MATHEMATICAL PHYSICS AND MATRIX REPRESENTATIONS

This book expounds three kinds of matrices that are of physical interest, centering on physical examples. Stochastic matrices describe dynamical systems of many different types, involving (or not) phenomena like transience, dissipation, ergodicity, nonequilibrium and hypersensitivity to initial conditions. The main characteristic is growth by agglomeration, as in glass formation. Circulants are the building blocks of elementary Fourier analysis and provide a natural gateway quantum mechanics, noncommutative geometry and random walks, including some phenomenology like diffusion-advection equation and prey-predator chains.

We also present two applications: chemical reaction and genetics. The last subject may seem specially out of place in 'Mathematical Physics'. Our excuse is that Mendelism and blood types are here presented by using just the same methods of the other chapters. Bell polynomials offer closed expressions for many formulas concerning Lie algebra invariants, differential geometry and real gases, and their matrices are instrumental in the study of chaotic mappings.

Contents:

• Foreword
• Preface
• Preface to the First Edition
• Some Fundamental Notions
• Stochastic Matrices:
  • Evolving Systems
  • Markov Chains
  • Glass Transition
  • The Kerner Model
  • Formal Developments
  • Equilibrium, Dissipation and Ergodicity
• Circulant Matrices:
  • Prelude
  • Definition and Main Properties
  • Random Walks
  • Applications I: Chemical Reactions
  • Applications II: Genetics
  • Discrete Quantum Mechanics
  • Quantum Symplectic Structure
• Bell Matrices:
  • Bell Polynomials
  • Determinants and Traces
  • Projectors and Iterates
  • Gases: Real and Ideal
  • Carleman Matrices
• Appendix A: Formulary:
  • General Formulas
  • General Matrices
  • Stochastic Matrices
  • Circulant Matrices
  • Bell Polynomials
  • Determinants, Minors and Traces
  • Bell Matrices
  • Statistical Mechanics
• Bibliography
• Index

Readership: Postgraduate and doctoral students, researchers and practitioners in the fields of mathematical physics, matrix theory, statistical mechanics, applied physics and biology. Advanced graduate students in Physics and Mathematics.