Gibbs Measures in Biology and Physics: The Potts Model

This book presents recently obtained mathematical results on Gibbs measures of the q-state Potts model on the integer lattice and on Cayley trees. It also illustrates many applications of the Potts model to real-world situations in biology, physics, financial engineering, medicine, and sociology, as well as in some examples of alloy behavior, cell sorting, flocking birds, flowing foams, and image segmentation.
Gibbs measure is one of the important measures in various problems of probability theory and statistical mechanics. It is a measure associated with the Hamiltonian of a biological or physical system. Each Gibbs measure gives a state of the system.
The main problem for a given Hamiltonian on a countable lattice is to describe all of its possible Gibbs measures. The existence of some values of parameters at which the uniqueness of Gibbs measure switches to non-uniqueness is interpreted as a phase transition.
This book informs the reader about what has been (mathematically) done in the theory of Gibbs measures of the Potts model and the numerous applications of the Potts model. The main aim is to facilitate the readers (in mathematical biology, statistical physics, applied mathematics, probability and measure theory) to progress into an in-depth understanding by giving a systematic review of the theory of Gibbs measures of the Potts model and its applications.

Contents:

• Introduction to Thermodynamics and Theory of Gibbs Measures:
  • Introduction
  • Gibbs Measures and Potts Model
• Thermodynamics in Biology: The Potts Model:
  • Hierarchy of DNAs and Holliday Junctions
  • Thermodynamics in a System of Interacting DNAs
  • Holliday Junctions for the Potts Model of DNA
  • Thermodynamics of DNA-RNA Renaturation
  • On Quadratic Stochastic Operators of Gibbs Measures
• Thermodynamics in Physics: The Potts Model:
  • Full Description of Translation-Invariant Gibbs Measures for the Potts Model on Trees
  • Extremality of Gibbs Measures
  • The Model with External Field
  • Periodic Gibbs Measures
  • Free Energies of the Potts Model

Readership: Advanced graduate students and researchers in mathematical biology, mathematical physics, statistical physics, probability and measure theory.