2004/10/08 by Qiuping A. Wang, Wei Li, Wang, Qiuping A. +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Fractional Differential Equations Solutions #Numerical methods for differential equations #Quantum chaos and dynamical systems #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/0410219
version2 with typing corrections
openalex publication_date 2004/10/08 · arxiv created 2004/10/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present certain mathematical aspects of an information method which was formulated in an attempt to investigate diffusion phenomena. We imagine a regular dynamical hamiltonian systems under the random perturbation of thermal (molecular) noise and chaotic motion. The random effect is taken into account via the uncertainty of irregular dynamic process produced in this way. This uncertainty due to different paths between two phase points is measured by a path information which is maximized in connection with the action defined originally for the unperturbed regular hamiltonian systems. The obtained transition probability depends exponentially on this action. The usefulness of this information method has been demonstrated by the derivation of diffusion laws without the usual assumptions. In this work, some essential mathematical aspects of this irregular dynamics is reviewed. It is emphasized that the classical action principle for single least action path is no more valid and the formalism of classical mechanics for regular hamiltonian systems is no more exact for irregular hamiltonian dynamics. There is violation of the fundamental laws of mechanics by randomly perturbed hamiltonian systems. However, the action principle is always present for the ensemble of paths through the average action. This average action principle leads to a formalism of stochastic mechanics in which, in spite of the violation of fundamental laws, the mathematical form of classical mechanics can be recovered by a consideration of the statistical averaging of the dynamics.