2002/04/25 by Demosthenes Ellinas, Ellinas, Demosthenes, I. Tsohantjis +2
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic structures and combinatorial models #Diffusion #FOS: Physical sciences #Geometry #Line (geometry) #Mathematics #Physics #Pure mathematics #Quantum Physics (quant-ph) #Quantum mechanics #Random Matrices and Applications #Random walk #Statistics #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0204148
24 pages, 1 figure attached in ps format
arxiv created 2002/04/25 · openalex publication_date 2002/04/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Working withing the framework of Hopf algebras, a random walk and the associated diffusion equation are constructed on a space that is algebraically described as the merging of the real line algebra with the anyonic line algebra. Technically this merged structure is a smash algebra, namely an algebra resulting by a braided tensoring of real with anyonic line algebras. The motivation of introducing the smashing results from the necessity of having non commuting increments in the random walk. Based on the observable-state duality provided by the underlying Hopf structure, the construction is cast into two dual forms: one using functionals determined by density probability functions and the other using the associated Markov transition operator. The ensuing diffusion equation is shown to possess triangular matrix realization. The study is completed by the incorporation of Hamiltonian dynamics in the above random walk model, and by the construction of the dynamical equation obeyed by statistical moments of the problem for generic entangled density functions.