Exactly Solvable Birth and Death Processes

Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable “matrix” quantum mechanics, which is recently proposed by Odake and the author [S. Odake and R. Sasaki, J. Math. Phys. 49, 053503 (2008)]. The (q-) Askey scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of qx (with x being the population) corresponding to the q-Racah polynomial.Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable “matrix” quantum mechanics, which is recently proposed by Odake and the author [S. Odake and R. Sasaki, J. Math. Phys. 49, 053503 (2008)]. The (q-) Askey scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of qx (with x being the population) corresponding to the q-Racah polynomial.

arXiv:0903.3097Journal of Mathematical PhysicsOct 1, 200935 citations

Quantum vs classical birth and death processes; exactly solvable examples

Ryu Sasaki B

A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable BD processes contain many adjustable free parameters which could be helpful for various simulation purposes.A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable BD processes contain many adjustable free parameters which could be helpful for various simulation purposes.

arXiv:2212.10710Journal of Mathematical PhysicsFeb 1, 20253 citations

Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes

Satoru Odake

Abstract We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)( $q$ -)Hahn and three kinds of $q$ -Krawtchouk and $q$ -Meixner). Based on them and the case-(1) multi-indexed orthogonal polynomials of Racah, $q$ -Racah, Meixner, little $q$ -Jacobi, and little $q$ -Laguerre types, exactly solvable continuous time birth and death processes are obtained. Their discrete time versions (Markov chains) are also obtained for finite types.Abstract We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)($q$-)Hahn and three kinds of $q$-Krawtchouk and $q$-Meixner). Based on them and the case-(1) multi-indexed orthogonal polynomials of Racah, $q$-Racah, Meixner, little $q$-Jacobi, and little $q$-Laguerre types, exactly solvable continuous time birth and death processes are obtained. Their discrete time versions (Markov chains) are also obtained for finite types.

arXiv:2501.02877Progress of Theoretical and Experimental PhysicsMar 12, 20251 citation

Quantum vs classical Markov chains; Exactly solvable examples

Ryu Sasaki B

A coinless quantisation procedure of general reversible Markov chains on graphs is presented. A quantum Hamiltonian H is obtained by a similarity transformation of the fundamental transition probability matrix K in terms of the square root of the reversible distribution. The evolution of the classical and quantum Markov chains is described by the solutions of the eigenvalue problem of the quantum Hamiltonian H. About twenty plus exactly solvable Markov chains based on the hypergeometric orthogonal polynomials of Askey scheme, derived by Odake-Sasaki, would provide a good window for scrutinising the quantum/classical contrast of Markov chains. Among them five explicit examples, related to the Krawtchouk, Hahn, q-Hahn, Charlier and Meixner, are demonstrated to illustrate the actual calculations.A coinless quantisation procedure of general reversible Markov chains on graphs is presented. A quantum Hamiltonian H is obtained by a similarity transformation of the fundamental transition probability matrix K in terms of the square root of the reversible distribution. The evolution of the classical and quantum Markov chains is described by the solutions of the eigenvalue problem of the quantum Hamiltonian H. About twenty plus exactly solvable Markov chains based on the hypergeometric orthogonal polynomials of Askey scheme, derived by Odake-Sasaki, would provide a good window for scrutinising the quantum/classical contrast of Markov chains. Among them five explicit examples, related to the Krawtchouk, Hahn, q-Hahn, Charlier and Meixner, are demonstrated to illustrate the actual calculations.

arXiv:2212.10713Dec 21, 20221 citation

Exactly Solvable Inhomogeneous Fermion Systems

Ryu Sasaki B

Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on 1D lattices are constructed explicitly based on the discrete orthogonal polynomials of the Askey scheme, e.g. Krawtchouk, Hahn, Racah, Meixner, and q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground-state two-point correlation functions are derived explicitly. The multipoint correlation functions are obtained by Wick’s theorem. 15 corresponding exactly solvable XX spin systems are also displayed. They all have nearest-neighbor interactions. The exact solvability of the Schrödinger equation means that the corresponding Fokker–Planck equation is also exactly solvable. This leads to 15 exactly solvable birth and death fermions and 15 birth and death spin models. These provide plenty of material for calculating interesting quantities, e.g. entanglement entropy, etc.Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on 1D lattices are constructed explicitly based on the discrete orthogonal polynomials of the Askey scheme, e.g. Krawtchouk, Hahn, Racah, Meixner, and q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground-state two-point correlation functions are derived explicitly. The multipoint correlation functions are obtained by Wick’s theorem. 15 corresponding exactly solvable XX spin systems are also displayed. They all have nearest-neighbor interactions. The exact solvability of the Schrödinger equation means that the corresponding Fokker–Planck equation is also exactly solvable. This leads to 15 exactly solvable birth and death fermions and 15 birth and death spin models. These provide plenty of material for calculating interesting quantities, e.g. entanglement entropy, etc.

arXiv:2410.07614Progress of Theoretical and Experimental PhysicsNov 14, 20244 citations

Exactly Solvable 'Discrete' Quantum Mechanics; Shape Invariance, Heisenberg Solutions, Annihilation-Creation Operators and Coherent States

Satoru Odake, Ryu Sasaki B

Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-) Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-) Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.

arXiv:0802.1075Progress of Theoretical PhysicsApr 1, 200863 citations

Abstract We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)(

q

-)Hahn and three kinds of

q

-Krawtchouk and

q

-Meixner). Based on them and the case-(1) multi-indexed orthogonal polynomials of Racah,

q

-Racah, Meixner, little

q

-Jacobi, and little

q

-Laguerre types, exactly solvable continuous time birth and death processes are obtained. Their discrete time versions (Markov chains) are also obtained for finite types.