## Special Matrices of Mathematical Physics: Stochastic, Circulant, and Bell MatricesThis book expounds three special 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 to quantum mechanics and noncommutative geometry. 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: Basics: 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; Discrete Quantum Mechanics; Quantum Symplectic Structure; Bell Matrices: An Organizing Tool; Bell Polynomials; Determinants and Traces; Projectors and Iterates; Gases: Real and Ideal. Readership: Mathematical physicists, statistical physicists and researchers in the field of combinatorics and graph theory. |

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### Contents

Some fundamental notions | 3 |

Evolving systems | 21 |

Glass transition | 31 |

Formal developments | 45 |

Equilibrium dissipation and ergodicity | 63 |

Prelude | 81 |

Discrete quantum mechanics | 99 |

Quantum symplectic structure | 127 |

An organizing tool | 149 |

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### Common terms and phrases

alphabet basis Bell matrices Bell polynomials braid group canonical partition function Cayley–Hamilton theorem characteristic polynomial circulant matrices circulant matrix classical cluster integrals column commutative components condition consequently continuum convolution corresponding cyclic defined derivative detailed balancing diagonal differential discrete distribution eigenvalues eigenvectors entries equation equilibrium evolution example factor fermions formalism formula Fourier transformations Fredholm geometry given glass grand canonical partition Hamiltonian Hopf algebras identity imprimitive invariant inverse irreducible iterate Ksas leads Lie algebra Markov chain noncommutative notation obtained operator particles permutation phase space Poisson bracket powers projectors properties Quantum Mechanics recursion representation stochastic matrix summation symmetric functions symmetric group symplectic Taylor coefficients theorem totally regular unitary values variables vector virial Weyl-Wigner Wigner functions