| Ricepile avalanche time series | |
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| Ricepile avalanche-size probability density | |
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| Ricepile modelling | |
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| Algorithm of the Oslo model | |
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| Transient and recurrent configurations | |
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| Avalanche time series | |
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| Avalanche-size probability | |
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| Earthquakes and the OFC Model | |
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| Earthquake mechanism | |
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| Earthquake time series | |
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| Earthquake-size frequency | |
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| Earthquake modelling | |
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| Algorithm of the OFC model | |
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| Steady state and the average avalanche size | |
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| Avalanche time series | |
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| Avalanche-size probability | |
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| Rainfall | |
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| Rainfall mechanism | |
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| Rainfall time series | |
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| Rainfall-size number density | |
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| Summary | |
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| Exercises | |
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| Taylor Expansion | |
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| Hyperbolic Functions | |
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| Homogeneous and Scaling Functions | |
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| Fractals | |
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| Data Binning | |
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| Boltzmann Distribution | |
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| Free Energy | |
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| Metropolis Algorithm | |
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| Bibliography | |
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| List of Symbols | |