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Mathematics of Complexity and Dynamical Systems pp 1400–1406Cite as

Phase Transitions on Fractals and Networks

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Article Outline

Glossary

Definition of the Subject

Introduction

Ising Model

Fractals

Diffusion on Fractals

Ising Model on Fractals

Networks

Future Directions

Bibliography

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Abbreviations

Cluster:

Clusters are sets of occupied neighboring sites.

Critical exponent:

At a critical point or second‐order phase transition, many quantities diverge or vanish with a power law of the distance from this critical point; the critical exponent is the exponent for this power law.

Diffusion:

A random walker decides at each time step randomly in which direction to proceed. The resulting mean square distance normally is linear in time.

Fractals:

Fractals have a mass varying with some power of their linear dimension. The exponent of this power law is called the fractal dimension and is smaller than the dimension of the space.

Ising model:

Each site carries a magnetic dipole which points up or down; neighboring dipoles “want” to be parallel.

Percolation:

Each site of a large lattice is randomly occupied or empty.

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Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, Cologne University, Köln, Germany

    Dietrich Stauffer

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  1. Dietrich Stauffer
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Stauffer, D. (2012). Phase Transitions on Fractals and Networks. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_88

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