Some aspects of the Liouville equation in mathematical physics and statistical mechanics

This paper presents some mathematical aspects of Classical Liouville theorem and we have noted some mathematical theorems about its initial value problem. Furthermore, we have implied on the formal frame work of Stochastic Liouville equation (SLE)

Stability of Systems with Random Initial Data

In this paper we consider a nonlinear non-autonomous system ordinary differential equations (ODE) and the corresponding Liouville equation. Initial data of the ODE system is random and lie in a given region with a known initial distribution law. For non-linear non-autonomous ODE system introduces the concept of ε is a statistical stability of the solution, which allows us to study the behavior of solutions of the system of ODE’s with nondeterministic initial data. Such a study is carried out using the probability density function of distribution of the ensemble of data points in the ODE system. The notion of ε is a statistical stability of the solution allows to operate directly from the set of trajectories movement of the ODE system, the initial values ??of which lie in a given area, as well as to test the criterion ε is a statistical stability rather a function of the probability density distribution of the ensemble of data points in the Gibbs ODE system, which, while satisfying partial differential equation, but it is a linear equation, and moreover sought not the total solution, and the solution of the Cauchy problem. To introduce the notion of ε is a statistical stability of the solution requires that the nonlinear ODE system has a solution as a whole, ie that the trajectories of the system does not go to infinity in finite time. In the general case, ε is a statistical stability is not equivalent to the asymptotic Lyapunov stability of solutions. However, between these concepts has close relationship allows us to formulate the necessary and sufficient condition ε is a statistical stability of the solution for a linear autonomous system of ODE and sufficient condition for the linear non-autonomous system of ODE (for homogeneous and inhomogeneous cases). The study of the dispersion of the nonlinear non-autonomous system of ODE was obtained A necessary and sufficient condition for ε is a statistical stability of the solution of the ODE system. All the results are illustrated in the examples of content