Book contents
- Frontmatter
- Contents
- Preface
- 1 A few well-known basic results
- 2 Introduction: order parameters, broken symmetries
- 3 Examples of physical situations modelled by the Ising model
- 4 A few results for the Ising model
- 5 High-temperature and low-temperature expansions
- 6 Some geometric problems related to phase transitions
- 7 Phenomenological description of critical behaviour
- 8 Mean field theory
- 9 Beyond the mean field theory
- 10 Introduction to the renormalization group
- 11 Renormalization group for the φ4 theory
- 12 Renormalized theory
- 13 Goldstone modes
- 14 Large n
- Index
2 - Introduction: order parameters, broken symmetries
Published online by Cambridge University Press: 05 July 2014
- Frontmatter
- Contents
- Preface
- 1 A few well-known basic results
- 2 Introduction: order parameters, broken symmetries
- 3 Examples of physical situations modelled by the Ising model
- 4 A few results for the Ising model
- 5 High-temperature and low-temperature expansions
- 6 Some geometric problems related to phase transitions
- 7 Phenomenological description of critical behaviour
- 8 Mean field theory
- 9 Beyond the mean field theory
- 10 Introduction to the renormalization group
- 11 Renormalization group for the φ4 theory
- 12 Renormalized theory
- 13 Goldstone modes
- 14 Large n
- Index
Summary
Can statistical mechanics be used to describe phase transitions?
A phenomenological description of a phase transition does not raise any special difficulty a priori. For instance, to describe the solidification of a gas under pressure, one can make a simple theory for the gaseous phase, e.g., an ideal gas corrected by a few terms of the virial expansion. Then, for the solid, one can use the extraction energies of the atoms, and the vibration energies around equilibrium positions. These calculations will provide a thermodynamic potential for each phase. The line of coexistence between the two phases in the pressure—temperature plane will be determined by imposing the equality of the two chemical potentials μI (T, P) = μII (T, P).
If this method may turn out to be useful in practice, it does not answer any of the questions that one can raise concerning the transition between the two states. Indeed the interactions between the molecules are not statistical in nature: they are independent of the temperature, or of the pressure; the Hamiltonian is a combination of kinetic energy and well-defined interaction potentials between pairs of molecules. How can one see in such a description, following the principles established by Boltzmann, Gibbs and their successors, that at equilibrium the same molecules can form a solid or a fluid, a superconductor, a ferromagnet, etc., without any modification of the interactions?
- Type
- Chapter
- Information
- Introduction to Statistical Field Theory , pp. 9 - 21Publisher: Cambridge University PressPrint publication year: 2010