Book contents
- Frontmatter
- Contents
- Preface
- 1 Collective behavior, from particles to fields
- 2 Statistical fields
- 3 Fluctuations
- 4 The scaling hypothesis
- 5 Perturbative renormalization group
- 6 Lattice systems
- 7 Series expansions
- 8 Beyond spin waves
- 9 Dissipative dynamics
- 10 Directed paths in random media
- Solutions to selected problems from chapter 1
- Solutions to selected problems from chapter 2
- Solutions to selected problems from chapter 3
- Solutions to selected problems from chapter 4
- Solutions to selected problems from chapter 5
- Solutions to selected problems from chapter 6
- Solutions to selected problems from chapter 7
- Solutions to selected problems from chapter 8
- Index
2 - Statistical fields
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Collective behavior, from particles to fields
- 2 Statistical fields
- 3 Fluctuations
- 4 The scaling hypothesis
- 5 Perturbative renormalization group
- 6 Lattice systems
- 7 Series expansions
- 8 Beyond spin waves
- 9 Dissipative dynamics
- 10 Directed paths in random media
- Solutions to selected problems from chapter 1
- Solutions to selected problems from chapter 2
- Solutions to selected problems from chapter 3
- Solutions to selected problems from chapter 4
- Solutions to selected problems from chapter 5
- Solutions to selected problems from chapter 6
- Solutions to selected problems from chapter 7
- Solutions to selected problems from chapter 8
- Index
Summary
Introduction
We noted in the previous chapter that the singular behavior of thermodynamic functions at a critical point (the termination of a coexistence line) can be characterized by a set of critical exponents {α, β, γ, …}. Experimental observations indicate that these exponents are quite universal, i.e. independent of the material under investigation, and to some extent, of the nature of the phase transition. For example, the vanishing of the coexistence boundary in the condensation of CO2 has the same singular behavior as that of the phase separation of protein solutions into dilute and dense components. This universality of behavior needs to be explained. We also noted that the divergence of the response functions, as well as direct observations of fluctuations via scattering studies, indicate that fluctuations have long wavelengths in the vicinity of the critical point, and are correlated over distances ξ ≫ a, where a is a typical interparticle spacing. Such correlated fluctuations involve many particles and a coarse-graining approach, in the spirit of the theory of elasticity, may be appropriate to their description. Here we shall construct such a statistical field theory.
We shall frame the discussion in the language of a magnetic system whose symmetries are more transparent, although the results are of more general applicability. Consider a material such as iron, which is experimentally observed to be ferromagnetic below a Curie temperature Tc, as in Fig. 1.4.
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- Chapter
- Information
- Statistical Physics of Fields , pp. 19 - 34Publisher: Cambridge University PressPrint publication year: 2007