1999/12/20 by Aristophanes Dimakis, A. Dimakis, Constantinos Tzanakis +1
Mathematics · Physics and Astronomy · #Classical mechanics #Differential equation #Equations of motion #Fokker–Planck equation #Lattice (music) #Master equation #Mathematical analysis #Mathematics #Observable #Phase space #Physics #Quantum #Quantum Mechanics and Applications #Quantum mechanics #Random Matrices and Applications #Smoluchowski coagulation equation #Statistical physics #Theoretical and Computational Physics #math-ph #math.MP
paper · pdf · doi:10.1088/0305-4470/33/30/301
published as J.Phys.A33:5267-5301,2000 · LaTeX2e, 40 pages, 1 Postscript figure, uses package epsfig
arxiv created 1999/12/20 · openalex publication_date 2000/07/19 · arxiv updated 2014/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
By considering a lattice model of extended phase space, and using techniques of non-commutative differential geometry, we are led to: (a) the concept of vector fields as generators of motion and transition probability distributions on the lattice; (b) the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure; (c) the general prescription for the evolution of the observables in analogy with classical dynamics. We show that, in the limit of a continuous description, these results lead to the time evolution of observables in terms of (the adjoint of) generalized Fokker-Planck equations having: (1) a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion; (2) a drift term given by the microscopic average of the dynamical equations in the present context. These results are applied to one- and two-dimensional problems. Specifically, we derive: (I) the equations of diffusion, Smoluchowski and Fokker-Planck in velocity space, thus indicating the way random-walk models are incorporated in the present context; (II) Kramers' equation, by further assuming that, motion is deterministic in coordinate space.