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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 | |