## 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 asymptotic basis Bell matrices Bell polynomials bosons braid group canonical partition function characteristic polynomial circulant matrices circulant matrix classical closed expression cluster integrals column commutative components condition configuration consequently continuum convolution corresponding cyclic defined derivative detailed balancing diagonal differential discrete dynamical eigenvalues eigenvectors entries ep+i equation ergodic evolution example factor fc=i fc=l fermions finite formalism formula Fourier transformations Fredholm geometry given glass grand canonical partition Hamiltonian Hopf algebras identity imprimitive interaction invariant inverse irreducible iterate leads Lie algebra Markov chain multinomial theorem noncommutative nondegenerate notation Notice obtained operator particles particular permutation phase space physical Poisson bracket powers projectors properties quantization Quantum Mechanics recursion representation roots simple stochastic matrix summation symmetric functions symmetric group Taylor coefficients theorem theory tion totally regular unitary values variables vector virial Weyl-Wigner Wigner functions